You want proof?

An image is worth a thousand words. And an equation is worth a thousand images. Yet, the value of pictorial proofs is astounding. I was reminded of this fact upon reading a delightful monograph by a brilliant mind, Sanjoy Mahajan, “Street-Fighting Mathematics”.

Let us examine one example (this comes from one of the exercises in Sanjoy’s book).

Consider the following sum:

You probably know that the general formula is: n(n+1)/2

But do you know why? Try it with a couple of examples to make sure that it works:

1+2+3=3x4/2=6

1+2+3+4+5=5x6/2=15

Proof 1: Group the first and last term, the 2^nd term and the one before last, etc.

q.e.d.

 (This is assuming that n is even. If n is odd, the last term in the second line is (n+1)/2+(n+1)/2).

Proof 2: By induction.

It is trivial to verify that S1=1.

Assume that 

is correct.

Then

q.e.d.

You probably have seen Proof 1 and/or Proof 2 at school at some point. And you may have forgotten about them. Here comes a neat graphical proof. Try to forget this one!

Proof 3: Pictorial

Here is another example.

You probably know Pythagoras theorem for right triangles:  where a,b are the sides and c is the hypotenuse. But do you remember the proof? Here is a pictorial proof. No words.

References

Sanjoy Mahajan. Street-Fighting Mathematics. The Art of Educated Guessing and Opportunistic Problem Solving. MIT Press. Cambridge, MA