#math proofs

[marigold-22]

Contrapositive are the same as contradiction, they shouldn’t have different names

“Suppose A, now for the sake of contradiction suppose -b then… -A, a contradiction, thus A implies b.”

Is the same thing as

“Suppose -b, then… -A. Therefore by contrapositive A implies b.”

Their just linguistically different but that doesn’t make them different, anything provable with the first technique is provable with the second technique and vice versa. Contrapositive adds nothing.

[knottybliss]

Good mornthing my darling thingmakers it is that day of days:

WIP WEDNESDAY

[phoebenpiperx]

Yes, 1³+2³+3³+4³+5³+6³+7³+8³+9³ = (1+2+3+4+5+6+7+8+9)^2 = 2025.

But I'm a major nerd so I just figured out that, for any set of consecutive integers starting from 1, the sum of their cubes is the square of their sum.

PROOF BELOW

1^2 = 1*1 = 1³

(1+2)^2 =(1+2)(1+2) = 1 + 1*2 + 2*1 + 2*2 = 1 + 2(1+1+2) = 1 + 2(2+2) = 1³+2³

(1+2+3)^2 = (1+2+3)(1+2+3) = 1 + 1*2 + 1*3 + 2*1 + 2*2 + 2*3 + 3*1 + 3*2 + 3*3 = 1 + 2(1+1+2) + 3(1+2+1+2+3) = 1 + 2(2+2) + 3(3+3+3) = 1³+2³+3³

(1+2+3+4)^2 = (1+2+3+4)(1+2+3+4) = 1 + 1*2 + 1*3 + 1*4 + 2*1 + 2*2 + 2*3 + 2*4 + 3*1 + 3*2 + 3*3 + 3*4 + 4*1 + 4*2 + 4*3 + 4*4 = 1 + 2(1+1+2) + 3(1+2+1+2+3) + 4(1+2+3+1+2+3+4) = 1 + 2(2+2) + 3(3+3+3) + 4(4+4+4+4) = 1³+2³+3³+4³

which generalizes to

(1+2+3+4+...+n)^2 = (1+2+3+4+...+n)*(1+2+3+4+...+n) = 1 + 1*2 + 1*3 + 1*4 + ... + 1*5 + 2*1 + 2* 2 + 2*3 + 2*4 + ... + 2*n + 3*1 + 3*2 + 3*3 + 3*4 + ... +3*n + 4*1 + 4*2 + 4*3 + 4*4 + ... + 4*n + n*1 + n*2 + n*3 + n*4 + ... + n*n) =1 + 2(1+1+2) + 3(1+2+1+2+3) + 4(1+2+3+1+2+3+4) + ... n(1+2+3+4+...+[n-1] + 1+2+3+4+...+[n-1] + n) =1 + 2(2+2) + 3(3+3+3) + 4(4+4+4+4) + ... + n(n+n+n+...n) = 1³+2³+3³+4³ + ... +n³

Note: 1+2+3+4+...+[n-1] + 1+2+3+4+...+[n-1] + n = 1+[n-1] + 2+[n-2] + 3+[n-3] + 4+[n-4] + ... + [n-4]+4 + [n-3]+3 + [n-2}+2 + [n-1]+1 + n = n[n-1] + n = n^2

ex: 1+2+3+4+5+6+7+1+2+3+4+5+6+7+8

not to be a number nerd on main but 2025 (45^2) will be the only square year most of us ever experience. the last one was 1936 and the next one will be 2116

[catscomputerscreativewriting]

It's so funny to me that people think of Math/Mathematicians as being hyper-logical and rational. Like, have you seen some of the wild things hiding in the Math?

Did you know there are non-computable numbers?? (https://en.wikipedia.org/wiki/Chaitin%27s_constant)

Did you know that there are things that are true, but we can't prove them??? (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)

Did you know that we can prove that something exists, and yet never actually figure out what that thing is?? (https://mathworld.wolfram.com/NonconstructiveProof.html)

Math is crazy. Math is wild. Math hardly makes sense, and when you think you understand the weirdest parts of it, everyone who hears you explain it to thinks you're a gibbering lunatic.

"In mathematics you don’t understand things. You just get used to them." - von Neumann

(please share more unhinged math with me, i want to see more scary math)

[tea-for-patchouli]

For your enjoyment, a lovely mathematical paper which can fit in a single image.

[mentalknot]

As a math major trying not to get bogged down in the hellscape of severe social anxiety once again, proofs are shockingly comforting.

How wonderful to behold a carefully chosen sequence of arguments and accompanying symbols that remains absolutely logical…

…while navigating a mind that does nothing but lie at every juncture of my life.

[mcnsieurfeuilly]

i just know kiryu is taking a selfie while on sakura’s back and waiting for the perfect moment to drop it in the group chat

[picsfortheday]

[secondwheel]

Proof in maths

[autism-industrial-complex]

Writing a math proof is a lot like the ending of the movie CLUE, where you have to run all over the house to describe your evidence before arriving at the identity of the killer

Mathematics people interested in either learning about the Peano axioms of natural numbers (how do you prove a+b = b+a) or interested in learning a proving assistant (Lean) should DEFINITELY check out the natural number game. This is a game which teaches you both of these things step by step, and has you construct the natural numbers and their most important properties from the very basic axioms. Even if you just like logical puzzles and are interested in a bit of mathematics you should check it out!

[pv1isalsoimportant]

anime girl proof complexity! ANIME GIRL PROOF COMPLEXITY!

Logic seminar at IM CAS; Metamathematics of Resolution Lower Bounds: A TFNP Perspective, Hanlin Ren, Oxford Univ.

[shadowyyyidk]

Isn't the Penrose tiling just an absolutely beautiful concept? The same bunch of shapes are being used, and similar patterns are being formed at multiple places, and yet they are never repeating, not even once, because of an infinite number of conformations.

[the-way-astray]

yeah so fitz is actually still sixteen in the latest book

[microqwavve]

math question for yall day 2

Prove or disprove the existence of an infinite sum M that diverges to infinity but converges to a finite value for every infinite (ordered) subsequence of M

[mentalknot]

Kids these days are quite spoiled… they get to add, subtract, multiply, and divide any numbers they want without specifying their field or proving that the operations they’re using are well defined… do they just expect that the teacher will do all the work ahead of time??

Utterly preposterous…

[algebraic-dualist]

One of my greatest proof writing tips is to believe in yourself enough to see that it does follow almost immediately from definition.

It's a little embarrassing the number of times I've been stuck on what is actually immediate just because I find the definition so visually intimidating.