What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?
Here are some I would like to share:
- Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
- Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)
The Busy Beaver function
Now this is the mind blowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.
- The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
- In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
- Σ(1) = 1
- Σ(4) = 13
- Σ(6) > 101010101010101010101010101010 (10s are stacked on each other)
- Σ(17) > Graham’s Number
- Σ(27) If you can compute this function the Goldbach conjecture is false.
- Σ(744) If you can compute this function the Riemann hypothesis is false.
Sources:
- YouTube - The Busy Beaver function by Mutual Information
- YouTube - Gödel’s incompleteness Theorem by Veritasium
- YouTube - Halting Problem by Computerphile
- YouTube - Graham’s Number by Numberphile
- YouTube - TREE(3) by Numberphile
- Wikipedia - Gödel’s incompleteness theorems
- Wikipedia - Halting Problem
- Wikipedia - Busy Beaver
- Wikipedia - Riemann hypothesis
- Wikipedia - Goldbach’s conjecture
- Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution
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Non-Euclidean geometry.
A triangle with three right angles (spherical).
A triangle whose sides are all infinite, whose angles are zero, and whose area is finite (hyperbolic).
I discovered this world 16 years ago - I’m still exploring the rabbit hole.
Spherical geometry isn’t even that weird because we experience it on earth at large scales.
I’m not sure Earth would be a correct analogy for spherical geometry. Correct me if I’m wrong, but spherical geometry is when the actual space curvature is a sphere, which is different from just living on a sphere.
CodeParade made a spherical/hyperbolic geometry game, and here’s one of his devlogs explaining spherical curvature: https://www.youtube.com/watch?v=yY9GAyJtuJ0
A sphere is a perfect model of spherical geometry. It’s just a 2-dimensional one, the spherical equivalent of a plane we might stand on as opposed to the space we live in. A sphere is locally flat (locally Euclidean/plane-like) but intrinsically curved, and indeed can have triangles with 3 right angles (with endpoints on a pole and the equator.)
Ah, gotcha! Spherical and hyberbolic geometries always mess with my mind a bit. Thanks for the explanation!
Oh yeah I’m subscribed to codeparade! I know it’s not a perfect analogue but since it’s such a large scale, our perspective makes it look flat. So at long distances it feels like you’re moving in a straight line when you’re really not.
Szemeredis regularity lemma is really cool. Basically if you desire a certain structure in your graph, you just have to make it really really (really) big and then you’re sure to find it. Or in other words you can find a really regular graph up to any positive error percentage as long as you make it really really (really really) big.
As someone who took maths in university for two years, this has successfully given me PTSD, well done Lemmy.
The birthday paradox
The one I bumped into recently: the Coastline Paradox
“The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension.”
Euler’s identity, which elegantly unites some of the most fundamental constants in a single equation:
e^(iπ)+1=0
Euler’s identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The number π (π = 3.1415…), the fundamental circle constant.
- The number e (e = 2.718…), also known as Euler’s number, which occurs widely in mathematical analysis.
- The number i, the imaginary unit of the complex numbers.
Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”. And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty”.
Mathematics writer Constance Reid has opined that Euler’s identity is “the most famous formula in all mathematics”. And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.
This is the one that made me say out loud, “math is fucking weird”
I started trying to read the explanations, and it just got more and more complicated. I minored in math. But the stuff I learned seems trivial by comparison. I have a friend who is about a year away from getting his PhD in math. I don’t even understand what he’s saying when he talks about math.
Recall the existence and uniqueness theorem(s) for initial value problems. With this, we conclude that e^(kx) is the unique function f such that f’(x) = k f(x) and f(0) = 1. Similarly, any solution to f’’ = -k^(2)f has the form f(x) = acos(kx) + bsin(kx). Now consider e^(ix). Differentiating it is the same as multiplying by i, so differentiating twice is the same as multiplying by i^(2) = -1. In other words, e^(ix) is a solution to f’’ = -f. Therefore, e^(ix) = a cos(x) + b sin(x) for some a, b. Plugging in x = 0 tells us a = 1. Differentiating both sides and plugging in x = 0 again tells us b = i. So e^(ix) = cos(x) + i sin(x).
We take for granted that the basic rules of calculus work for complex numbers: the chain rule, and the derivative of the exponential function, and the existence/uniqueness theorem, and so on. But these are all proved in much the same way as for real numbers, there’s nothing special behind the scenes.
Time for a deep dive. Wish me luck, lads.
Let us know what you find in the depths.
e^(pi i) = -1
like, what?
If you think of complex numbers in their polar form, everything is much simpler. If you know basic calculus, it can be intuitive.
Instead of z = + iy, write z = (r, t) where r is the distance from the origin and t is the angle from the positive x-axis. Now addition is trickier to write, but multiplication is simple: (a,b) * (c,d) = (ab, b + d). That is, the lengths multiply and the angles add. Multiplication by a number (1, t) simply adds t to the angle. That is, multiplying a point by (1, t) is the same as rotating it counterclockwise about the origin by an angle t.
The function f(t) = (1, t) then parameterizes a circular motion with a constant radial velocity t. The tangential velocity of a circular motion is perpendicular to the current position, and so the derivative of our function is a constant 90 degree multiple of itself. In radians, that means f’(t) = (1, pi/2)f(t). And now we have one of the simplest differential equations whose solution can only be f(t) = k * e^(t* (1, pi/2)) = ke^(it) for some k. Given f(0) = 1, we have k = 1.
All that said, we now know that f(t) = e^(it) is a circular motion passing through f(0) = 1 with a rate of 1 radian per unit time, and e^(i pi) is halfway through a full rotation, which is -1.
If you don’t know calculus, then consider the relationship between exponentiation and multiplication. We learn that when you take an interest rate of a fixed annual percent r and compound it n times a year, as you compound more and more frequently (i.e. as n gets larger and larger), the formula turns from multiplication (P(1+r/n)^(nt)) to exponentiation (Pe^(rt)). Thus, exponentiation is like a continuous series of tiny multiplications. Since, geometrically speaking, multiplying by a complex number (z, z^(2), z^(3), …) causes us to rotate by a fixed amount each time, then complex exponentiation by a continuous real variable (z^t for t in [0,1]) causes us to rotate continuously over time. Now the precise nature of the numbers e and pi here might not be apparent, but that is the intuition behind why I say e^(it) draws a circular motion, and hopefully it’s believable that e^(i pi) = -1.
All explanations will tend to have an algebraic component (the exponential and the number e arise from an algebraic relationship in a fundamental geometric equation) and a geometric component (the number pi and its relationship to circles). The previous explanations are somewhat more geometric in nature. Here is a more algebraic one.
The real-valued function e^(x) arises naturally in many contexts. It’s natural to wonder if it can be extended to the complex plane, and how. To tackle this, we can fall back on a tool we often use to calculate values of smooth functions, which is the Taylor series. Knowing that the derivative of e^(x) is itself immediately tells us that e^(x) = 1 + x + x^(2)/2! + x^(3)/3! + …, and now can simply plug in a complex value for x and see what happens (although we don’t yet know if the result is even well-defined.)
Let x = iy be a purely imaginary number, where y is a real number. Then substitution gives e^x = e^(iy) = 1 + iy + i(2)y(2)/2! + i(3)y(3)/3! + …, and of course since i^(2) = -1, this can be simplified:
e^(iy) = 1 + iy - y^(2)/2! - iy^(3)/3! + y^(4)/4! + iy^(5)/5! - y^(6)/6! + …
So we’re alternating between real/imaginary and positive/negative. Let’s factor it into a real and imaginary component: e^(iy) = a + bi, where
a = 1 - y^(2)/2! + y^(4)/4! - y^(6)/6! + …
b = y - y^(3)/3! + y^(5)/5! - y^(7)/7! + …
And here’s the kicker: from our prolific experience with calculus of the real numbers, we instantly recognize these as the Taylor series a = cos(y) and b = sin(y), and thus conclude that if anything, e^(iy) = a + bi = cos(y) + i sin(y). Finally, we have e^(i pi) = cos(pi) + i sin(pi) = -1.
Damn that’s actually pretty simple way to use differential equations to show this.
Edit. The series one is also pretty simple.
3Blue 1Brown actually explains that one in a way that makes it seem less coincidental and black magic. Totally worth a watch
That you can have 5 apples, divide them zero times, and somehow end up with math shitting itself inside-out at you even though the apples are still just sitting there.
You try having 5 apples and divide them into 0 equal groups and you’ll shit yourself too.
Except that by dividing the total number zero times means you’re not dividing them at all, and therefore by doing nothing you are still left with 5 apples.
Not dividing at all is dividing by 1.
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11 X 11 = 121
111 X 111 = 12321
1111 X 1111 = 1234321
11111 X 11111 = 123454321
111111 X 1111111 = 12345654321
Amazing in deed!
Just a small typo in the very last factor
1111111.You could include 1 x 1 = 1
But thats so cool. Maths is crazy.
holt shit
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Which one?
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Integrals. I can have an area function, integrate it, and then have a volume.
And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.
Geometric interpretation of integration is really fun, it’s the analytic interpretation that most people (and I) find harder to understand.
If you work in numerically solving integrals using computers, you realise that it’s all just adding tiny areas.
I finally understand what divergent integrals are intuitively when I encountered one while trying to do a calculation on a computer.
Incompleteness is great… internal consistency is incompatible with universality… goes hand in hand with Relativity… they both are trying to lift us toward higher dimensional understanding…
Goldbach’s Conjecture: Every even natural number > 2 is a sum of 2 prime numbers. Eg: 8=5+3, 20=13+7.
https://en.m.wikipedia.org/wiki/Goldbach’s_conjecture
Such a simple construct right? Notice the word “conjecture”. The above has been verified till 4x10^18 numbers BUT no one has been able to prove it mathematically till date! It’s one of the best known unsolved problems in mathematics.
How can you prove something in math when numbers are infinite? That number you gave if it works up to there we can call it proven no? I’m not sure I understand
There are many structures of proof. A simple one might be to prove a statement is true for all cases, by simply examining each case and demonstrating it, but as you point out this won’t be useful for proving statements about infinite cases.
Instead you could assume, for the sake of argument, that the statement is false, and show how this leads to a logical inconsistency, which is called proof by contradiction. For example, Georg Cantor used a proof by contradiction to demonstrate that the set of Natural Numbers (1,2,3,4…) are smaller than the set of Real Numbers (which includes the Naturals and all decimal numbers like pi and 69.6969696969…), and so there exist different “sizes” of infinity!
For a method explicitly concerned with proofs about infinite numbers of things, you can try Proof by Mathematical Induction. It’s a bit tricky to describe…
- First demonstrate that a statement is true in some 1st base case.
- Then demonstrate that if it holds true for the abstract Nth case, then it necessarily holds true for the (N+1)th case (by doing some clever rearranging of algebra terms or something)
- Therefore since it holds true for the 1th case, it must hold true for the (1+1)th case = the 2th case. And since it holds true for the 2th case it must hold true for the (2+1)=3th case. And so on ad infinitum.
Wikipedia says:
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes.
Bear in mind, in formal terms a “proof” is simply a list of true statements, that begin with axioms (which are true by default) and rules of inference that show how each line is derived from the line above.
Very cool and fascinating world of mathematics!
Just to add to this. Another way could be to find a specific construction. If you could for example find an algorithm that given any even integer returns two primes that add up to it and you showed this algorithm always works. Then that would be a proof of the Goldbach conjecture.
As you said, we have infinite numbers so the fact that something works till 4x10^18 doesn’t prove that it will work for all numbers. It will take only one counterexample to disprove this conjecture, even if it is found at 10^100. Because then we wouldn’t be able to say that “all” even numbers > 2 are a sum of 2 prime numbers.
So mathematicians strive for general proofs. You start with something like: Let n be any even number > 2. Now using the known axioms of mathematics, you need to prove that for every n, there always exists two prime numbers p,q such that n=p+q.
Would recommend watching the following short and simple video on the Pythagoras theorem, it’d make it perfectly clear how proofs work in mathematics. You know the theorem right? For any right angled triangle, the square of the hypotenuse is equal to the sum of squares of both the sides. Now we can verify this for billions of different right angled triangles but it wouldn’t make it a theorem. It is a theorem because we have proved it mathematically for the general case using other known axioms of mathematics.
Wtf !
This is my silly contribution: 70% of 30 is equal to 30% of 70. This applies to other numbers and can be really helpful when doing percentages in your head. 15% of 77 is equal to 77% of 15.
I’ve seen this one used in the news when they want to make one side of a statistic stand out.
A/100×B=A×B/100
Multiply 9 times any number and it always “reduces” back down to 9 (add up the individual numbers in the result)
For example: 9 x 872 = 7848, so you take 7848 and split it into 7 + 8 + 4 + 8 = 27, then do it again 2 + 7 = 9 and we’re back to 9
It can be a huge number and it still works:
9 x 987345734 = 8886111606
8+8+8+6+1+1+1+6+0+6 = 45
4+5 = 9
Also here’s a cool video about some more mind blowing math facts
I suspect this holds true to any base x numbering where you take the highest valued digit and multiply it by any number. Try it with base 2 (1), 4 (3), 16 (F) or whatever.
