There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

There’s a lot of math out there. Some things show up all the time on the ACT. Other things don’t. I need to know this information in order to make the questions and question-selection algorithms for Mathchops. So I went through every question from the last 10 ACTs and figured out which skills showed up. Then one of my partners helped me make a Python script and we did a bunch of data analysis. What follows is a list of the 75 most common skills, along with an estimate of how likely they are to appear on your actual test.
Guaranteed To Show Up: These have to be rock solid because A) they’ll definitely show up and B) they’ll often be combined with other skills.
Fractions – All four operations. Mixed numbers.
Average – Also called the arithmetic mean. There is always a basic version and usually an advanced one, like the average sum trick (see below).
Probability – Know the basic part:whole versions. There is usually a harder one also (like one with two events).
Percents – Know all basic variations. More advanced ones are common also.
Exponents – All operations. Fractional and negative exponents are very common too (see below).
Linear Equations/Slope – Find the slope when given two points. Be able to isolate y (to create y = mx + b). All the standard stuff from 8th grade Algebra.
Solving Equations – Be very comfortable with ax + b = cx + d. Distribute. Combine like terms. You also need to be able to create these equations based on word problems.
Picking Numbers – You never have to use this but it will be a useful option on every test.
Ratio – Part:part, part:whole.
Quadratic skills – Factor. FOIL. Set parenthesis equal to zero. Graph parabolas.
Area/Perimeter of basic shapes – Triangles, rectangles, circles.
Negatives – Be comfortable with all operations.
SOHCAHTOA – Every variation of right triangle trig, including word problems.
Plug in answers – Like picking numbers, it’s not required but it’s often helpful.
Extremely Likely (> 80% chance):
Function shifts – Horizontal shifts, vertical shifts. Stretches. You should recognize y = 2(x+1)^2 - 5 right away and know exactly what to do.
Average sum trick – 5 tests, average is 80. After the 6th test, the average is 82. What was 6th test score?
MPH – The concept of speed in miles per hour shows up every time.
Median – Middle when organized from low to high. Even number of numbers. What happens when you make the highest number higher or the lowest number lower?
Radicals – Basic operations. Translate to fractional exponents.
System of Equations – Elimination. Substitution. Word problems.
Angle chasing – 180 in a line. 180 in a triangle. Corresponding angles. Vertical angles.
Time – Hours to minutes, minutes to seconds
Pythagorean Theorem – Sometimes asked directly, other times required as part of something else (like SOHCAHTOA or finding the distance between two points).
Apply formula – they give you a formula (sometimes in the context of a word problem) and you have to plug stuff in.
Composite function – As in g(f(x)).
Factoring – Mostly the basics. Almost never involves a leading coefficient.
Matrices – Adding, subtracting, multiplying. Knowing when products are possible.
Very Likely (> 50% chance):
Absolute Value – Sometimes basic arithmetic, sometimes an algebraic equation or inequality.
Fractional Exponents – Rewrite radicals as fractional exponents and vice versa.
Multistep conversion – For example, they might give you a mph and a cost/gallon and then ask for the total cost.
Probability, two events – If there's a .4 probability of rain and a .6 probability of tacos, what is the probability of rain and tacos?
Remainders – Can be simple or pattern based, as in “If 1/7 is written as a repeating decimal, what is the 400th digit to the right of the decimal point?”
Midpoint – Given two ordered pairs, find the midpoint. Sometimes they’ll give you the midpoint and ask for one of the pairs.
Weird shape area – It’s an unusual shape but you can use rectangles and triangles to find the area.
Periodic function graph – The basics of sine and cosine graphs (shifts, amplitude, period).
Circle equations – (x-h)^2 + (y-k)^2 = r^2. Sometimes you have to complete the square.
Negative exponents – Know what they do and how to combine them with other exponents.
Shaded area – The classic one has a square with a circle inside.
Counting principle – License plate questions.
Logarithms – Rewrite in exponential form. Basic operations.
Imaginary numbers – Powers of i. What is i^2? The complex plane.
LCM – Straight up. In word problems. In algebraic fractions.
FOIL – This has to be automatic.
Worth Knowing (>25% chance):
Ellipses – Know how to graph basic versions.
Scientific notation – Go back and forth between standard and scientific notation. All four operations.
Vectors – Add, subtract, multiply (scalar), i and j notation.
Permutation – You have 5 plants and 3 spots. How many ways can you arrange them?
Volume of a prism – Know that the volume = area of something x height. Sometimes the base will be a weird shape.
c = product of roots, -b = sum of roots – Use when in x^2 + bx + c form. Usually not required but often helpful.
Difference of two squares – (x + y)(x - y) = x^2 - y^2
Arithmetic sequence – Usually asks you to find a specific term, sometimes asks you to find the formula.
Law of Cosines – They almost always give you the formula. Then you just have to plug things in.
Triangle opposite side rule – There is a relationship between an angle and the side across from that angle?
Change the base – If 9^x = 27^5, what is x?
Similar triangles – Relate the sides with a proportion.
Probability with “or” – 3 reds, 5 blue, 6 green. Probability of picking a red or blue?
Probability with “not” – 3 reds, 5 blue, 6 green. Probability of picking one that’s not red?
Factors – The basic concept and greatest common factor, with numbers and variables.
30:60:90 – Know the basic relationships. Sometimes required for advanced trig questions.
Volume of a cylinder – They’ll usually give it to you but not always.
Trapezoid – Usually basic area questions.
Domain – Usually you can think of it as “possible x values”.
Conjugates – Rationalize denominators that include radicals or imaginary numbers. Know that imaginary roots come in pairs.
Exponential Growth/Decay – Be comfortable with this: Final = Initial(1+/- rate)^time.
Weighted average – Class A has 8 kids and an average of 70. Class B has 12 kids and an average of 94. What is the combined average of the two classes?
Inverse trig – Use right triangle ratios to find angles.
Parallelogram – Know that adjacent angles add to 180. Area formula.
Use the radius – A circle will be combined with another shape and you have to use the radius to find the essential info about that other shape.
Value/frequency charts – They’ll tell you the value and frequency and then ask about mean or median.
3:4:5 – Recognize 3:4:5 right triangle relationships.
Algebra LCD – Find the lowest common denominator, then combine the numerators.
5:12:13 – Recognize 5:12:13 right triangle relationships.
System of equations with three equations – Usually a word problem. Involves substitution.
Compare numbers – Radicals, fractions, decimals, absolute value.
Translate points – Images, reflections.

Hi,
I'm currently enrolled in a M.Sc. course in Data Science & Engineering, and I have to decide which optional course to take among the following (I can choose only 1), please take a brief look at the syllabus.
1) OOP
syllabus:

• Basic features (1 credit) • Object-oriented programming, java, eclipse • Classes, attributes, methods and constructors, objects • Packages and visibility rules • Strings, wrapper classes • Arrays
• Inheritance and interfaces (2 credits) • Inheritance • Abstract classes, interfaces • Functional interfaces, lambda expressions • Exceptions • Generic types
• Standard libraries (3 credits) • Collections: sets, lists, maps • Streams • Files • Dates • Threads • Graphical interfaces, Swing, JavaFX
• Software Engineering principles (2 credits) - Software life cycle - Design using UML - Design Patterns - Configuration management - Testing

2) Numerical Optimization & Stochastic Optimization
syllabus:

• Convex optimization: - gradient descent method; conjugate gradient method - Numerical differentiation - Newton and quasi-Newton methods - Globalization techniques - Alternating direction method of multipliers (ADMM)
• Constrained optimization: - Interior point methods - Projected gradient method - Active set
• Stochastic optimization - Static simulation-based optimization (parametric optimization) - Dynamic simulation-based optimization (control optimization)

3) Decision Making & Optimization
syllabus:

1. Linear programming: modeling techniques, basic concepts of the Simplex Method, and duality (10% of the course).

• 2. Computational complexity: problem classes P, NP, NP-complete, and CoNP-complete (10% of the course).
• 3. Exact optimization methods: Branch and Bound, Cutting Planes, and Dynamic Programming (20% of the course).
• 4. Heuristic optimization methods: greedy algorithms, GRASP, Beam Search, meta-heuristics (Tabu Search, Simulated Annealing, Genetic Algorithms, ACO, VNS, RBS), and math-heuristics (30% of the course).
• 5. Decision making under uncertainty: Stochastic Programming with recourse, Measures for Stochastic Programming, Progressive Hedging method (10% of the course).
• 6. Nonlinear Programming: theoretical conditions for unconstrained and constrained optimization, algorithms for unconstrained and constrained optimization (20%).

Of course, the best answer is "it depends on what you have already done, and what you would like to do", so I try to give a brief introduction. I come from a B.Sc. in Electronic Engineering, and this is the only reason why I'm considering taking OOP. I don't have much problem with programming, but I feel like I don't have some skills because my Bsc was not in CS.
Regrading the other 2 courses, they are not the only math courses in my degree, I have many others ( such as ML&DL, Math for ML, Statistics for Data Science, Network Dynamics & Learning, Computational Linear Algebra), but still, they might be interesting.
I don't want to work as a software developer, I'm more interested in research, but do you think I should take OOP anyway to fill some gaps? Can you give me some examples where Decision Making and Numerical/Stochastic Optimization could be useful? (As I said the most important topics of both courses are also covered partially in other courses)

I took calculus I for the first time last semester as a freshman and barely scraped by with a borderline C after having a low D mid-semester. At this point I'm already going to have to take 5 years if I want to get my bachelor's in engineering, so I'm retaking it this semester (both virtual, no other option) to give myself a chance to get a better grip before taking calculus II so I don't crash and burn completely. I am determined to learn calculus if I have to spend 20 hours a week on this single goddamn class.
I think I have 2 problems:

1. I don't have a natural aptitude for math at all and tend to get frustrated and feel hopeless with it easily, because I often don't even know what I'm doing wrong, so I can't even figure out what I need to do to fix it. The problem solving strategies we're meant to use often don't make any sense to me. How am I supposed to know that I even can do things like multiply by the conjugate, and how do I know when to use those problem solving methods?
2. I went to a low-income, low-performing high school, and got to calculus I horribly unprepared--I think the most useful things I learned last semester were basic trig functions (sin, cos, and tan) and the logarithm and exponent rules. I had never learned the exponent rules in high school, and had never encountered logarithms or trig functions until the first week of calculus I, which was supposed to be a review. I still don't understand either.

Basically, I need to learn how to learn math, and I need to learn everything through calculus I by the time I take calculus II next August. Right now, I've got a TI84 plus, graph and lined paper, some pens and pencils, a PDF of Calculus Early Transcendentals 5th ed, and 5-10 practice problems and answers for every major topic in calculus from my professor. I've bookmarked Paul's Online Math Notes and Prof. Leonard's youtube channel, but I am otherwise completely lost. I seemed to have the most trouble last semester with simplifying and problem solving. I don't know how everyone else seems to intuitively know when and how to do things to manipulate a complicated equation into a simple one, and I get really lost when I have to do things like simplify derivatives that used the quotient rule, or take the derivative of something like cos(3x). Why is that not just -sin(3x), when the formula clearly says that the derivative of cos(x) is -sin(x)?
I know I am capable of being at least average at math if I put in enough work, and I need to know how to do calculus properly if I ever want to be a mildly competent engineer, so I need to know how to learn math first, because everyone else seems to get something I just don't, and I want to get it.
So my questions are: how do I study calculus, and what strategies do I need to be using to learn this material? I can listen to a million videos and take notes, sure, but how do I go from doing that to a) understanding the concepts and b) being able to look at a problem, know what I need to do to solve it, and then solve it?

Last night while binge drinking and reading through chains of articles on Wikipedia I had the best idea. I realized there are sections of the site I never explored. So I immediately closed List of animals with fraudulent diplomas and the n+1 articles on Permian fauna I had open, then went to the Wikipedia reference desk. Now the reference desk is pretty cool and it's one more reason why Wikipedia is an internet gem, anyone can ask a question about any topic, and anyone can answer.
Most of the questions in the mathematics section are what one would expect; people asking about things in homework assignments they don't understand, people asking for help deciphering arcane mathematics articles on the site, and people questioning their own understanding of things. Every now and then you'd find people asking why some proof of [insert famous conjecture here] published in [insert obscure journal here] wasn't cited on Wikipedia and why it wasn't accepted as a proof by the mathematical community. But I found something a little more spicy than that, I found a user that claims to have proven the Riemann hypothesis, the Collatz conjecture, the Goldbach conjecture, and created an elementary proof of Fermat's last theorem.
Is the following proof of Riemann Hypothesis correct?

Riemann Hypothesis states that the real part of all non-trivial zeros of the Riemann zeta function, or ζ(s) = Σ(k=1 to ∞) 1/k^s = 0, equals one-half. For the non-trivial zero, s, a complex number, we have s = a + bi where Re(s)= a = 1/2.

If I had $1 for every "proof" of the Riemann hypothesis I've seen where the writer starts by trying to find s such that 1+1/2^s+1/3^s+1/4^s+⋯=0 I'd probably be lounging on a beach in the Caribbean right now. The problem here is that the Dirichlet series for ζ(s) only converges when the real part of s is greater than 1, and this series is never 0 where it converges. So analyzing only this series will not be helpful.

Fact III: The sum of the complex conjugate pairs of non-trivial zeros, s = a + bi and s' = c + di where ζ(s) = Σ(k=1 to ∞) 1/k^s = 0 and ζ(s') = Σ(k=1 to ∞) 1/k^s' = 0, of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and the Harmonic Series (H):(Note: Euler and others have proven that there exists an infinite set of primes in H. And that the divergence of H is a key reason for that result.)

If s' is the complex conjugate of s=a+bi then why not just write s'=a-bi instead of s'=c+di? Or why not write s=σ+it instead, as this is a fairly standard way to write a non-trivial zero in literature on the topic? Sure, this isn't bad math per say, but it's pretty bad notation. Also, s+s'=1 always only if the Riemann hypothesis is true and this would have nothing to do with the fundamental theorem of arithmetic or the harmonic series! They have already assumed the Riemann hypothesis is true before they've done anything!
The bit where they talk about primes in the harmonic series is somewhat odd. It looks like they think the divergence of the harmonic series implies the divergence of the sum of reciprocal primes (which it doesn't, the implication is the other way around) and they seem to treat the harmonic series like a set.
After this our writer slaps his four facts together in some convoluted way that I can't decipher and declares victory.

Therefore, according to Facts I, II, III, and IV, we have:
k^(1/2) ≤ k^a ≤ k, k^(1/2) ≤ k^c ≤ k, and a + c = 1.
Hence, k^a = k^c = k^(1/2) which implies a = c = 1/2. Riemann Hypothesis is true! Riemann was right!

Then they make some final notes where they try to rewrite the harmonic series using some underexplained ideas about prime gaps and says

There are infinitely many more positive integers than there are prime numbers, or prime numbers have a zero density relative to the positive integers, and prime numbers generate the positive even integers efficiently so that gaps between two consecutive prime numbers increase without bound.

which is true in the sense of natural density for sure, so why not just say that? Using the phrase "infinitely many more" makes it sound like cardinality. Saying "so that gaps between two consecutive prime numbers increase without bound" makes it look like they're saying all prime gaps become larger as we increase through the sequence of primes, this isn't necessarily true although it's statistically something we should expect. The existence of arbitrarily large prime gaps is true though and isn't hard to prove, but they did not prove it in any of what was written and it's not the same as what they said.
Is the following proof of Goldbach Conjecture correct?

Keywords: π(*):= Odd Prime Counting Function and Fundamental Theorem of Arithmetic (FTA) Goldbach conjecture states every positive even integer is the sum of two prime numbers. (We count one as prime in the sense of additive number theory outside of the FTA.)

What? The parenthetical here is so strange. Additive number theorists don’t take 1 to be prime and they have no reason to do so.
The writer then tries to make a probabilistic argument from a system of linear equations defined over a set of odd primes less than an even number e>2,

Therefore, e ≠ p + q over S, (p,q є S) , implies the following system of equations over S, 1 = e - n1 * q1, 3 = e - n2 * q2, ..., pk = e - nk * qk, according to the Fundamental Theorem of Arithmetic where 1 < qj ≤ (nj * qj)^.5 ≤ nj for 1 ≤ j ≤ k where pj, qj є S and nj is a positive integer. Note: If qj = 1, then nj є S, or nj is an odd prime less than e.

and this last sentence is what they try to base their argument on. They attempt argue that for every even number e>2, the probability that an equation of the form p=e-1q doesn't show up goes to 0. Which would mean that it's likely that e=p+q.
Even if their probabilistic manipulations made sense this obviously still wouldn't prove the Goldbach conjecture. Showing that it's "probably true" isn't a proof that it's true. As if to attest to the writer's own doubt,

In addition, empirical evidence has confirmed the validity of the conjecture for all positive even integers up to at least an order of 10^18. Therefore, we conclude the conjecture is true.

If you proved it, why do you need to test it empirically?
Is the following elementary proof of Fermat's Last Theorem correct?

1. x^n+y^n=z^n for n > 2. I begin the proof by assuming there exists an integral (positive integer) solution to equation one for some n > 2. Equation one becomes with some algebraic manipulation, 2. x^n=z^n-y^n = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2)).

Okay.

Now that I have factored the right side of equation two, Fermat, the great French mathematician and respectable jurist, made I believe the next logical and crucial step.

Any evidence that Fermat did what you're about to do?

He factored the left side as well, x^n, with the help of an extra real variable, Ɛ, such that 0 < Ɛ < n . I have the following equation, x^n = x^(n/2+Ɛ/2)* x^(n/2-Ɛ/2) = (z^(n/2)+y^(n/2))*(z^(n/2)-y^(n/2) ). This equation implies x^(n/2+Ɛ/2)= z^(n/2)+y^(n/2) and x^(n/2-Ɛ/2) = z^(n/2)-y^(n/2).

Ah yes, if ab=cd then a=c and b=d. Everyone knows that! Eventually, after a few more lines, the author concludes

However, (1/4)^(1/n) is not a rational number, a ratio of two whole numbers, for n > 2. This implies the right side of equation five is not a positive integer. This contradicts my assumption that y is a positive integer. Thus, Fermat’s Last Theorem is true, and Fermat was right!

It's so easy now, Fermat's last theorem obviously just reduces to knowing 1/4^1/n is irrational for n>2. How did nobody see this before?
Is the following proof of the Collatz Conjecture correct?

Proof of the Collatz Conjecture: Suppose there exists a sequence, S’={n0, n1, n2, …} that does not converge to one, or nk ≠ 1 or nsub(k-r) ≠ 2^µ over S’ for all kϵ ℕ where r

It's obvious that hailstone sequences don't converge, so the “does not converge to one” bit is irrelevant. Here the fundamental error is same error as in their attempted proof of the Goldbach conjecture; they think making a probabilistic argument in favor of the conjecture being true is the same thing as proving it. Lots of other basic little details are also wrong, but I'll just look at one:

From a given positive integer, n, we obtain the maximum positive odd integer, n0 > 7, by repeated division of n by 2.

What is n here, the starting number? What if n is odd? We'd have to 3n+1 it first, not divide by 2. Even if n is even the first odd number we hit once we finish dividing by 2 is not the maximum odd number in its hailstone sequence, this is easy to see starting with n=22.

Here is the implementation in python of the algorithm in this article:

#! /usbin/python #---------- # This unusual and intriguing algorithm was originally invented # by Michael V. Klibanov, Professor, Department of Mathematics and Statistics, # University of North Carolina at Charlotte. It is published in the following # paper: # M.V. Klibanov, A.V. Kuzhuget and K.V. Golubnichiy, # "An ill-posed problem for the Black-Scholes equation # for a profitable forecast of prices of stock options on real market data", # Inverse Problems, 32 (2016) 015010. #---------- # Script assumes it's called by crontab, at the opening of the market #----- import numpy as np import pause, datetime from bs4 import BeautifulSoup import requests # Quadratic interpolation of the bid and ask option prices, and linear interpolation in between (https://people.math.sc.edu/kellerlv/Quadratic_Interpolation.pdf) def funcQuadraticInterpolationCoef(values): # There is 'scipy.interpolate.interp1d' too y = np.array(values) A = np.array([[1,0,0],[1,-1,1],[1,-2,4]]) return np.linalg.solve(A,y) # https://en.wikipedia.org/wiki/Polynomial_regression def funcUab(t,coef): return coef[2]*t**2 + coef[1]*t + coef[0] def funcF(s, sa, sb, ua, ub): return (s-sb)*(ua-ub)/(sa-sb) + ub # Initialize the volatility and option lists of 3 values optionBid = [0] # dummy value to pop in the loop optionAsk = [0] # dummy value to pop in the loop volatility = [0] # dummy value to pop in the loop # Initalization for the loop Nt = 4 # even number greater than 2: 4, 6, ... Ns = 2 # even number greater than 0: 2, 4, ... twotau = 2 # not a parameter... alpha = 0.01 # not a parameter... dt = twotau / Nt # time grid step dimA = ( (Nt+1)*(Ns+1), (Nt+1)*(Ns+1) ) # Matrix A dimensions dimb = ( (Nt+1)*(Ns+1), 1 ) # Vector b dimensions A = np.zeros( dimA ) # Matrix A b = np.zeros( dimb ) # Vector b portfolio = 1000000 # Money 'available' securityMargin = 0.00083 # EMPIRICAL: needs to be adjusted when taking into account the transaction fees (should rise, see the article p.8) # Wait 10mn after the opening of the market datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) # Record the stock and option values and wait 10mn more def funcRetrieveStockOptionVolatility(): # Stock stock_data_url = "https://finance.yahoo.com/quote/MSFT?p=MSFT" stock_data_html = requests.get(data_url).content stock_content = BeautifulSoup(stock_data_html, "html.parser") stock_bid = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "BID-value"}) print(stock_bid) stock_ask = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "ASK-value"}) print(stock_ask) stockOptVol[0] = stock_bid.text.split()[0] stockOptVol[1] = stock_ask.text.split()[0] # Option option_data_url = "https://finance.yahoo.com/quote/MSFT/options?p=MSFT&date=1631836800" option_data_html = requests.get(option_data_url).content option_content = BeautifulSoup(option_data_html, "html.parser") call_option_table = content.find("table", {'class': 'calls W(100%) Pos(r) Bd(0) Pt(0) list-options'}) calls = call_option_table.find_all("tr")[1:] it = 0 for call_option in calls: it+=1 print("it = ", it) if "in-the-money " in str(call_option): itm_calls.append(call_option) print("in the money") itm_put_data = [] for td in BeautifulSoup(str(itm_calls[-1]), "html.parser").find_all("td"): itm_put_data.append(td.text) print(itm_put_data) if itm_put_data[0] == 'MSFT210917C00220000': # One single option stockOptVol[2] = float(itm_put_data[4]) stockOptVol[3] = float(itm_put_data[5]) stockOptVol[4] = float(itm_put_data[-1].strip('%')) else: otm_calls.append(call_option) print("out the money") print("bid = ", option_bid, "\nask = ", option_ask, "\nvol = ",option_vol) return stockOptVol # Record option and volatility stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) # Wait another 10mn to record a second value for the quadratic interpolation datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) tradeAtTimeTau = False tradeAtTimeTwoTau = False # Run the loop until 30mn before closure datet = datetime.datetime.now() datetend = datetime.datetime(datet.year, datet.month, datet.day, datet.hour + 6, datet.minute + 10) while datet <= datetend: datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) optionBid.pop(0) optionAsk.pop(0) optionVol.pop(0) stockOptVol = funcRetrieveStockOptionVolatility() stockBid = stockOptVol[0] stockAsk = stockOptVol[1] optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[5]) # Trade if required if tradeAtTimeTau == True or tradeAtTimeTwoTau == True: # sell if tradeAtTimeTau == True: portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # sell 140 options bought 10mn ago tradeAtTimeTau = tradeAtTimeTwoTau sellingPriceAtTimeTau = sellingPriceAtTimeTwoTau sellingPriceAtTimeTwoTau = false else: # forecast the option when no trading # Interpolation coefa = funcQuadraticInterpolationCoef(optionAsk) # quadratic interpolation of the option ask price coefb = funcQuadraticInterpolationCoef(optionBid) # quadratic interpolation of the option bid price coefs = funcQuadraticInterpolationCoef(optionVol) # quadratic interpolation of the volatility sigma sa = stockAsk # stock ask price sb = stockBid # stock bid price ds = (sa - sb) / Ns # stock grid step for k in range (0, Ns+1): # fill the matrix and the vector for j in range (0, Nt+1): Atemp = np.zeros( dimA ) btemp = np.zeros( dimb ) print("k = {k}, j = {j}".format(k=k,j=j)) if k == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefb) elif k == Ns: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefa) elif j == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) elif j == Nt: # do nothing pass else: # main case akj = 0.5*(255*13*3)* funcUab(j*dt, coefs)**2 * (k*ds + sb)**2 dts = (twotau-dt)/Nt * (sa-sb-ds)/Ns #---------- #----- Integral of the generator L #---------- #----- time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] = dts / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = dts / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] = - dts / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] = - dts / dt**2 # k,j-1 ~ k,j+1 #---------- #----- stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = 4 * akj**2 * dts / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k+1,j #----- Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k-1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k+1,j #---------- #----- time and stock derivatives #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k+1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k+1,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] = -2 * akj * dts / (dt*ds**2) # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = -2 * akj * dts / (dt*ds**2) # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] = 2 * akj * dts / (dt*ds**2) # k,j-1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = 2 * akj * dts / (dt*ds**2) # k,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k-1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k-1,j ~ k,j-1 #---------- #---------- #----- Regularisation term - using alpha = 0.01 #---------- #---------- #----- H2 norm: 0 derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += alpha # k,j ~ k,j #----- coef = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) btemp[ (k+0)*(Nt+1)+(j+0) ] += alpha * 2 * coef #---------- #----- H2 norm: time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += -alpha / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += -alpha / dt**2 # k,j-1 ~ k,j+1 #----- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef btemp[ (k+0)*(Nt+1)+(j-1) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k+1,j ~ k+1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k-1,j ~ k+1,j #----- coef = ( funcUab(j*dt,coefa) - funcUab(j*dt,coefb) ) / (sa - sb) btemp[ (k+1)*(Nt+1)+(j+0) ] += alpha * 2 * coef btemp[ (k-1)*(Nt+1)+(j+0) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock and time derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j+1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j-1 ~ k-1,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j-1 ~ k+1,j-1 #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k-1,j+1 Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k+1,j-1 Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j+1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j-1 ~ k+1,j+1 #---------- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j+1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k-1,j-1 #----- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j-1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k-1,j+1 #---------- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k+1,j-1 #---------- coef = ( funcUab((j+1)*dt,coefa) - funcUab((j+1)*dt,coefb) \ - funcUab((j-1)*dt,coefa) + funcUab((j-1)*dt,coefb) ) / (dt * (sa - sb)) btemp[ (k+1)*(Nt+1)+(j+1) ] += alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j+1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j-1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k+1)*(Nt+1)+(j-1) ] += alpha * 2 * coef / (ds*dt) #---------- #----- H2 norm: stock second derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / dt**4 # k,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] += -2 * alpha / dt**4 # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j-1 ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] += -2 * alpha / dt**4 # k,j ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j-1 ~ k,j #---------- #----- H2 norm: time second derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / ds**4 # k,j ~ k,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k+1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k-1,j ~ k,j #---------- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - 2 * funcF( k*ds+sb, sa, sb, funcUab((j+0)*dt,coefa), funcUab((j+0)*dt,coefb) ) \ + funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt**2 btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j+0) ] += - alpha * 4 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j-1) ] += alpha * 2 * coef / dt**2 #---------- #---------- #----- Boundary de-computation #---------- if k+1 == Ns: Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 btemp[ (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j btemp[ (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 if k-1 == 0: Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j btemp[ (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 if j-1 == 0: Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 ~ k,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 #---------- pass print("-----") print("Atemp = ") print(Atemp) print("-----") print("btemp = ") print(btemp) print("-----") print("-----") A = A + Atemp b = b + btemp print("-----") print("A = ") print(A) print("-----") print("b = ") print(b) print("-----") print("-----") input("Press Enter to continue...") # Conjugate gradient algorithm: https://en.wikipedia.org/wiki/Conjugate_gradient_method x = np.zeros(N).reshape(N,1) r = b - np.matmul(A,x) p = r rsold = np.dot(r.transpose(),r) for i in range(len(b)): Ap = np.matmul(A,p) alpha = rsold / np.matmul(p.transpose(),Ap) x = x + alpha * p r = r - alpha * Ap rsnew = np.dot(r.transpose(),r) if np.sqrt(rsnew) < 1e-16: break p = r + (rsnew / rsold) * p rsold = rsnew print("it = ", i) print("rsold = ", rsold) # Trading strategy sm = (sa + sb)/2 if x[Ns/2*(Nt+1)+Nt/2] >= optionAsk[0] + securityMargin: tradeAtTimeTau = True sellingPriceAtTimeTau = x[Ns/2*(Nt+1)+Nt/2] portfolio -= 140 * optionAsk # buy 140 options if x[Ns/2*(Nt+1)+Nt] >= optionAsk[0] + securityMargin: tradeAtTimeTwoTau = True sellingPriceAtTimeTwoTau = x[Ns/2*(Nt+1)+Nt] portfolio -= 140 * optionAsk # buy 140 options pause.until(datet) # Wait 10mn before the next loop pause.until(datet) datet = datetime.datetime.now() # Time should be around 20mn before closure datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) if tradeAtTimeTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # Wait 10mn more to sell the last options pause.until(datet) # it should be around 10mn before closure if tradeAtTimeTwoTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTwoTau) * 140 # Market closure 

Don't put money on this as I'm still debugging (I bet you half a bitcoin I have mistaken a few indices in the H_2 norm)... Here is the discretisation formula I used, to copy-paste on latexbase:

\documentclass[12pt]{article} \usepackage{amsmath} \usepackage[latin1]{inputenc} \title{Klibanov algorithm} \author{Discretisation formula} \date{\today} \begin{document} \maketitle Let $$ a_{k,j} = \frac12\sigma(j\delta_\tau)^2\times(255\times13\times3)\times(k\delta_s+s_a)^2, $$ then \begin{alignat*}{3} J_\alpha(u) = & \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_\tau} + a_{k,j} \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2\frac{2\tau - \delta_\tau}{N_t}\frac{s_a - s_b - \delta_s}{N_s}\\ & + \alpha \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| u_{k,j} - F_{k,j}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_t} - \frac{F_{k,j+1} - F_{k,j-1}}{\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - u_{k-1,j}}{\delta_s} - \frac{u_{a,j} - u_{b,j}}{s_a - s_b}\right|^2 \\ & \qquad + \left| \frac{(u_{k+1,j+1} - u_{k-1,j+1}) - (u_{k+1,j-1} - u_{k-1,j-1})}{\delta_s\delta_t} \right. \\ & \qquad \qquad \left. - \frac{(u_{a,j+1} - u_{b,j+1}) - (u_{a,j-1} - u_{b,j-1})}{(s_a-s_b)\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - 2u_{k,j} + u_{k,j-1}}{\delta_\tau^2} - \frac{F_{k,j+1} - 2F_{k,j} + F_{k,j-1}}{\delta_\tau^2} \right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2 \end{alignat*} %% \left| \right|^2 with $\tau = 1$ unit of time (for example 10mn). \end{document} 

Let me know if you see something wrong... And if you want to contribute, feel free

Hi,
I'm currently enrolled in a M.Sc. course in Data Science & Engineering, and I have to decide which optional course to take among the following (I can choose only 1), please take a brief look at the syllabus.
1) OOP
syllabus:

• Basic features (1 credit) • Object-oriented programming, java, eclipse • Classes, attributes, methods and constructors, objects • Packages and visibility rules • Strings, wrapper classes • Arrays
• Inheritance and interfaces (2 credits) • Inheritance • Abstract classes, interfaces • Functional interfaces, lambda expressions • Exceptions • Generic types
• Standard libraries (3 credits) • Collections: sets, lists, maps • Streams • Files • Dates • Threads • Graphical interfaces, Swing, JavaFX
• Software Engineering principles (2 credits) - Software life cycle - Design using UML - Design Patterns - Configuration management - Testing

2) Numerical Optimization & Stochastic Optimization
syllabus:

• Convex optimization: - gradient descent method; conjugate gradient method - Numerical differentiation - Newton and quasi-Newton methods - Globalization techniques - Alternating direction method of multipliers (ADMM)
• Constrained optimization: - Interior point methods - Projected gradient method - Active set
• Stochastic optimization - Static simulation-based optimization (parametric optimization) - Dynamic simulation-based optimization (control optimization)

3) Decision Making & Optimization
syllabus:

1. Linear programming: modeling techniques, basic concepts of the Simplex Method, and duality (10% of the course).

• 2. Computational complexity: problem classes P, NP, NP-complete, and CoNP-complete (10% of the course).
• 3. Exact optimization methods: Branch and Bound, Cutting Planes, and Dynamic Programming (20% of the course).
• 4. Heuristic optimization methods: greedy algorithms, GRASP, Beam Search, meta-heuristics (Tabu Search, Simulated Annealing, Genetic Algorithms, ACO, VNS, RBS), and math-heuristics (30% of the course).
• 5. Decision making under uncertainty: Stochastic Programming with recourse, Measures for Stochastic Programming, Progressive Hedging method (10% of the course).
• 6. Nonlinear Programming: theoretical conditions for unconstrained and constrained optimization, algorithms for unconstrained and constrained optimization (20%).

Of course, the best answer is "it depends on what you have already done, and what you would like to do", so I try to give a brief introduction. I come from a B.Sc. in Electronic Engineering, and this is the only reason why I'm considering taking OOP. I don't have much problem with programming, but I feel like I don't have some skills because my Bsc was not in CS.
Regrading the other 2 courses, they are not the only math courses in my degree, I have many others ( such as ML&DL, Math for ML, Statistics for Data Science, Network Dynamics & Learning, Computational Linear Algebra), but still, they might be interesting.
I don't want to work as a software developer, I'm more interested in research, but do you think I should take OOP anyway to fill some gaps? Can you give me some examples where Decision Making and Numerical/Stochastic Optimization could be useful? (As I said the most important topics of both courses are also covered partially in other courses)

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[https://pubchem.ncbi.nlm.nih.gov/periodic-table/png/Periodic_Table_of_Elements_w_Chemical_Group_Block_PubChem.png ] or [https://ptable.com/#Properties ]
In the last 14 posts, I have attempted to present the main points/useful information from a whole academic year of general chemistry. A significant fraction of the material taught in general chemistry is obsolete, but I am also skipping over any of the information that is actually beneficial to have somewhat memorized, all of the math, etc. Generally speaking, people don’t seem to have much trouble retaining information that is useful to them, so unless you’re having to pass a series of exams I would not worry about any of the details if you don’t want to. Maintaining a degree of rigor and intellectual honesty is important, but at the same time knowing a theory should enhance your understanding of the real world instead of detracting from it.
In any case, we have atomic nuclei with positively charged protons and non-charged neutrons surrounded by somewhat amorphous clouds of negatively charged electron density generated by a discrete number of negatively charged electrons moving around at high speed. How nuclei, orbitals, and electrons interact is chemistry, and given the complexity in chemical reactions that is evident (particularly in biology) it should come as no surprise that the behavior of electrons, elements, and molecules is also extremely complex. We as a species have spent many centuries of unified time and uncountable person-millennia of effort grappling with aspects of the complexity of chemical behavior, before discovering relatively recently that everything is derived from quantum mechanics and none of the simple mathematical models are particularly valid. The discovery of quantum mechanics started in the early 1900s to the 1920s or so in the physics community and has led to a progressive series of major improvements in the way we think about the world that is still underway. The information gained has led to our disastrous exploration of nuclear fission in heavy elements but also to the development of much more potent instrumentation, semiconductors, computers, and a better, if not necessarily more comforting, understanding of the universe that we live in.
Looking at chemistry specifically, our goal as a species needs to be to do as little chemistry as possible while still ensuring our survival. Where chemical reactions are unavoidable, we need to take care to ensure that the resulting waste is as non-toxic, biodegradable, and/or easily denaturable as possible. Simple molecules such as carbon dioxide can cause problems when emitted in bulk, and more complex molecules tend to be nastier in much lower quantities and concentrations (eg polychlorinated biphenyls/PCBs). As creatures with cellular machinery that is mostly made of organic molecules, we are going to be most interested in organic reactions despite our historical inability to make much sense of the complicated electronics and molecular orbitals of organic reactions. Unfortunately, this means that we will not be able to skip as many of the details, and if I want to try for complete coverage I would expect to see a few tens of posts. The main difference between general and organic chemistry is that a significant fraction (possibly even most) of the general chemistry material is obsolete and/or irrelevant, while the majority of organic chemistry material is both important and relevant. So this may take a while, and I’m going to wish that I still had access to the ChemDoodle software that is set up for organic structures. On ubuntu linux, the GChemPaint program seems similar and is free, and I guess that I’m about to find out how well that it works.
I will do my best to relate concepts back to the mental picture of how chemical compounds interact that you are hopefully building up as I introduce them, but as always things are usually going to be messy. The list of high level topics in organic chemistry as defined by my undergraduate study guide is as follows: structure, bonding, intermolecular forces of organic molecules, acids and bases in organic reactions, nomenclature, isomers, principles of kinetics and energy in organic reactions, preparation and reactions of (alkenes, alkynes, aldehydes, ketones, alcohols, sulfides, carboxylic acids, amines, aromatic compounds), organic reaction mechanisms, principles of conjugation and aromaticity, and spectroscopy. I have not yet decided if this is the order in which I would like to present these concepts, but hopefully you can see that this is a large amount of material. As a final note, organic chemistry is mostly the chemistry of hydrogen, carbon, nitrogen, and oxygen with trace quantities of several other elements participating at times. Organic molecules are interesting both because of the wide range of properties and behaviors that they exhibit and also because of our desire to understand our biology, and we are studying mainly the chemistry of the 1s, 2s, and 2p valence orbitals in small atoms.

First of all I hope you all are doing great!
It might sound kind of obsessive but I've been trying to find my type from a long time ago, and I've considered even the strangest possibilities (according to close friends) just because I overthink the functions a lot and I consider I become more and more biased with time, changing the meanings of things just to make them fit into different functions whenever I come up with a new theory. It sounds convincing to everyone, but not to me... I really have no one I can talk about mbti in this way because people close to me don't understand or are very into it haha, I believe I don't know myself enough (which I should work on, and that's why I really like mbti but I feel I worsen things instead of finding some light hahaha).
Anyways, so sorry for the long introduction and thanks a lot for reading! <3
• How old are you? What's your gender? Give us a general description of yourself.
I'm a 22 year old female and I'm from Colombia. I am interested in several things though I am not good at all of them haha. I played tennis almost during 9 years (or 10) and people don't know why since I am not very good for sports but I've always been very disciplined. I love languages and I think I have a knack for them, I love to paint, draw, sculpt or anything I can do with my hands since it helps me to focus only on what I'm doing and stops my thoughts or helps me clarifying them, I love to write and read (yeah yeah, infp haha) and love to learn new things. My childhood dream was to become astronomer but I am terrible at math and teachers never understood how I always got the right answers with the wrong (or not indicated) methods haha.
• Is there a medical diagnosis that may impact your mental stability somehow?
I don't think so... I've had problems like everyone else and hard times as well, but I'm ok I guess haha.
• Describe your upbringing. Did it have any kind of religious or structured influence? How did you respond to it?
I've always said that my dad taught me to love knowledge and my mom taught me to spread it. My dad always told me Jesus was a philosopher whose ideas were distorted since I was very little and up until now I fiercely believe in that haha, there was a very difficult time in which I used to get upset when my mom talked about God (she's very catholic) but now I like to listen to everyone and get ideas from their beliefs. I believe God is a misunderstood artist whose plans we don't fully understand but I like to compare him to those writers who always imagine tragic endings for their characters... (?)
• What do you do as a job or as a career (if you have one)? Do you like it? Why or why not?
I currently work in a call center, even thought I hated the idea 4 years ago I've learn a lot from it! Now I don't feel anxious with calls and I am very thankful that because of my situation I was forced to work in one hahaha, I can express myself better and explain my ideas to other without fear of rejection or making them feel upset with me. I don't plan to work in one of them my whole life, I am actually saving money to start my university studies in something related to languages and teaching! :)
• If you had to spend an entire weekend by yourself, how would you feel? Would you feel lonely or refreshed?
I would definitely love it! Even though I am pretty talkative I always have a hard time when spending a lot of time with my family or friends because I feel I am more productive when by myself. I feel very refreshed after having a long time by myself and I can learn a lot of things!
• What kinds of activities do you prefer? Do you like, and are you good at sports? Do you enjoy any other outdoor or indoor activities?
I love spending time in nature, places like parks or mountains are my favorite ones! As I mentioned I played tennis and through that experience I realized how important it is to take care of your body and your mind at the same time, but I am not naturally skilled for sports haha. It'd be redundant if I talked about my hobbies once again.
• How curious are you? Do you have more ideas then you can execute? What are your curiosities about? What are your ideas about - is it environmental or conceptual, and can you please elaborate?
I've always said my mind fluctuates in a lot of hypocognitions and I can never accomplish them entirely because there are just a lot of them! Most of the time I need to create something in order to understand a concept, an idea and/or felling I have that I don't understand in the very moment. For example I like to draw things that come to my mind when feeling something I don't understand and I'll understand that feeling three months later (or more)... Not a very practical technique, but getting better at it hahaha.
• Would you enjoy taking on a leadership position? Do you think you would be good at it? What would your leadership style be?
I love to help people in general, but being in charge is a different thing, a work I admire but wouldn't be able to execute. Still, if I ever had to be in a leadership position under stress I would be kind of bossy to be honest, when normal I'd be very active and funny as well but would burn out easily.
• Are you coordinated? Why do you feel as if you are or are not? Do you enjoy working with your hands in some form? Describe your activity?
I am very coordinated with specific tasks, not very multitasking, when investing a lot of time on certain skill I can be very coordinated, though but yes! I adore working with my hands! (Going back to the hobbies thing again).
• Are you artistic? If yes, describe your art? If you are not particular artistic but can appreciate art please likewise describe what forums of art you enjoy. Please explain your answer.
I am very artistic, I guess, specially because creating things is my way to understand myself, I always end up drawing like cute things that have a weird aspect with a lot of self-references I don't like to explain to people. Other thing I like to do is to ask people what do they thing a tale/drawing/sculpture means and try to figure out how I feel or what I meant based on that.
• What's your opinion about the past, present, and future? How do you deal with them?
Ugh, I am very nostalgic when thinking about the past, I wish I could get some sensations back but I understand that I cannot retrocede so that ideas makes me feel better at times.
Present moment? Whaaaat is thaaaaat? Haha, kidding; I've learnt to appreciate what I have in this very moment because it might change at any moment and I am afraid of that!
Future is basically the reason why I wake up and think I need to work harder to save money and be in the place I want to be... Not very good to think a it every single moment, but working on that.
• How do you act when others request your help to do something (anything)? If you would decide to help them, why would you do so?
Before, it was very hard for me to say no to my immediate family and it still is but I've become better at it * makes uncomfortable noises *. I feel I owe what I am to my parents and sisters so that's why I help them. I help people in general if I feel I am able to do it and it will really make their situation better. No regrets when helping, because if you regret why did you do it in the first place?
• Do you need logical consistency in your life?
I feel logic is the most importan thing in my life, not because I consider I am logical at all, but because it needs to be my main focus for being able to accomplish things in life and not losing track. Logic is as beautiful as feelings when wisely used.
• How important is efficiency and productivity to you?
I don't think I am naturally a productive or efficient person but life has pushed me to be so, currently I think those are amazing traits people should work on!
• Do you control others, even if indirectly? How and why do you do that?
I do it all the time, most of the time because I want to have a balance between personal and collective goals and I want things to occur in the most convenient way.
• What are your hobbies? Why do you like them?
Uhm, I've written a lot of stuff I know, I already talked about them haha, artistic thingies that might make people thing I am an ixfp haha. I love to watch youtube videos as well and memes hehe.
• What is your learning style? What kind of learning environments do you struggle with most? Why do you like/struggle with these learning styles? Do you prefer classes involving memorization, logic, creativity, or your physical senses?
I love learning by context and I have that very clear since I've noticed it in my languages classes, if I cannot get (pseudo)philosophical with it, I don't wanna learn it haha if it does not lead everyone to a conversation where they express their ideas about it it's not working. But maybe I need to thing more about it. For certain stuff I like repetition, conjugation and stuff is better when repeating it, that depends on what your learning, several techniques can be used for different results... (?)
• How good are you at strategizing? Do you easily break up projects into manageable tasks? Or do you have a tendency to wing projects and improvise as you go?
I have a talent for planning but not for executing haha, recently I have learnt that executing is important and maybe have gotten better at it, but definitely not natural for me.
• What are your aspirations in life, professionally and personally?
I want to have mental, health and economic stability to help my family and myself as well, if after that I can help others I'll be happy to! I want to know more every day and be excellent at the things I like to do and have to do.
• What are your fears? What makes you uncomfortable? What do you hate? Why?
I hate not knowing things, it always turns out to be something awful when I don't know how a person is going to react, what can I expect from a job interview, what is making me uncomfortable, and so on and so on... Not knowing what needs to be known is awful though ignorance saves me at times from getting depressed haha. Besides, who am I to decide what needs to be known and what does not?
• What do the "highs" in your life look like?
Peace, love, philanthropy, rainbows and tons of knowledge.
• What do the "lows" in your life look like?
Desire for absolute destruction, when helpless wanting to control everything, very bitter and cynical.
(I currently feel a bit of both since I am feeling fine and neutral, the examples on the highs is when I am very very high that it becomes something bad).
• How attached are you to reality? Do you daydream often, or do you pay attention to what's around you? If you do daydream, are you aware of your surroundings while you do so?
What is surroundings? Haha, I have a very hard time focusing in the real moment and when I finally can do it I forget to think haha, have to do one of those at the same time. Daydreaming 85% to be honest.
• Imagine you are alone in a blank, empty room. There is nothing for you to do and no one to talk to. What do you think about?
First thing I would think about would be the love of my life haha, cheesyyyy. After that I would start overthinking about everything else, when on the empty room for hours I'd start thinking about what others are going to do to me or my family or the reasons why I ended up in there, then ways to escape after years of being there haha... Not really sure. (?)
• How long do you take to make an important decision? And do you change your mind once you've made it?
Woah, that's related to the last thing I've said hahaha, it takes years or months for me to decide something, but when decided there is no coming back.
• How long do you take to process your emotions? How important are emotions in your life?
I guess I explained that as well before, understanding feelings and thoughts (which I like to combine) through tangible things like drawings, writings and others.
• Do you ever catch yourself agreeing with others just to appease them and keep the conversation going? How often? Why?
That happens to me often with strangers or people I am not very close to. When confident in the conversation or if I see the person is getting my point and is respectful I can give a little bit more of info, if my family, most of the time I will speak up my mind regardless of the outcome of it hahaha.
• Do you break rules often? Do you think authority should be challenged, or that they know better? If you do break rules, why?
I don't break rules often just in case I break them once for solid and valid reasons I will be heard and maybe that specific rule will be reviewed and changed hahaha. The fact that someone is in a position of authority does not make them being right every single time, it means a lot of stuff: they either worked very hard for it, or they bribed someone or whatever but one has to be very attentive and aware of the person that is leading, they are humans after all, haha.
​
Well, I think this was a lot and maybe no one is going to read it but thanks a lot for giving us this space, maybe after writing so much about myself I learnt something new about me!
​
Cheers!!! :)
​
Sorry for the typos and for getting more and more concise in the latest answers, I got tired hahahaha. :(

I’m taking College Trigonometry right now. As we’re nearing the end of the semester, we’re getting into trigonometric identities and solving trig equations. I am REALLY struggling with this.
It seems every concept in math I’ve learned so far involves recognizing the problem before you, and then executing a repeatable process in order to solve that problem. And then you can undo said process to check your work.
With Trig identities I feel like there’s no set way to solve a problem, since multiple identities can be applied in multiple places. I never know when to apply a Pythagorean Property, change TAN or COT into SIN/COS (or vice versa), or change the trig functions into their reciprocals, just to name a few. This is even before I get into the mess of multiplying by conjugates or trying to manipulate denominators to be common, which usually messes up my expression beyond all recognition. All the options just lead me into dead ends with incompatible terms that don’t go together with no further identities able to be applied.
Then when I look over the problem’s solution in my book it’s always something that I easily know how to execute once I see it, but wouldn’t have been able to recognize needed done, if that makes sense.
I guess what I’m asking overall is do you guys have any sort of Steps or Conditions to check to sort of nail down a process for simplifying trigonometric expressions beyond just trial and error (sort of like a problem solving flowchart)? Furthermore, what are some key features to look for in a problem that may hint which direction you can go with the simplification via identities?

So I recently came across this tik tok video. It’s about a girl who works in retail and she basically makes a joke about how when a customer gives her extra cash to round up the customers change to a dollar, she has an extreme amount of difficulty to understand that the customer wants a dollar back.
And soooooo many people in the comments on tik tok were like “OMG THIS IS SO FUCKING RELATABLE OMG!!! I HATE WHEN THEY GIVE ME CHANGE LAST MINUTE LIKE IT SUCKS!!!! I CAN’T COMPUTE THIS UNDER PRESSURE LIKE OMG!!! I SUCK AT MATH AND THERE’S NOTHING I CAN DO ABOUT IT!!” With all these insane amounts of likes. So clearly lots of people agree with them.
I even saw people say “I just tell them I can’t give them anything back, I already put it in the register, simple as that. I don’t have time to deal with that. Oh well, their fault for bringing change in last minute!”
Now, I will say this. I understand that people melt under pressure and have anxiety about performance. I also understand how frustrating it can be for customers to bring out change last minute. And I also understand that some people have learning disabilities. Also sometimes people just have straight up brain farts. But I am truly appalled at all the people who claim that this is math that is too difficult to understand, so they just dismiss the extra change the customer tries to give them because they “just don’t get it”.
For god’s sake it’s not even hard math. It’s not like they’re asking you to find the complex conjugate of a function or to find all the real and imaginary roots of a polynomial. They’re fucking asking you to count. Where is the difficulty in this?
I started my first job when I was 16. I am currently 22, turning 23 this year. I have never in my life had this problem even till this say and I am shocked at the amount of young people that were in that comment section of that video, confessing that they struggle with basic counting.
If I asked you to count all the letters in your name, that’s an easy task right?
If I asked you to count how many keys are on your keychain, that’s an easy task right?
But if a customers change is like....95 cents, and they give you a nickel, you can’t comprehend that they want a dollar back? Like you seriously cannot tell that 95 cents and 5 cents make a dollar? You can’t count from 95 to 100 and realize that’s 5 cents?
I can understand if they give you a random set of numbers like if 67 cents is their change and they give you 33 cents. Like i can tell how that would take a little moment because you can’t computer 67 and 33 off the top of your head and realize that’s 100, as easily as you can count, say, 95 and 5 and realize that’s 100. But USUALLY the customer will give you change that rounds up to a dollar. Or, they’ll give you a certain amount of dollars and the exact amount of cents so they’ll only get bills back and no change (such as having a total of $5.30 and the customer gives you $20.30 so they get back only $15.00 in bills and no change).
But for god’s sake, asking people to do basic counting math is apparently asking for the world now. Why would you find not being able to fucking COUNT as “relatable?” What the actual fuck? Im appalled that so many who have worked in retail for fucking years struggle with this.
Certain skills are life skills that you will have serious difficulty with if you don’t understand them. You cannot go through life not knowing how to count. That is insane. You should seriously study math and counting and youtube tutorials on counting math if you seriously have trouble with this task.

I wrote a quaternion library using help from the following sources:
https://github.com/JujuAdams/basic-quaternions/tree/mastescripts
http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/arithmetic/index.htm
But when I try to use the quaternions to rotate vectors, weird stuff happens. The following video shows this. The first half uses rotation matrices to rotate the vectors and the model properly rotates around the X axis. The second half uses quaternions, and the model is clearly rotating around some different axis. Not only is the rotation wrong, but it appears to mess up the backface culling too.
https://www.youtube.com/watch?v=Zf0Zkncj0mY&feature=youtu.be
​
So I obviously don't understand quaternions, but here is my quaternion library, in case that helps.
​

function quat4d() constructor{ X = 0.00000001; Y = 0; Z = 0; W = 1; static copy = function(){ var n = new quat4d(); n.X = X; n.y = Y; n.Z = Z; n.W = W; return n; } static normalize_self = function(){ var n = sqrt(X * X + Y * Y + Z * Z + W * W); X /= n; Y /= n; Z /= n; W /= n; } static conjugate_self = function(){ X = -X; Y = -Y; Z = -Z; } static conjugate = function(){ var n = new quat4d(); n.X = -X; n.Y = -Y; n.Z = -Z; n.W = W; } static add_self = function(_other){ X += _other.X; Y += _other.Y; Z += _other.Z; W += _other.W; } static add = function(_other){ var n = new quat4d(); n.X = X + _other.X; n.Y = Y + _other.Y; n.Z = Z + _other.Z; n.W = W + _other.W; return n; } static subtract_self = function(_other){ X -= _other.X; Y -= _other.Y; Z -= _other.Z; W -= _other.W; } static add = function(_other){ var n = new quat4d(); n.X = X - _other.X; n.Y = Y - _other.Y; n.Z = Z - _other.Z; n.W = W - _other.W; return n; } static multiply_self = function(_other){ var XX = X * _other.W + W * _other.X + Y * _other.Z - Z * _other.Y; var YY = W * _other.Y - X * _other.Z + Y * _other.W + Z * _other.X; var ZZ = W * _other.Z + X * _other.Y - Y * _other.X + Z * _other.W; var WW = W * _other.W - X * _other.X - Y * _other.Y - Z * _other.Z; X = XX; Y = YY; Z = ZZ; W = WW; } static multiply = function(_other){ var XX = X * _other.W + W * _other.X + Y * _other.Z - Z * _other.Y; var YY = W * _other.Y - X * _other.Z + Y * _other.W + Z * _other.X; var ZZ = W * _other.Z + X * _other.Y - Y * _other.X + Z * _other.W; var WW = W * _other.W - X * _other.X - Y * _other.Y - Z * _other.Z; var n = new quat4d(); n.X = XX; n.Y = YY; n.Z = ZZ; n.W = WW; return n; } static xrotate_self = function(_a){ var _other = new quat4d() _other.X = dsin(_a/2); _other.W = dcos(_a/2); self.multiply_self(_other); } static yrotate_self = function(_a){ var _other = new quat4d() _other.Y = dsin(_a/2); _other.W = dcos(_a/2); self.multiply_self(_other); } static zrotate_self = function(_a){ var _other = new quat4d() _other.Z = dsin(_a/2); _other.W = dcos(_a/2); self.multiply_self(_other); } static rotate_self = function(_a, _v){ var _other = new quat4d() _other.X = _v.X * dsin(_a/2); _other.Y = _v.Y * dsin(_a/2); _other.Z = _v.Z * dsin(_a/2); _other.W = dcos(_a/2); self.multiply_self(_other); } static rotate_vec = function(_v){ var v_quat = new quat4d(); v_quat.X = _v.X; v_quat.Y = _v.Y; v_quat.Z = _v.Z; v_quat.W = 0; v_quat = self.multiply(v_quat); v_quat.conjugate_self(); v_quat = self.multiply(v_quat); return new vec3d(v_quat.X, v_quat.Y, v_quat.Z); } } 

And here is my vector rotation code. Right now it has code for both rotation matrix and quaternion rotation, and I just comment or the other out depending on what I want to use.
​

static rotate_self = function(_a, _b, _c){ var q = new quat4d(); q.xrotate_self(_a); q.yrotate_self(_b); q.zrotate_self(_c); var temp = q.rotate_vec(self); self.set(temp.X, temp.Y, temp.Z); /* var mat = matrix_build(0, 0, 0, _a, _b, _c, 1, 1, 1); var array = matrix_transform_vertex(mat, X, Y, Z); X = array[0]; Y = array[1]; Z = array[2]; */ } 

EDIT: I have solved the problem. It was in my quaternion rotate_vector function. I was taking the conjugate of the vector and multiplying by the quaternion, when I needed to take the conjugate of the quaternion and multiply by the vector. Here is the corrected code:
​

 static rotate_vec = function(_v){ var v_quat = new quat4d(); v_quat.X = _v.X; v_quat.Y = _v.Y; v_quat.Z = _v.Z; v_quat.W = 0; var conj = self.conjugate(); v_quat = self.multiply(v_quat); v_quat = v_quat.multiply(conj); return new vec3d(v_quat.X, v_quat.Y, v_quat.Z); } 

​

Lisryuzaki is what some people call me, others say L. What the future lies is here, the world of SAO AND YUGIOH, chronologically and rationally is what the world will be. Behold,
https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.pinterest.com%2Fpin%2F635570566132254776%2F&psig=AOvVaw1W6JQLI9XmkhDvmHiGJJiT&ust=1606803205807000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCICIzcfOqe0CFQAAAAAdAAAAABAD
https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.deviantart.com%2Fliara-halsey%2Fart%2FMirco-Duel-Disks-2-205488184&psig=AOvVaw3N-PeIfsF1NL9X1dDd9lwU&ust=1606803309476000&source=images&cd=vfe&ved=0CAIQjRxqFwoTCLjo9ffOqe0CFQAAAAAdAAAAABAD
Science
The Invention titled Nerve gear is a biopic utensil from the neural relay of the brain. The main course of it is to connect neurons across the brain to form a digital world. It works like this, it requires the computer of the brain to be harnessed by the machine which works as an input device which secretes a computer signal in to the brain at over 1/500 seconds. This, then causes the outlay of the brain to process, the inner lay of the machine and backwards, vice-versa. The visual interface is the display of the neuronic signals merging with the machine's plugin feature as, the bios runs the software at a console rate. The four pieces of a brain determine the reasonable amount of power and data analysis such as the computer brain merging with the computer machine giving an easier runtime and speed while a 'hitman' brain gives an easier framerate and data feed. The power supply from the Nerve gear is a sustainable battery from the exposure to light as it uses light-emitting diodes to charge itself from a fluorescent or LED Light with Brain neuro-electricity to back it up. It gives a total of 16 hours of use under Battery N and 24 hour use under Battery LED and 36 hour use under Both N and LED Battery.
CONSTRUCTION
Central processing unit (CPU): AMD Ryzen Threadripper 3960X Cores: 24 | Threads: 48 | Base clock: 3.8GHz | Boost clock: 4.5GHz | L3 cache: 128MB | TDP: 280W
Motherboard: Single Board Computers CIRCUIT MODULE, AMD T40R MINI-ITX, VGA/LVDS/HDMI/4GbE/2COM
Memory (RAM): Crucial 64GB Kit (32GBx2) DDR4 2666 MT/S CL19 SODIMM 260-Pin Memory - CT2K32G4SFD8266
Graphics processing unit (GPU): nVidia Jetson NANO Developer Kit, Arm A57 4-core CPU, Maxwell 128-core GPU, 4GB DDR4, M.2 Key E, 4x USB 3.0, USB 2.0, DisplayPort, HDMI2.0b, Gb LAN, MicroSD slot: Memory Card Connectors MICRO SD PUSH/PUSH NORMAL 1.28MM 8C Storage: (240 GB, mSATA) - Kingston SUV500MS/240G SSD UV500 mSATA Power supply unit (PSU): Synology 250W Power Supply for Selected NAS Models System cooling: Sabrent M.2 2280 SSD Rocket Heatsink (SB-HTSK) Operating system(OS): Ubuntu
Engineering
The core of the projector is a small 4K LCD panel, which is from a modified Sony smartphone. [Matt] disassembled the phone, removed the backlight from the LCD, which leaves it semi-transparent, and mounted it at a right angle to the rest of the phone body. The battery was also replaced with a voltage regulator to simulate a full battery. To create a practical projector, a much brighter backlight is needed. [Matt] used a 100W 10 mm diameter LED for this purpose. The LED needs some serious cooling to prevent it from burning itself out, and a large CPU cooler does the job perfectly. Two Fresnel lenses in series are used to turn the diverging light from the LED into a converging light source to pass through the LCD. An old 135 mm large format camera lens is placed at the focal point of light to act as a projection lens. The entire assembly is mounted on a vertical frame of threaded rods, nuts, and aluminium plates. [Matt] also used these threaded rods with GT2 pulleys to create a simple but effective moving platform for the projection lens that allows the focus of the projected image to be adjusted. The frame is topped off by a 45-degree mirror to project the image against a wall instead of the roof, and the frame is covered with aluminium panels.
Compared side by side, the DIY projector beats a $2000 commercial 4K projector in terms of image sharpness and colour. The DIY version only falls short in terms of brightness, because it uses a lower output light source. It requires a very dark room to see the projected image, but it also means that less active cooling is needed, making it quieter than the commercial projector.
Method: By building a 3-D printed Neve Gear in design form and putting the pieces together, with the science and maths, then you've got the nerve gear.
And By assembling a duel disk with tested projector technology and putting the pieces together, with the maths and science, TA-DA Nerve Gear
Science: Nerve Gear and Duel Disk
Maths: Hodge Conjecture, P = NP, Navier Stokes Equation.
MATHS
A complexity of combinations between a simple equation of 2n² which states that a computational tree is created as the problem is administered as 1(n): A Quantum Computer Logical Framework: Non-Binary Computational Maths
The level of third-dimensional rendering of reality, henceforth light giving the equation of n(6*6)² understood as a cubiouc structure with atomic points of interception: A hologram visual perception; Holographic Geometry
It percuss as a gravitational equation: created by a conjugal of several equations lead to develop a single equation based on four parts: 1; Air pressure, Spatial Mass, Depth Mass and Downward force: An deferral gravitational spectrum; Gravitational Mathematics
Coding
https://github.com/ellisdg/3DUnetCNN : NEURAL SCANNING AND PROCESSING SOFTWARE https://github.com/mrdoob/three.js : 3D LIBRARY https://github.com/alicevision/meshroom :3D CAPTURING AND DESIGNING SOFTWARE https://github.com/godotengine/godot :3D VIDEO GAME MAKING SOFTWARE https://github.com/slic3Slic3r :3D PRINTING SOFTWARE FINAL PIECE: https://github.com/microsoft/AI :PERFECT A.I
Business
A very comprehensive model of such an integrated and networked fifth generation innovation process model is given by Galanakis [8]. He proposes an innovation process description using a systems thinking approach (which he terms the “the creative factory concept”) (refer to Fig. 8). This model has at its centre the firm (enterprise), which is the generator and promoter of innovations in the market, the industrial sector and the nation. The model’s overall innovation process is constructed of three main innovation processes: 1. the knowledge creation process from public or industrial research; 2. the new product development process, which transforms knowledge into a new product, and 3. the product success in the market, which depends on the product’s functional competencies and the organisational competencies of the firm to produce it at a reasonable price and quality and place it adequately in the market. This process is affected by internal factors of the firm (e.g. corporate strategy, organisational structure, etc.), as well as by external factors in the National Innovation Environment (e.g. regulations, national infrastructure, etc.).
There you have it, GO BUILD AND MAKE HISTORY.

Hi,
I have studied physics for one year as a part of my philosophy degree. I am not especially talented in physics or math but I really enjoy it when I make progress. I believe it's super important for a philosophers understanding of metaphysical problems to understand physics. So in QM I really struggle to understand the use of complex functions because I don't really get what the complex plane is. I know you get rid of the imaginary part by multiplying psi with it's complex conjugate. That's all fine, but it still bugs me not to understand complex numbers or the role they have in calculating the probability distribution. I'm not english speaking, if I seem dumb, I blame the language hehe.
If some of you could send me in the direction of some good lectures or such on the topic I'll be super thankful. :)

I dropped my calc class because I kept running into problems that I fundamentally couldn't do, and they were all typically fractions. I want to be able understand the math logically.
My problem is that I think I could just multiply both sides by the denominator, but instead I should have conjugated... or it was a squared binomial, or a polynomial.. and I should have done something else. I dont know what this something else is.
I purchased a college algebra book, but even then it looks like maybe I need to go to an earlier math. I'm not even sure where I should start looking.
Is there a pre-algebra book or something else I should work through? I think I should just work may way back up from a base level, but don't know what is a good source.

https://www.howtostudykorean.com

2

있다 when used as "have" or "in/at location" is an adjective
Example of difference between 이/가 and 은/는 - 는/은 has a role of indicating that something is being compared with something else. 고양이는 집 뒤에 있다 = The cat is behind the house The speaker is saying that the cat is behind the house (in comparison to something else that is not behind the house).

3

Remove 하다 from verb/adj to create nouns e.g. 해복하다 (happy) -> 행복 (happiness)
Adjectives cannot act on object (no 을/를)

4

Adjective - add ㄴ for vowel (큰); add 은 for constant (좋은); add 는 if adj ends in 있다 (맛있는)
~도 particle (meaning "also") replaces other particles 은/는 을/를

5

Attached subject particle to 저/나 (I/me) changes them to 제가/내가
How to avoid saying 당신:

• [Name] 선생님; 고객님 (customer); 손님 (guest); 부장님 (boss)
• 형/누나
• 할머니/할아버지 if very old
• 아저씨/아주머니/아줌마/아가씨 for stranger
• 너 is acceptable for informal (changes to 네가, pronounced "knee-ga")

Conjugating verbs in plain form:

• Past - add 았/었/였 e.g. 먹었다 (note - 했다 is shorted 하였다, but in some situations e.g. street signs, it is left as 하였다)
• Present - add ㄴ/는다 depending on consonant/vowel e.g 먹는다, 배운다
• Future - add 겠다 e.g. 먹겠다

Conjugating adjectives in plain form:

• Past - add 았/었/였
• Present - do nothing! e.g. 크다
• Future - 겠다

Note that sometimes 있다 is a verb (e.g. -고 있다), so 있다 can be conjugated both ways depending on whether it is a verb or adj

6

Conjugating in informal form: Past - 봤어, 먹었어, 했어 Present - 봐, 먹어, 해 Future - 보겠어, 먹겠어, 하겠어
Conjugating in polite form: Past - 봤어요, 먹었어요, 했어요 Present - 봐요, 먹어요, 해요 Future - 보겠어요, 먹겠어요, 하겠어요
Conjugating in formal form: Past - 봤습니다, 먹었습니다, 했습니다 Present - 봅니다, 먹습니다, 합니다 Future - 보겠습니다, 먹겠습니다, 하겠습니다

7

Irregulars - almost all apply when adding vowel to stem (very common):
ㅅ - e.g. 짓다 (to build) - ㅅ is removed, so becomes 지었어, 지어요 etc. [씻다, 웃다 are regular though]
ㄷ - e.g. 걷다 (to walk) - ㄷ becomes ㄹ e.g. 걸어, 걸었어요 [받다, 닫다 are regular though]
ㅂ - e.g. 쉽다 (easy) - ㅂ changes to 우 e.g. 쉬워, 어려워, 귀여워 ㅂ - e.g. 돕다 (to help) - ㅂ changes to 오 e.g. 도와요, 고와 The ㅂ irregular is common in adjectives, so when inside sentence, will change to e.g. 귀여운, 새로운 [잡다, 넓다 are regular though]
ㅡ - when conjugating, need to decide how to conjugate (with 아/어) based on vowel in 2nd-last syllable (if there is one) e.g. 바쁘다 becomes 바빠 예쁘다 becomes 예뻐 잠그다 becomes 잠가 슬프다 becomes 슬퍼 크다 becomes 커
르 - when final syllable is 르, another ㄹ is added to preceding syllable, and 르 becomes 라/러 e.g. 다르다 becomes 달라요 빠르다 becomes 발라요 부르다 becomes 불러요
ㄹ - mainly concerned with creating adjectives and the plain & formal conjugations When final letter is ㄹ and you have a choice of ㄴ/는, ㄴ/은, ㅂ/습, ㄹ/을, always choose the first and remove the ㄹ! e.g. 길다 (long) -> 긴 거리 (long street), 깁니다 (formal) 멀다 (far away) -> 먼 병원 (far away hospital), 멉니다 (formal) 열다 (to open) -> 연다 (plain), 엽니다 (formal)
Can get confusing with 듣다 and 들다 due to conjugations and irregulars - have to tell difference from context
End of chapter 7 also has link to irregular quick reference guide.

8

~에 used for when/where e.g. 저는 가을에 공원 옆에 병원을 지었어요 (x2 에) Exceptions - 오늘, 내일, and 어제 do not use 에
Adverbs - a lot can be made by adding 게 on verb stem (e.g. 다르게 - differently), but 하다 verbs sometimes use 히 to replace 하다 e.g. 조용하다 -> 조용히. Confusingly, 게 can also usually be used - 조용하게. Adverbs like 많이 and 빨리 don't use 게/히. 저는 많은 밥을 먹었어요 and 저는 밥을 많이 먹었어요 have the same meaning.
On 안 and ~지 않다: For 하다 verbs it is common to separate the noun and 하다 and put 안 in between (e.g. 저는 공부를 안 했어요), you can do the same with -지 않다 but not quite so common. But you absolutely can't do it with adjectives. 안 is effectively a negative adverb, and having two adverbs act on the same verb is awkward, so should use ~지 않다 (or better yet, use the adverb with the opposite meaning i.e. 'slowly' instead of 'not quickly')
아니다 is separate word and requires 이/가 particle, in contrast to 이다
싫다 has a stronger meaning as "don't like" so is fine to use in place of 싫어하다. As a result, if you want to say something is bad, 나쁘다 is more common.

9

Conjugating 이다: Plain - 이다 (or even just 다 of noun ends with vowel) Informal - 이야 for consonant, (이)야 for vowel (can drop the 이 if you want). 아니다 becomes 아니야. Polite - 이에요 for consonant, 예요 for vowel. 아니다 should be 아니에요, but is a common mistake to be written as 아니예요 Formal - 입니다 (although 이 can be dropped for nouns ending in vowels e.g. 의삽니다)
Past: Plain - 이었다 (or can be shortened to 였다 if noun ends in vowel, but don't have to) Informal - 이었어 (optional contraction to 였어 for vowel-ending word) Polite - 이었어요 (optional 였어요 for vowel) Formal - 이었습니다 (optional 였습니다 for vowel). 아니다 is 아니었[다,어,어요,습니다], but 아니였… is common mistake
Rest of lesson 9 is about ㄹ/을 것이다 future tense. 것 often shortened to 거, so 것이에요 and 거예요 are both fine. 할 거에요 is a common mistake. For the ㄹ irregular, it looks like nothing happens e.g. 열+ㄹ/을 거예요 = 열 거예요.
이다 can be future-conjugated e.g. 선생님일 거예요, but using 되다 is quite common e.g. 선생님이 될 거예요. Both are fine. However, when talking about someone else, using 일 거예요 is often a guess in the present tense e.g. 그 사람이 의사일 거예요 = That person is probably a doctor.

10

Sino-korean numbers 백 - 100; 천 - 1000; 억 - 100 million. Weirdly have to say 일억 for 100mil, for 십/백/천/만 don't need 일. 조 (일조) is 1 trillion. Used in Money Measuring (?) Maths Phone numbers Time, except hours Names of months Can count months, but so can pure korean
For pure korean, even though it goes up to 99, rarely used above 60 (switch to sino-korean). Used in Counting things/people/actions Hours Sometimes months
Common counters - 개 (thing), 명 (people), 번 (behaviours/actions - often for how many times something was done, or 'this/last/next time'). If you can't remember correct counter, can usually get away with 개.
The words 1, 2, 3, 4 and 20 change when adding a counter: 1 = 하나 -> 한 2 = 둘 -> 두 3 = 셋 -> 세 4 = 넷 -> 네 20 = 스물 -> 스무
Use of counter is noun-number-counter e.g. 사람 두 명. Can use a less common way of number-counter의-noun e.g. 두 명의 사람
영 and 공 are both Chinese origin, but 영 is more like the number, whereas 공 is like 'nothing'. Usually use 영, but 공 is used in phone numbers.
두시 삼십분 = 2시 30분 = 2:30
살 is counter for ages 몇 살이에요? 저는 열하나 살이에요
번째 is counter for first/second/third etc. Never say 한 번째 though, it's always 첫 번째. In certain situations, gets contracted - e.g. 첫 번째 = 첫째 = first 두 번째 = 둘째 = second 세 번째 = 셋째 = third 네 번째 = 넷째 = fourth Usually when talking about children (우리 둘째 아들 or just 우리 둘째예요) or making a list (first I have to do this, secondly...)
마지막 means "last", but only when referring to the last thing in a sequence, so it's more like final/finally e.g. 이것은 저의 마지막 수업이에요 = This is my last class
처음 - first time Two most common use cases: 1. 처음에 … (at first/in the beginning) e.g. 처음에 그 여자를 싫어했어요 = I didn’t like that girl at first 2. As an adverb e.g. 저는 어제 선생님을 처음 만났어요 = I met my teacher for the first time yesterday

11

개월 is Chinese origin, so use sino-korean numbers. 달 is pure korean, so use korean numbers
동안 - for a duration of time e.g. 10분 동안. No -에 attached.
시간 - hour(s). 시 - o'clock
지난 - previous e.g. 지난 주에 영화를 봤어요
Weirdly, 지난 시간 can be uses to mean "last (previous) time" e.g. 저는 그것을 지난 시간에 배웠어요 = I learned that (thing) last time Can also use 번 to mean the time something happened though e.g. 저는 그것을 지난 번에 배웠어요 = I learned that (thing) last time
다음 - next
Days - 일/날/하루 일 is never used alone, it's always attached to another word (alone it usually means 'work') e.g. 현충일 = Memorial day 일 is also the counter for days e.g. 나는 3일 동안 공부했어 = I studied for 3 days
There are also separate words for 1 day, 2 days, etc. up to 10, but only 하루 (1 day) is used more often than 1일. e.g. 저는 하루 동안 여행했어요 = I traveled for 1 day
일 also used for day of the month e.g. 3월 2일 = March 2nd
날 cannot be counted, is used only for talking about a specific day e.g. 저는 그 날에 갔어요 = I went on that day e.g. 우리는 두 번째 날에 서울에 갔어요 = We went to Seoul on the second day 첫날 refers to the first day
주 - week(s) e.g. 저는 다음 주에 미국에 갈 거예요 e.g. 저는 2주 동안 한식을 안 먹었어요 = I didn’t eat Korean food for 2 weeks 주일 can also be used as counter for weeks, but not common
Month(s) - 달/개월 달 and 개월 can be used interchangeably, just that 달 is pure korean and 개월 is sino-korean so respective number systems must be used.
이번 is similar to 지난 and 다음, meaning "this" e.g. 저는 이번 주에 계획이 없어요 = I have no plans this week Similarly to 지난 시간, can use 이번 시간 to mean this time, but usually just use 이번에 instead. 다음 번에 and 지난 번에 to mean next/previous time also common.
년 - year(s) If you want to say “last/next year” in Korean, you can’t use “지난/다음/이번 년.” Instead, you must use separate words: 작년 = last year 내년 = next year 올해 = this year e.g. 나는 내년에 한국에 갈 거야 = I will go to Korea next year For some reason it is common to omit ~에 when using 올해
"Per" - add ~에 to the unit of time e.g. 저는 이 약을 하루에 두 번 먹어요 = I take this medicine twice per day

12

몇 can take place of a number to mean "some" e.g. 사과 몇 개를 샀어 = I bought some apples
Adding ~들 (plural) is unnatural unless referring to a person (i.e. 사람들 or actors, doctors, etc.)
~만 - only Can separate 하다 verbs and use 만 e.g. 저는 어제 일만 했어요 = Yesterday, I only worked 만 will replace 을/를 and 은/는, but gets appended on everything else e.g. 우리는 학교에만 갔어요 = We only went to school
~에서 - location in which subject is doing something. Other main useage is as "from". 여기에서 gets shortened to 여기서 (and same for 거기 and 저기.
~부터 seems similar to ~에서, but 부터 specifically indicates the time/place that something starts from (example is ~에서 for a bus departing from a stop, ~부터 for departing from bus garage at start of route), and seems for location ~에서 will generally be more natural. Main use of 부터 is for time, including to mean "since".
~까지 used with 에서 or 부터 (or neither) to mean "to/until a place/time" e.g. 저는 그 책을 처음부터 끝까지 읽었어요 = I read that book from start to finish
~(으)로 - mainly to indicate with what tool/device/method/material something is carried out. Other uses:

• language used 한국어로
• what you ate for specific meal 아침식사로 밥을 먹었어요
• order in which you did smthg 저는 학교에 두 번째로 왔어요 = I came to school second (I was the second person to come to school)

Other main use is direction e.g. 집으로 갈 거예요 = I will go home e.g. 저의 친구는 저 쪽으로 갔어요 = My friend went that way

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~과/와, ~랑/이랑 and ~하고 can all be used interchangeably to mean “and” e.g. 사과와 바나나를 샀어 They also all mean "with" e.g. 친구와 집에 갔어요 = I went home with my friend
같이 and 함께 also mean "with", and can be used with 과/와 etc. or without e.g. 빵을 같이 먹었어요 = We ate bread together e.g 빵을 친구와 함께 먹었어요 = I ate bread with a friend 와/과 more likely to be used for writing and formal speech; (이)랑 more likely for other speech. 함께 more likely for writing and formal, but 같이 is somehow the much more common word. So pick the correct combination! (It's ok to use 같이 with ~어요 endings)
~에게 (written, formal), 한테 (converstaional), 께 (honorific) used for doing something (usually giving) to someone
~에게서/한테서 is used when somebody receives something from somebody. The “thing” that is being received doesn’t need to be something physical. It could be something abstract like stories or explanations. e.g. 저는 교감선생님에게서 한국어를 배웠어요= I learned Korean from my vice principal e.g 저는 그것을 친구한테서 들었어요 = I heard that from my friend ~(으)로부터 can also be attached to the person from whom one receives something, but a difference from 에게서/한테서 is that it can also be attached to a non-person e.g. 나는 돈을 정부로부터 받았어 = I received money from the government
~을/를 위해(서) - doing something for (the benefit of) somebody e.g. 나의 여자 친구를 위해(서) 꽃을 샀어= I bought flowers for my girlfriend The ~서 is completely optional, there is no difference in meaning Can also be used on non-person (e.g. for company)
~에 대해 - "about", seems to be "(verb) about..." e.g. 나는 너에 대해 생각했어 = I thought about you e.g.나는 나의 아버지에 대해 말했어 = I spoke about my father Again, ~서 is completely optional

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Passive verbs Passive sentences indicate that an action is performed on the subject. For example: I was kicked The door was opened The hamburger was eaten Because passive verbs cannot act on an object, you will never see ~을/를 in a sentence predicated by a passive verb. Note that passive verbs feel a bit unnatural in Korean, but they need to be understood.
When dealing with 하다 verbs, most of the time you can simply exchange 하다 with 되다, to make that verb passive.
You will always need to be careful with your particles. To indicate the person doing the action, can use ~에게. To indicate non-person (company etc.), can use ~에 or ~에 의해.
Active: 학생들은 그것을 이해했어요 = The students understood that Passive: 그것은 학생들에게 이해되었어 = It was understood by the students Active: 아버지는 집을 청소했어요 = My dad cleaned the house Passive: 집은 아버지에게 청소되었어 = The house was cleaned by my dad
Passive: 점심이 학교에 준비되었어요 = The lunch was provided by the school
Using ~(으)로 might also be appropriate. Passive: 집은 청소기로 청소되었어요 = the house was cleaned by a vacuum cleaner
The other option for making a 하다 verb passive is to make it 받다 (to receive). Active: 저는 저의 형을 존경해요 = I respect my brother Passive: 저의 형은 존경 받아요 = My brother is respected
Non-하다 verbs: Often the difference is the addition/removal of an extra 지/이/히, but there is no pattern to which is the longer word 켜다 to turn on/켜지다 to be turned on 끓이다 to boil/끓다 to be boiled
Need to think whether passive verb is also indicating the state of something, in which case need to add ~아/어 있다 to the passive verb e.g. 컴퓨터가 켜져 있어요 = The computer is on e.g. 문이 닫혀 있어요 = The door is closed
A lot of active verbs end in 내다 (e.g. 끝내다, to finish). The passive form is 나다 (e.g. 끝나다) e.g. 숙제는 끝났어요 = My homework is finished e.g 컴퓨터는 고장 났어요 = The computer is broken Sometimes the passive 나다 verb is conjugated in the past when in English it is in the present e.g. 아! 그것이 기억났다! = Ah! I remember that! e.g. 땀이 났어요 = I’m sweating But it makes sense - you remembesweat just before, not as you are speaking But if the sentence is negative it is present tense: 나는 그것이 기억 안 나 = I don’t remember that
For other verbs except 하다/내다 and verbs that can be in a state, can add ~아/어지다 to make it passive: 주다 = to give 주어지다 = to be given e.g. 기회가 주어졌어요 = I was given a chance

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An explanation on the use of 좋다 to mean "I like" - the key is the use of particles 김치는 좋아요 = Kimchi is good 저는 김치를 좋아해요 = I like kimchi 저는 김치가 좋아요 = I like kimchi Another example is 그립다 (to miss a non-person, but describes the feeling so is actually an adjective) 저는 한국 음식이 그리워요 = I miss Korean food
들다 - meaning depends massively on context Two of the most common usages are: 들다 = to carry/hold something 들다 = to entego into something/somewhere e.g. 나는 동아리에 들었어 = I joined a club (I “entered” a club) e.g. 잠이 들었다 = I fell asleep (I “entered” sleep) e.g. 저는 그 그림이 마음에 들어요 = I like that picture (That picture enters my heart) e.g. 저는 손을 들었어요 = I raised my hand (I “held up” my hand/carried my hand) e.g. 저는 가방을 들었어요 = I carried the/my bag
Compound verbs - one one verb to another with 아/어 e.g. 들다 (enter) + 가다 = 들어가다 - to go into to e.g. 들다 + 오다 = 들어오다 - to come into e.g. 나다 (come out) + 가다 = 나가다 - to go out e.g 가지다 (to own/have) + 오다 = 가져오다 - to bring e.g. 돌다 (to turn) + 가다 = 돌아가다 - to go back e.g. 돌리다 (to turn smthng) + 주다 = 돌려주다 - to give back
같다 translates to "same" but when nothing is being compared, usually use 똑같다 e.g. 우리는 똑같아요 = We are exactly the same When comparing things with 다르다/비슷다/같다, can use any of the particles 와/과/랑/이랑/하고 attached to one of the things e.g. 저는 친구와 비슷해요 = I am similar to my friend e.g. 그 건물은 어제와 달라요 = That building is different from yesterday e.g. 우리 학교와 이 학교는 똑같아요 = Our school and this school are exactly the same 다르다 often used to mean "other". If you want to say "another" (like "additional") use 또 다르다 (또 is adverb for when something happens again). e.g. 또 다른 문제는 그것이 비싸요 = Another problem is it (that thing) is expensive
Homonyms Some have very distinct meanings (쓰다 = write/use/wear a hat) Some have similar meanings of you break it down 걸리다 = to be in state of hanging /caught/stuck/trapped /to take a certain amount of time /to catch a cold/sickness But think - The picture is caught (hanging) on the wall I was caught (tripped) over the line 2 hours are caught (taken) to get from Seoul to Incheon
Sickness e.g. 팔이 아파요 = My arm is sore (아프다 is an adjective) e.g. 저는 감기에 걸렸어요 = I caught a cold/I have a cold (~에 is used, nuance of being stuck 'in' something. Also past tense even though currently have cold)

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~적/적으로/적이다 적 is attached to some nouns to change into a descriptive word e.g. 역사 = history, 역사적 = historical e.g. 과학 = science, 과학적 = scientific
Adding ~적으로 makes it an adverb e.g. 경제적으로 = economically e.g. 그것은 경제적으로 가능하지 않아요 = That isn’t economically possible
Adding ~적이다 makes it an adjective e.g. 저 학교는 역사적인 건물이에요 = That school is a historical building Can technically rearrange sentence to just use ~적 instead of ~적인 (e.g. 이 건물은 역사적 건물이다 = This building is a historical building), but unnatural
~스럽다 is similar to ~적 - it can be attached to some nouns to transform into adjective e.g. 사랑스럽다 = lovely Then goes on to have a complicated distiction of 실망하다 (verb describimg feelings of disappointment) and 실망스럽다 (adjective describing something disappointing) e.g. 저는 친구에게 실망했어요 = I was disappointed in my friend e.g. 저는 영화에 실망했어요 = I was disappointed in the movie e.g. 그 영화는 조금 실망스러웠어요 = The movie was a little bit disappointing e.g. 저는 그 영화가 실망스러웠어요 = I was disappointed in that movie
Using as adjective in middle of sentence: e.g. 우리 딸은 사랑스러운 여자예요 = Our daughter is a loving/lovely girl Using as adverb: e.g. 저는 그를 사랑스럽게 봤어요 = I looked at him lovingly

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~고 Most basic - one action occurs then something else happens e.g. 저는 밥을 먹고 갈 거예요 = I will eat then go Can also use ~고 나서 e.g. 저는 숙제를 끝내고 나서 집으로 갈 거예요= I will finish my homework and then go home ~고 can also be used to connect two similar clauses - it kind of serves as "and" for verbs/adjectives e.g. 저는 과일도 좋아하고 야채도 좋아해요 (but can also just say 저는 과일과 야채도 좋아해요) e.g. 저의 여자 친구는 귀엽고 예뻐요 Usually first adj/verb is not conjugated, but in some instances they are: A long time passed between connected actions e.g. 저는 열심히 공부했고 의사가 되었어요 = I studied hard and (then) became a doctor Both past tense, but no real relation (i.e. didn't happen one after the other) e.g. 저는 방학 동안 영어 문법을 많이 공부했고 영어 신문도 많이 읽었어요 = During vacation I studied a lot of English grammar, and I also read a lot of English newspapers
Common to see 은/는 in both clauses connected with ~고 for comparison e.g. 이 산은 높고 저 산은 낮아요 = This mountain is high, but that mountain is low
~아/어서 Another way of indicating one happens after another, the first verb is never conjugated to past tense ~아/어서 more likely to be used than ~고 when the first action is "intrinsically linked" with second. Example given is with 오다 and 가다 - if in first clause, will always use 아/어서. e.g. 저는 학교에 가서 공부할 거예요 = I will go to school and then study e.g. 우리는 집에 와서 바로 잤어요 = We came home and went to sleep immediately Also always have to use ~아/어서 with "positional verbs" e.g. 앉다, 서다 (to stand), and 눕다 (to lie down)
Note that ~고 싶다 can only be used with verbs (adjectives use ~아/어지다, "become adjective"). For wanting an object, use 원하다 e.g. 책을 원해요 Negating ~고 싶다: e.g. 저는 술을 안 마시고 싶어요 = I don’t want to drink alcohol e.g. 저는 울고 싶지 않아요 = I don’t want to cry
More differences between 는/은 and 이/가 (aside from 는 being for comparison): 는/은 is used for general statements, 이/가 used for specifics, usually that you are experiencing e.g. 산은 높다 = mountains are high e.g. 산이 높다 = that mountain that I've noticed is high e.g. 여름 날씨는 좋아요 = summer weather is good e.g. 날씨가 좋아요 = the weather is good That's why we say 비가 오다, since it's not generally raining, you're 'experiencing' the rain

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~고 있다 - present progressive (있다 is verb here) e.g. 여자들은 지금 영화를 보고 있어요 = The girls are watching a movie now Can be conjugated to past tense, but normally would just conjugate to past without 고 있다. Same for future tense. 살고 있디 is common for saying where you live. Interesting that in that context you can use ~에 or ~에서, but 에서 maybe more common. 알고 있다 is just as common as 알다 for saying "I know" To say ok/I understand, 알겠다 or 알았다 are equally common 가지다 is to have/possess, but using it as 가지고 있다 is more natural(?).
Typically don't use ~고 있다 with positional verbs - instead use ~아/어 있다 to indicate state e.g. 나는 앉아 있어 = I’m sitting e.g. 그 돼지는 살아 있어요 = That pig is living (alive)
Cannot use ~고 있다 with adjectives
Can add ~아/어지다 to adjectives to say you are becoming (adjective) e.g. 행복해지다 = to get/become happy e.g. 날씨가 매일 밤에 추워져요 = The weather gets cold every night e.g. 대학교 수업은 내년에 어려워질 거예요 = University classes will get difficult next year
Adding ~아/어지다 turns it into a verb which means you can then add ~고 있다 or ~고 싶다! e.g. 집 값은 비싸지고 있어 = House prices are getting expensive e.g. 나는 행복해지고 싶어 = I want to become happy (I want to be happy)

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더 - more (Adjectives) e.g. 라면은 더 매워요 = Ramen is spicier e.g. 저는 더 넓은 집에서 살고 싶어요 = I want to live in a bigger (wider) house (Verbs) e.g. 저는 밥을 더 먹을 거예요 = I will eat more (Adverbs) e.g. 저는 더 열심히 공부할 거예요 = I will study harder (조금) e.g. 이 방은 조금 더 좁아요 = This room is a little bit smaller (Counters) e.g. 사람 두 명이 더 올 거예요 = Two more people will come (좋다) e.g. 그 학생의 태도가 더 좋아요 = That student’s attitude is better (많다) e.g. 저는 돈이 더 많아요 = I have more money
~보다 for specific comparisons The 더 is generally optional (Adjectives) e.g. 선생님들은 학생들보다 더 똑똑해요 = Teachers are smarter than students (Verbs) e.g. 나는 어제보다 밥을 더 먹었어 = I ate more than yesterday (Adverbs) e.g. 저는 작년보다 더 열심히 공부할 거예요 = I will study harder than last year (Counters) e.g. 나는 친구보다 펜이 두 개 더 있어 = I have two more pens than my friend
낫다 is very much like 좋다, but it is more naturally used when a specified comparison is being made. Therefore, it is common to see 낫다 used in sentences with ~보다. e.g. 라면보다 밥이 더 나아요 = Rice is better than Ramen 낫다 also common when saying you got better after being ill e.g. 병은 나았어 = I’m better
~보다 also commonly attached to 평소 (usual) and 생각 (thought) e.g. 나는 평소보다 더 공부하고 있어 = I am studying more than usual e.g. 도심은 생각보다 멀어요 = Downtown is further than I thought
덜 means "less" but it's more natural to form the sentence a different way and use 더
Superlatives - 가장 or 제일 (latter more often for speech) e.g. 가족은 가장 중요해요 = Family is the most important e.g. 나는 수학을 가장 좋아해 = I like math most (math is my favorite) e.g. 저는 사과를 제일 싫어해요 = I dislike apples the most

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잘 & 잘하다 - to do something well e.g. 나는 공부를 잘해 = I study well e.g. 나는 잘 공부해 = I study well The former has the nuance that you are good at it, the latter that you are studying hard because of some situation e.g. upcoming test e.g. 나는 공부를 잘 해 (separating 공부 from 하다)
못 & 못하다 - to do something poorly e.g. 나는 수영을 못해 = I am bad at swimming
~지 못하다 is identical to 못
Beware 못 though, because it also means you can't do something (like something is preventing you from doing it) e.g. 저는 어제 못 잤어요 = I didn't sleep well yesterday OR I couldn't sleep yesterday Also contrast it with 전 어제 안 잤어요 = I didn't sleep yesterday - there's no implication of something preventing you from sleeping
Another example of 안/못 difference: “Did you hear what I say?” (내 말을 들었어?) "아니. 안 들었어." - Incorrect (probably) "아니. 못 들었어." - Correct
To remove ambiguity over 못, can use 잘 못 (space between words) which is specifically not doing something well e.g. 저는 어제 시험을 잘 못 봤어요 = I didn’t do good on the exam yesterday
Be careful of 질못 (no space) which is a noun meaning "mistake"
잘 and 못 are quite common with superlatives and comparitives e.g. 저는 수영을 작년보다 더 잘해요 = I am better at swimming than last year e.g. 우리 아들은 축구를 가장 잘해요 = Our (my) son is the best at soccer

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~ㅂ/습니까 and ~니 are the question endings for formal and informal respectively. (Although ~니 makes it sound feminine, so a man might just keep ~아/어 ending)
~ㄴ/은가(요) is another question ending for informal/(polite) but only for adjectives, amd can only be used for present tense. It is considered a "soft" way of asking a question. Can also be attached to 이다, and 아닌가요 is a common way of ending a sentence to doubt yourself or ask for confirmation, agreement e.g. 너무 복잡해! 아닌가? = This is too complicated. Isn’t it?
~나(요) is another informal/polite ending, is soft, and common with verbs, 있다, 없다. Can be used in past tense, and the ~겠다 future tense. Unatural to use with 이다.
왜 - why. Usually goes before verb. Remember its use as "what" in English e.g. 어제 학교에 왜 안 갔어요? = Why didn’t you go to school yesterday?
언제 - when e.g. 집에 언제 갔어? = When did you go home? ~부터 and ~까지 can be attached to indicate form/until when e.g. 언제부터 아팠어요? = Since when have you been sick? 어제 can also be attached to 이다 to ask when something (a noun) is: e.g. 결혼식은 언제야? = When is the wedding?
어디 - where e.g. 어디 살아요? = Where do you live? The 에 in ~에 or ~에서 is often omitted: e.g. 그것을 어디서 하고 싶어요? = Where do you want to do that? Used with ~까지: e.g. 어디까지 가고 싶어요? = How far do you want to go? Use with 이다 to ask a person directly where they are or where a place is: e.g. 어디야? = Where are you? e.g. 너의 집이 어디야? = Where is your house? Use with 있다 to ask where another person or object is: e.g. 친구가 어디에 있어요? = Where is your friend?
누구 - who e.g. 너는 내일 누구(를) 만날 거야? = Who will you meet tomorrow? e.g. 그 사람은 누구야? = Who is that person? When 누구 is subject, it becomes 누가: e.g. 누가 내일 한국어를 공부할 거야? = Who will study Korean tomorrow?

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어떻게 - how (adverb form of 어떻다) e.g. 그 파일을 어떻게 보낼 거예요? = How are you going to send that file? Note in English you say "what do you think about...". In Korean, use 어떻게 e.g. 그 여자에 대해 어떻게 생각해요? = What do you think about that girl?
어떻다 gets conjugated to 어때(요) e.g. 남자 친구 어때? = What's your boyfriend like? e.g. 이 사진(이) 어때? = How about this picture?
뭐 and 무엇 - what. Both pronouns, ~을 more likely to be attached to 무엇, and 뭐 more likely to be used with 이다 e.g. 뭐 먹었어요? = What did you eat? e.g. 무엇을 먹었어? = What did you eat? e.g. 이름이 뭐예요? = What is your name?
무슨 - what. Used like an adjective but isn't technically one e.g. 무슨 영화를 보고 싶어요? = What movie do you want to see? e.g. 그것이 무슨 냄새야? = What is that smell?
무슨 vs 어떤 vs 어느 무슨 - almost unlimited choice of options, have no idea what answerer might say - "what" 어떤 - answerer has a limited number of choices - "which" 어떤 (2) - to ask about the properties or characteristics - could be translated as "what kind of" 어느 - limited choices - "which"
e.g. 무슨 영화를 보고 싶어요? - speaker has no idea what answer could be e.g. 어떤 영화를 보고 싶어요? - speaker might have listed a few choices just before e.g. 어떤 영화를 보고 싶어요? - speaker asking the genre of movie e.g. 어느 영화를 보고 싶어요? - speaker might have listed a few choices just before Note with 어떤 and 어느, the speaker doesn't have to have limited the options, there might already exist a limited number e.g. the cinema is only showing 3 films
One more unrelated use of 어떤 is to mean "some" e.g. 저는 어떤 책을 읽고 있었어요 = I was reading some book (the speaker doesn’t know exactly what book he was reading)
몇 - how many. Have to place 몇 before counter. e.g. 차가 몇 대 있어요? = How many cars do you have? e.g. 어제 학교에 몇 번 갔어요? = How many times did you go to school yesterday? Remember when used in a statement, 몇 has the meaning of some as in "a few" e.g. 친구를 몇 명 만났어요 = I met some friends 몇 also used to ask time and age: e.g. 몇 시예요? = What time is it? e.g. 몇 살이에요? = How old are you?
얼마나 - how many/much/to what extent e.g. 축구를 얼마나 잘 해요? = How well do you play soccer? e.g. 한국어를 얼마나 자주 공부해요? = How often do you study Korean? e.g. 돈을 얼마나 가져갈 거예요? = How much money will you take? e.g. 물을 얼마나 마셨어요? = How much water did you drink? Can also include 많이 when speaker knows the answer is 'a lot' bit wants to know just how much e.g. 그 여자를 얼마나 많이 사랑해요? = How much do you love that girl? If the question has a counter in it, use 몇 not 얼마나
얼마 - for asking how much something costs e.g. 이게 얼마예요? = How much is this?
이/가 vs 은/는 again 이/가 often used to give stress to the subject - seems linked to saying "X specifically/in particular" Summary of all uses so far: 은/는

• to compare something
• to state a general fact

이/가

• to indicate something based on a recent experience/observation
• to stress that the subject does the action (or is the adjective)
• objects predictaed by adjectives e.g. 나는 학교가 싫다
• objects before 아니다 e.g. 나는 학생이 아니다
• objects predicated by 되다 e.g. 나는 의사가 되고 싶다

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Colours commonly have a ㅎ irregular, and when conjugating it the ㅎ is dropped e.g. 노랗다 -> 노란 (in-sentence adjective), 노래 (informal), 노랬어 (past informal) With colours, can either say e.g. 빨간 or 빨간색, it makes no difference Note some colour words do not have a normal adjective form e.g. 초록색 (green), 보라색 (purple) You can add ~의 to all 색 colours e.g. 저는 초록색(의) 펜으로 쓰고 싶어요 e.g. 저 노란색의 집이 예뻐요 But it's considered more natural not to do this Conjugating colour adjective normally or using 이다 are both acceptable e.g. 얼굴이 왜 빨개요? e.g. 얼굴이 왜 빨간색이에요? But 이다 in the above example has the nuance that your face is red due to having something on it, normal adjective conjugation is like you're blushing
이렇다 - like this e.g. 이런 일은 위험하다 = This type of work is dangerous e.g. 저는 이렇게 하고 싶어요 = I want to do it like this Sometimes the 이렇게 looks superfluous e.g. 이 일은 왜 이렇게 어려워요? = Why is this so hard but it's used to stress the 'why'
그렇다 - like that e.g. 저는 그런 사람을 믿지 않아요 = I don’t trust that type of person/those types of people e.g. 저는 그렇게 생각하지 않아요 = I don’t think like that
저렇다 - also 'like that', but further away than 그… e.g. 저는 저런 여자를 좋아하지 않아요 = I don’t like that kind of girl e.g. 저 사람이 왜 저렇게 걸어요? = Why is that person walking like that?
The endless uses of 그렇다: e.g. 왜 그래? = What's wrong/what are you doing? e.g. 그래요. 같이 가요 = Sure (like that is fine). Let’s go together e.g. 그래요? 어디에 갔어요? = Really!? (It’s like that?) Where did you go? e.g. 그럼 = “Yes, like that.” e.g. 왜 그런지 몰라요 = “I don’t know why it is like that” e.g. 그럴 것 같아요 = “It is probably like that” e.g. 그래서 = “It is like that, so…” (Therefore) e.g. 그렇기 때문에 = “It is like that, so…" (Therefore) e.g. 그러면 = “If it is like that…” e.g. 그렇지만 = “Even though it is like that” e.g. 그랬으면 좋겠다 = “It would be nice if it is like that” e.g. 그럴까? = “Do you think it is like that?” e.g. 그런데 = “It is like that… so…” e.g. 그렇구나 = “Oh! It is like that” e.g. 그러네 = “Oh! It is like that” e.g. 그렇죠 = Sure, yep, it is like that
Lesson ends with brief discussion of adding ~적 (changes noun to adjective) vs ~의 remains a noun. Doesn't look important.

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전 - When placed after time word, "ago", need to include ~에 e.g. 저는 2주 전에 남동생을 만났어요 = I met my brother 2 weeks ago 전 - after verb, "before". Need to use format ~기 전에. Note that the verb is not conjugated for tense. e.g 엄마가 오기 전에 나는 먹었어 = Before mom came, I ate e.g. 엄마가 오기 전에 나는 먹을 거야 = Before mom comes, I will eat ~이/가 typically used on first clause, ~은/는 on main clause, although depends on context Can drop subject in main clause sometimes: e.g. 나는 오기 전에 밥을 먹었어 = Before I came, I ate
후 - latefrom now e.g. 수업은 2분 후에 끝날 거예요 = Class will finish 2 minutes from now 후 - after (format is -ㄴ/은 후에) e.g. 밥을 먹은 후에 친구를 만났어요 = After I ate I met a friend 다음 can so be used in place of 후: e.g. 숙제가 끝난 다음에 나는 집에 갈 거예요 = After my homework is finished, I will go home
직전 and 직후 - 직 emphasises sometimes was done immediately before/after
~ㄴ/은 이래로 can be used in place of …후… to mean "since", but 후 seems to be the preferred choice so don't bother e.g. 한국에 온 후에 한국어를 배우고 있어요 = Aftesince coming to Korea, I have been learning Korean
안/이내 = within (time) 안 means inside (location) but can aslo be used with time. 이내 synonymous e.g. 나는 5년 이내에 외국어 다섯 개를 배우고 싶어 = I want to learn five languages within 5 years e.g. 나는 5년 안에 외국어 다섯 개를 배우고 싶어 = I want to learn five languages within 5 years

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모든 - every (basically an adjective, but has no stem) 모든 것 = everything 모든 사람 = all people/every person
다 - all (adverb, indicates all of something is done) e.g. 저는 소설을 다 읽었어요 = I read the whole book OR I read all the books
다 is more about doing one action to completion and leaving nothing behind. 모든 is indicating that the action was performed on all possible nouns
다 can also mean the same as 모든 though e.g. 사람들이 다 왔어요 = All the people have come 다 can also be used as a noun: e.g. 그게 다야? = Is that all?
모두 when used as adverb means same as 다: e.g. 선생님들은 모두 똑똑해요 = All teachers are smart 모두 when used as a pronoun means everybody/everthing: e.g. 모두가 이해했어요 = Everybody understood e.g. 나는 모두 이해했어 = I understood everything Note in latter example, object suffix 를 is omitted because it's technically still an adverb
~나 can be added to the words ‘where,’ ‘when,’ and ‘who’ to mean ‘everywhere,’ ‘every time,’ and ‘everyone.’ Particles are usually not added to these words. e.g. 밥은 어디나 맛이 똑같아요 = Rice tastes the same everywhere e.g. 그녀는 언제나 늦게 와요 = She comes late every time e.g. 누구나 그 여자를 알아요 = Everybody knows that girl
~ㄴ가 can be added to the words ‘what,’ ‘where,’ ‘when,’ and ‘who’ to mean ‘something,’ ‘somewhere,’ ‘sometime,’ and ‘somebody.’ Subject and object particles commonly omitted. e.g. 나는 방금 뭔가(를) 봤어 = I just saw something a minute ago e.g. 피가 어딘가에서 나오고 있어요 = Blood is coming out of somewhere e.g. 저는 언젠가 중국어도 배우고 싶어요 = I want to learn Chinese as well some day e.g. 누군가(는) 너를 찾고 있어 = Somebody is looking for you
어느 ('which' question word) can also be used to mean 'some', when the particular place etc. is not important, typically in stories e.g. 어느 마을에서 애기 두 명이 태어났다 = Two babies were born in some village e.g. 어느 날 = some day/a day
Other question words that can be used to mean "some": e.g. 어디 갔어 = (They) went somewhere e.g. 우리가 이미 뭐 먹었어요 = We already ate something e.g. 나는 내일 누구 만날 거야 = Tomorrow I’m going to meet somebody
아무 - anybody Typically used with -나 particle (emphasises indifference to who) e.g. 나는 아무와나 사귀고 싶어 = I want to go out with anybody
아무도 - nobody (or could be still considered 'anybody' but negative). Needs to be cojugated in negative way or use negative word e.g. 집에 아무도 없어요 = There is nobody at home/There isn’t anybody at home
아무 can also be used before nouns (e.g. 거 (thing), 데 (place), 때 (time)) to mean anything etc. e.g. 아무 때나 좋아요 = Anytime is good
마다 - each/(every) e.g. 학생마다 달라요 = Each student is different e.g. 그 버스는 10분마다 와요 = That bus comes each/every 10 minutes

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~는 것

Present tense

• add -ㄴ/은 to adjectives (way of creating adjectives we already know about)
• add 는 to verbs

e.g. 저는 빨리 걷는 여자를 만났어요 = I met a girl who walks fast e.g. 저는 과학을 좋아하는 여자들을 좋아해요 = I like girls that like science e.g. 그 사람은 내가 가르치는 학생이다 = That person is a student that I teach (I teach that student)
Past tense

• add -ㄴ/은 to verbs

e.g. 엄마가 요리한 음식은 너무 맛있어요 = The food my mom cooked is delicious e.g. 내가 회사에 가지 않은 날에 병원에 갔어 = On the day I didn’t go to work, I went to the hospital
Future tense (note link to future tense ㄹ/을 것이다)

• add -ㄹ/을 to verbs

e.g. 제가 갈 곳은 제주도예요 = The place I will go is Jeju-do e.g. 제가 받을 점수는 중요해요 = The score I will receive is important
Bringing it back to ~는 것 - changing any verb to a noun I want (apples) -> 저는 사과를 원해요 I want (my friend to bring apples) -> 저는 (my friend to bring apples)를 원해요 친구가 사과를 가져오는 것 = my friend to bring apples i.e. 저는 친구가 사과를 가져오는 것을 원해요= I want my friend to bring apples
e.g. 저는 영화를 보는 것을 좋아해요 = I like watching movies

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Instead of the simple ~ㄴ/은 for past tense 는 것, can add in ~더~. it signifies that the speaker is recalling/remembering some fact from the past that was experienced. The speaker is explicitly expressing that this thought is coming from memory. In addition to having this “memory from experience” feeling, the use of ~던 in the construction indicates that an action occurred repeatedly in the past Can still add the ㄴ/은 after 더 to complete the past tense form, which means it's always ~던
Subtleties: 내가 입던 옷 vs 내가 사던 옷 You can wear clothes repeatedly, but not buy repeatedly - the latter would not be used
It is also possible that this “repeated” action is still reoccurring into the present (or whatever current time is being described in the sentence). Unless otherwise specified in other parts of the sentence, this repeated action hasn’t been stopped and is still re-occurring.
e.g. 내가 지금까지 입던 옷을 내일 버릴 거야 = Tomorrow, I am going to throw out the clothes that I have been wearing until now
When talking about non-self: e.g. 이 빵은 슬기가 자주 먹던 빵이야 = This bread is bread that Seulgi eats often It is specifically “bread that I specifically rememberecall/experienced Seulgi eating often.”
Can also add in ~았/었 e.g. 내가 입었던 옷 = The clothes I wore When ~았/었던 is added to a verb to describe an upcoming noun, the speaker is indicating that the action has completely finished and is not currently occurring.
Summary (verbs): ~ㄴ/은: Attached to a verb to describe a noun where the action occurred sometime in the past. There is no additional meaning given to it. All we know is that at some point in the past, the action happened.
~던: Attached to a verb to describe a noun where the action is recalled to have occurred repeatedly in the past, and is continually repeating to the present (or to the time described in the sentence).
~았/었던: Attached to a verb to describe a noun where the action is recalled to have occurred in the past, but has finished occurring and currently does not occur.
~던, 았/었던 with adjectives With 는 것, there is was past and future tense. Can make past tense with this. i.e. 예뻤던 여자 = The girl who I recall being pretty, but is not pretty anymore
~던 can be added to adjectives in certain situations, but also risks sounding very unnatural e.g. 시끄러운 교실 = the noisy classroom
시끄러웠던 교실 = the classroom that I recall being noisy, but is not noisy anymore
시끄럽던 교실 = the classroom was loud up until the present, but it just stopped being loud
Adding ~던 to 예쁘다 doesn't really make sense - typically getting ugly happens slowly over time

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Irregulars with 는 것 (see chapter 7 for earlier irregulars) In the present tense, only need to worry about is for stems ending in ㄹ - the ㄹ just gets completely removed e.g. 열다 -> 여는 것
In the past and future tenses have these 4: ㅅ - ㅅ gets removed when adding vowel e.g. 짓디 -> 지은 것, 지을 것 ㄷ - ㄷ changes to ㄹ when adding vowel e.g. 걷다 -> 걸은 것, 걸을 것 ㅂ - ㅂ changes to 우 when adding vowel e.g. 쉽다 -> 쉬운 것, 쉬울 것 ㄹ - when adding ~ㄴ/은 or ~ㄹ/을 to the stem of a verb or adjective where the stem ends in ㄹ, ~ㄴor ~ㄹ replaces the ㄹin the stem. e.g. 길다 -> 긴 것, 길 것 e.g. 열다 -> 연 것, 열 것
Favourite - "가장(/제일) 좋아하는 것" e.g. 제가 가장 좋아하는 것은 음식이에요 = My favorite thing is food e.g. 제가 가장 좋아하는 음식은 김치예요 = My favorite food is kimchi (Double 는 것): e.g. 내가 가장 좋아하는 것은 영화를 보는 것이다 = My favorite thing is watching movies
Favourite thing about - attach ~에 있어서 to noun (literally "when it comes to...") e.g. 한국에 있어서 내가 가장 좋아하는 것은 한식이야 = My favorite thing about Korea is Korean food