/sci/ - Science & Math

Are visual proofs valid?

[math]1+2+\ldots+(n-1) = \binom{n}{2} [/math]

>>7091790
>my god, proof is not that hard of that.
OP didn't ask for the most efficient proof for a theorem, or proofs of hard theorems, but nice proofs.
You reciting the Gauss proof like the smartass you are doesn't contribute to much - it also doesn't touch the fact that n+1 choose 2 is equal to it too.

>>7093055
A proof is any convincing argument, it always depends on the recipient. I've seen type theorists on math overflow say that they don't consider 97% of math papers to contain any proof, by their standards, just outlines of arguments.
My post is a proof indeed and mathematical papers do show simple result also like that. I understand, though, the notion of "proof" as used by the mathematical community and 2d-geometric arguments are not proofs under this perspective.

>>6976232
pic related

The sum of all numbers up to n, i.e. 1+2+3+...+(n-1)+n
is 'n+1 choose 2', i.e. the number of possibilities to choose 2 out of n+1.

You can also view the yellow pyramid as a packed rectangle of height n+1 and width n, divided by two, i.e. the above expressions are n(n+1)/2.

To see the latter algebraically,
2·(1+2+3+...+(n-1)+n)
can be reordered as
(1+n)+(2+(n-1))+(3+(n-2))+...+(n+1)
which is
n times n+1

and on the other hand, by definition
'n+1 choose 2' is
(n+1)! / (2!(n-2)!)
which is
(n+1)·n / 2

Beautiful proof of <div class="math"> 1+2+3+\dots+(n-1) = \binom{n}{2}</div>