3 Visual Mathematical Proofs

Math is beautiful for a lot of reasons. Some would appreciate the little details in a rigorous proof and others might be fascinated with complex patterns that emerge from seemingly simple equations.

In this article, we are going to concentrate on a different aspect that makes math beautiful, at least in my opinion, and it’s the ability to prove mathematical statements without saying a word — just with a drawing.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. — G. H. Hardy

1. Every odd integer is the difference of 2 squares

Let’s pick an odd number together, for example, 13.

Next, we will “bend” it in half

Lastly, we fill the blank space

Now let’s count the squares.
The whole square has an area of 7², and just the blue square has an area of 6².
Meaning that, 7²-6² = 49–36 = 13.

Can you think of a way to prove this algebraically? It can be done in a single line — I’ll leave a solution at the bottom.

2. 1/2 + 1/4 + 1/8 + 1/16 + … = 1

If you are not familiar with it, it may sound weird at first that an infinite sum of numbers is equal to 1 — but a single picture can make you understand.

Just think of a square with size 1×1, now slice it into half — and then slice one of the halves into another half and so on…

3. Sum of cubes equals the sum squared

1³+2³+…+n³ = (1+2+…+n)², or in other words

This equation is easily proved with the following drawing

Regarding the first proof — if you were curious about how to prove it algebraically I got you covered

Every odd number can be written as 2n+1 and

Now we also know what would be the squared numbers that would satisfy the statement for every odd number we choose.
For example, if we take 53 — reducing 1 we get 52 — dividing by 2 we get 26 so we know that n=26.
Meaning that 27²-26²=53 as expected.

Although the article is about visual proofs, I would like it to be complete, and when I say complete I mean to say “here are the two missing proofs”.

Let’s go through them in order, starting with

By definition, an infinite sum equals the limit of a finite sum (of the same sequence of course) while n approaches infinity, simply meaning that

To make life a bit easier, let’s annotate the finite sum with a symbol

And now, by simply multiplying the above equation by 2, we get

From the above equation, we conclude

So now we just have to calculate the limit of Sn when n approaches infinity.
The right term is clearly approaching zero as n approaches infinity — so we remain with 1.

And we are done — I assume that the visual was a bit easier to comprehend.

Moving on to the last proof, this time we are going to prove that

This statement can be proved with simple induction.

Base: n=1, 1³ = 1² — and that’s it.

Assumption: We assume that the statement is correct for n.

Step:

In this proof, we will use a known result, that states that

With that in mind, the induction step is

To conclude, I will cite Karl Weierstrass

“A mathematician who is not also something of a poet will never be a complete mathematician”