5 Unusual Proofs | Infinite Series

Find out how logic, induction, visuals, bijections, and a little algebra can prove some surprising math theorems.

Tweet at us! @pbsinfinite
Email us! pbsinfiniteseries [at] gmail [dot] com

Previous Episode - Infinite Chess

Hierarchy of Infinities

Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux

Sources and further references:

Proofs Without Words by Roger B. Nelsen

Thanks: Kelly Delp, Joe Gallagher

Comments answered by Kelsey:

MeloDeathKT

TheDruidKing

rayfangrui

[bits]

My favorite tool for proofs is
"It just came to me" - Ramanujan

[davejermy]

When I was in high school, I was challenged by my maths teacher to prove that, for any four consecutive fibonacci numbers a, b, c & d, the value of ad - bc is either 1 or -1.

He gave me no hints on how to do it, so I played around with some examples and noticed that the result switched between the two values. I.e. if ad - bc = 1 then the next four numbers, starting at b, gives b(d+c) - cd = -1

That lead me onto proving it by induction, using two base cases rather than the usual one. Writing QED at the end of that proof was very satisfying.

[Swiftclaw]

For the second proof, you have to still show that the density of points is uniformly distributed. In theory a bijection can send half of all cuts to the middle triangle, for example, and we should show this is not the case.

[BurakBagdatli]

I love bijections because at first it feels like you're making a really interesting connection between seemingly unrelated concepts and that by itself is very satisfactory.

But then you realize that you actually showed that these connected concepts are not just connected, they are actually the same thing.

[skylardeslypere]

I totally love the N choose 2 one with the triangles
It is such a clever idea to prove it

[pkeros]

In high school, my precalc teacher taught us proof by induction. Didn't think much of it at the time. Fast forward to differential equations in college, and one of our homework problems required us to use proof by induction to solve it. Everyone else in the class freaked out because they didn't know how to do it, and the TAs pretty much spent all their office hours teaching proof by induction. Glad I had that when I needed it.

[WombatSamurai]

since these tools are so standard, shouldnt the video be called "5 usual proofs"

[heyitsme]

Im so happy that you're doing this! This is awesome keep it up.

[kasperbrohusallerslev]

Some of my favorite, intuitive proofs are:
- There's an infinite number of primes
- Deriving the formula for the zero points on a parabola by drawing a bunch of rectangles

[alexandercolefield]

I love Euclid visual proof of the Pythagorean theorem, very elegant

[zairaner]

My favourite tool for proofs is probably uniqueness statements. Proving something about an object without knowing how exactly it looks, just knowing that it is unique in some way is satisfying

[devluz]

This is my favorite video so far. Very good combination of useful information and easy to understand

[oosh]

This channel does an amazing job at presenting math to people of any background

[morgengabe]

This video is very timely for me! Been struggling to understand where to start and stop with proofs, just started them at uni.
Thanks!

[Quasarbooster]

My favorite proof is the one I discovered in middle school: for a and b (preferably a > b), a^2 = b^2 + (a+b)(a-b). I was trying to find a way to "slingshot" from squares I already knew to squares just a little bigger that I didn't know. For example, if I want to find 302^2 but don't want to take time multiplying it by itself, I can just use 300^2+(602)(2) to get 90000+1204 or 91204.

[Kalobi]

Nice video, but the title was a bit misleading. Those are quite usual proofs.

[andrewandrew]

I contrived a proof, but this comment section is too small to contain it. :)

[marcusscience]

3:10 can be proven with an alternate method:
Draw in n dots. Connect every node with every other node with lines by doing this: First look at the first dot and connect it with all the other dots, making (n-1) lines. Then connect the second dot with all others except the first as it is already connected, which makes (n-2) lines. Repeat, each time connecting the next dot with all others excluding the previous already taken care of ones, each time adding one less than the previous number of lines. Do this until you get down to the last 2 dots that you connect together. All the dots will be connected with every other, and the number of lines you have is (n-1) add every smaller positive integer, which is the (n-1)th triangle number. However, as each line connects 2 dots the number of lines is n choose 2. Both T(n-1) and n choose 2 are the number of lines, therefore they are equal.
QED.

[douggwyn]

Besides other issues for the three-stick problem, there is also a problem in precisely specifying *how* the two break points are chosen. Martin Gardner had a column about that.

[shubhamshinde]

One of the best maths channels on youtube!!! No proof required, it's an axiom...