[britchard]

'Pure mathematics is, in its way, the poetry of logical ideas.' ~Albert Einstein

I was surprised that when I searched for one, there was no dedicated thread to mathematics! My intention for this thread is to discuss mathematical ideas, problems and proofs from a (relative) beginner level to the highest level.

I plan to post several math problems during a week, at varying difficulties. Please also ask any questions, I am sure that someone here will be skilled enough to answer it.

Why you should take an interest in mathematics:

• It is truly 'red pill'- With maths, there is no bullshit. There is the truth, and only the truth.
• You are learning a language- The language of the universe. You will need to have a grasp of math to comprehend how the universe behaves and is formed as we know it.
• It is masculine- It is no coincidence that in several Anglosphere countries where girls perform better at school than boys due to the feminisation of schooling, the only subject in which boys do better is mathematics. The application of logic and understanding to number is key to a healthy male mind.
• You will be a better problem solver- As my maths skills have improved, so has my reasoning and concentration.
  

I am not a skilled enough mathematician to know anything about topics such as number theory, game theory or anything that is at the upper end of degree level. Hence, the questions and topics I talk about will usually be from: Calculus, Geometry, Statistics, Algebra, Number, Mechanics/Physics, Probability/Combinatorics.

[Wahawahwah]

Interested.
Maybe it might make sense to pick a topic every week and then try and post problems / applications / theories related to that particular topic for the week?

[ElFlaco]

Personally I'm most interested in the kind of math(s) problems that relate to everyday reasoning, brain teasers, that reveal something interesting or unexpected, and that are accessible with high school math (basic algebra) only.

Here's a famous problem from elementary probability with a counterintuitive answer:

Quote:

How many people do you have to have in a room in order for the probability to be over 50% that at least two of them share a birthday (not necessarily the same year)?

As I was reminded in the thread that spawned this one, it is difficult to word a math problem unambiguously. That's why I added "at least" and "not necessarily the same year" here. SAT questions (USA college entrance exam) get challenged and recalled from time to time, despite having been substantially reviewed and tested beforehand.

Let's avoid people posting their math homework here. Also, calculus and many math subjects require special symbols that aren't easy to construct and post here. I have a masters in pure (not applied) math, have taught math at several levels (high school and university calculus; and to adults going for their high school equivalency) but haven't done much with it in a long time.

For everyone who stopped at calculus or basic statistics or earlier, math gets a lot more interesting after that: 'abstract' algebra (group theory) is fascinating; so is topology. It can be a very creative endeavor, trying to demonstrate something without knowing where to begin. Higher mathematics has nothing to do with adding up very long lists of numbers (as idiots seem to assume when they find out you majored in math).

There have been a lot of interesting mathematicians, unfortunately many of them borderline insane.

[britchard]

(10-21-2016 02:01 PM)Wahawahwah Wrote:  

Interested.
Maybe it might make sense to pick a topic every week and then try and post problems / applications / theories related to that particular topic for the week?

Yes, that seems sensible. How about a new topic is put forward every Friday, I will post the first one this evening (GMT). Topic decision can be by any member with 1 or more rep points, or by an un-repped member who has contributed to the thread.

(10-21-2016 02:06 PM)ElFlaco Wrote:  

Here's a famous problem from elementary probability with a counterintuitive answer:

Quote:

How many people do you have to have in a room in order for the probability to be over 50% that at least two of them share a birthday (not necessarily the same year)?

Agree completely with all your points, but didn't want to quote a huge block of text. I think once the average level of the people participating is established, then it will be far easier for people to post new problems.

In response to the problem you posted:

Let's say that we the people A,B,C and D. It might appear that the probability of any of them sharing a birthday is 4 in 365, yet that is incorrect as that would be equal to the probability of A with B, A with C, A with D and say B with D.

What you actually need to compare is how many combinations can be made with a set of values (people A,B,C,D,E...).

[MMX2010]

My brother and I concocted a story that rabbits are unfathomably more intelligent than humans, telepathically connected (that's why they twitch their noses), and highly frustrated that they have no thumbs.

While reading up on non-Euclidean geometry (which every rabbit is born knowing), I read that scientists do now know whether a triangle comprised of three points on the edge of the universe would contain less than 180°, exactly 180°, or more than 180°. Reading that, I imagined every rabbit laughing, because they know. And they won't tell us.

-----

(Non-Euclidean geometry posits that lines can be curved, so triangles comprised of at least one curved line are NOT guaranteed to have 180°. Example: Triangles comprised of at least one of earth's longitudinal lines usually have more than 180°.)

[Edmund Ironside]

(10-21-2016 01:50 PM)britchard Wrote:  

Why you should take an interest in mathematics:

• It is truly 'red pill'- With maths, there is no bullshit. There is the truth, and only the truth.
  

Adding to that, if you are young and haven't decided on a career and/or major yet, think long and hard about the significance of how much subjectivity you want in the field you choose. Whatever you choose, there will be more politics and general BS than you expect. Our society could benefit from a lot more logical people in the social sciences, and ceding those areas to the SJWs has very negative consequences, but on an individual level going into anything with room for BS can be very frustrating if you are a person with good hard logical ability.

[britchard]

Problem 1- Trigonometry

Going to start off with a nice, simple-ish proof. I will post my results tomorrow morning, or failing that at some point this weekend.

Prove that sin^2 (x) + cos^2 (x) = 1

[ShotgunUppercuts]

I'm studying math own my on this weekend. I even hired a tutor but I'm gonna hold out on that this weekend to see if I can learn what I need to again.

[Architekt]

(10-21-2016 05:28 PM)britchard Wrote:  

Problem 1- Trigonometry

Going to start off with a nice, simple-ish proof. I will post my results tomorrow morning, or failing that at some point this weekend.

Prove that sin^2 (x) + cos^2 (x) = 1

Definition of unit circle..

[RexImperator]

If you like this kind of stuff I recommend looking up Steve Strogatz.

[britchard]

(10-22-2016 08:58 AM)ShotgunUppercuts Wrote:  

I'm studying math own my on this weekend. I even hired a tutor but I'm gonna hold out on that this weekend to see if I can learn what I need to again.

What stuff are you studying?

(10-23-2016 09:15 AM)Architekt Wrote:  

(10-21-2016 05:28 PM)britchard Wrote:  

Problem 1- Trigonometry

Going to start off with a nice, simple-ish proof. I will post my results tomorrow morning, or failing that at some point this weekend.

Prove that sin^2 (x) + cos^2 (x) = 1

Definition of unit circle..

Quite. If you wanted to add more detail, I'd say that if you drew a triangle with a hypotenuse of length '1', then since sin x= O/H, since H=1 then sin x= O (opposite side), and then the adjacent side is equal to cos x. Therefore using pythagoras' theorem 1= cos^2 (x) + sin^2 (x).

Looking back that was a bit too easy.

(10-23-2016 09:25 AM)RexImperator Wrote:  

If you like this kind of stuff I recommend looking up Steve Strogatz.

Will do!

[ShotgunUppercuts]

Well the basics ....a little bit of everything really.

Im studying so I can score really well I. The asvab (I'd like to go to the air force) . Math is my weakest subject and I've been outta school for almost 6 years now. I've learned it before so I'm sure I can do it again but this time around it'll be a bit harder because of my studying time is slim during the week .

[ElFlaco]

(10-23-2016 09:15 AM)Architekt Wrote:  

(10-21-2016 05:28 PM)britchard Wrote:  

Problem 1- Trigonometry

Going to start off with a nice, simple-ish proof. I will post my results tomorrow morning, or failing that at some point this weekend.

Prove that sin^2 (x) + cos^2 (x) = 1

Definition of unit circle..

Shall we call this a circular definition?

[Soma]

.

[Isaac Jordan]

I'm currently reading a very interesting book called Algorithms to Live By, which sort of occupies the nexus of computer science, math and human psychology:

Quote:

All our lives are constrained by limited space and time, limits that give rise to a particular set of problems. What should we do, or leave undone, in a day or a lifetime? How much messiness should we accept? What balance of new activities and familiar favorites is the most fulfilling?

These may seem like uniquely human quandaries, but they are not: computers, too, face the same constraints, so computer scientists have been grappling with their version of such problems for decades. And the solutions they've found have much to teach us.

In a dazzlingly interdisciplinary work, acclaimed author Brian Christian and cognitive scientist Tom Griffiths show how the simple, precise algorithms used by computers can also untangle very human questions. They explain how to have better hunches and when to leave things to chance, how to deal with overwhelming choices and how best to connect with others.

From finding a spouse to finding a parking spot, from organizing one's inbox to understanding the workings of human memory, Algorithms to Live By transforms the wisdom of computer science into strategies for human living.

I'm only halfway through, but already I'd highly recommend it.

[ivansirko]

^ Mathematical Logic....when Philosophy and Math overlap. Math can be very red pill but Mathematical Logic is shitlord.

Most people in this forum use this form of logic......Game.

Data Science can also work along those lines however that can border on voodoo.

[player]

I love mathematics! I had a few years without doing much but starting to get back into it to prepare for the GMAT.

(10-21-2016 02:06 PM)ElFlaco Wrote:  

How many people do you have to have in a room in order for the probability to be over 50% that at least two of them share a birthday (not necessarily the same year)?

Consider a group of n people. Let us ask the related question: what is the probability that none of them share a birthday?

The first person's birthday can be any of 365 days (probability 365/365)
The second person's birthday must not be the first person's birthday, but can be any of the other 364 days (probability 364/365)
The third person's birthday must not be either of the first two people's birthdays, but can be any of the other 363 days (probability 364/365)
...
Generally, the nth person's birthday can be any of 366-n days (probability [366-n] / 365).

Assuming that none of these people are twins, their birthdays are independent. Therefore, the probability that none of them share a birthday is

365/365 * 364/365 * 363/365 * ... * (366-n) / 365

Using Excel for convenience to calculate this for various n, we find that the probability that 22 people do not share a birthday is 0.524 while the probability that 23 people do not share a birthday is 0.493 (to 3 significant figures).

Therefore, if you have a group of 23 people, there is a 1 - 0.493 = 0.507 probability that two or more share a birthday.


Here's a fairly challenging question (but can be elegantly solved with high school-level mathematics) that starts off easy.

The arithmetic mean of n numbers, x1, ..., xn, is defined as (x1+x2+...+xn)/n (add them all up and divide by n)
The geometric mean of n numbers, x1, ..., xn, is defined as (x1*x2*...*xn)^(1/n) (multiply them all together and take the nth root)

When you have just two numbers, which is larger, the arithmetic mean or the geometric mean? Prove it. When you have three numbers, which is larger, the arithmetic mean or the geometric mean? Prove it. Based on the cases with n = 2 and n = 3, what do you conjecture to be larger in the general case with n numbers, the arithmetic mean or the geometric mean? Prove your conjecture. (This is the AM-GM inequality.)

[Wutang]

If you're into mathematical logic and looking for some entertainment I would suggest reading the graphic novel 'LogiComix'

https://www.logicomix.com/en/index.html

The story features Bertrand Russell as the main character and follows his journey to base logic on mathematics which would make logic as sure in it's knowledge as math is in it's knowledge. This would therefore give humanity a tool a way discover absolute truth. Among the way other figures who were important to the development of mathematical logic in the 20th century such as Gottlob Frege (founder of set theory), Kurt Godel (discover of the incompleteness theorem), and David Hilbert are encountered. The co-author of the comic is a theoretical computer science professor and he's also featured as a character in the book in parts to explain the more technical details.

I can honestly say this book was a huge motivator in my decision to head back to school to earn a degree in computer science. I had to take a discrete mathematics course as part of the program and the textbook my class used would in the footnotes often feature brief biographies of some of the people discussed in the comic which gave me a sense of the importance of the work they did.