Pythagorean Arithmetic

I’ve recently learned the possible name of a subject that I’ve always been interested in: Pythagorean Arithmetic. In short, this involves ‘proving’ theorems in number theory using mathematical thinking from approximately the 6th century BC, involving for the most very visual proofs, which can be explained using pebbles on a sandy beach.

I write ‘proving’ as most of these ‘proofs’ are not rigorous and require some hand-waving. Nevertheless, they show more insight into why some theorems in number theory are true. They are also useful to explain numerical properties to those with an interest in numbers, but with no formal training in mathematics.

Take for example the following theorem: For any $n \in \mathbb{N}$ , we have

$1 + 2 + 3 + \ldots + n = \sum_{i = 1}^n i = \frac{n^2 + n}{2}.$

Anyone familiar with mathematical induction knows that it is easy to find a proof, and the famous anecdote of Gauss too constitutes a formal proof. Let’s show how to prove such a statement using Pythagorean Arithmetic.

We name the sum of every natural number up to a certain $n \in \mathbb{N}$ as $S$ . Then $S$ can be visualised as $n$ columns of pebbles, starting with height 1 all the way up to $n$ .

How many pebbles are there? Let’s simplify this by adding $S$ again, but rotated 180 degrees. This gives the following picture:

The sum S two times, one time in red and one in gray

There are now six columns of five pebbles, giving us 30 pebbles in total. Thus, $2 S = 30$ and therefore $S = 15$ . And then you see quite easily that this holds for all $n \in \mathbb{N}$ : we get $n+1$ columns of $n$ pebbles, by arranging $2S$ correctly. In general, we get

$S = 1 + 2 + 3 + \ldots + n = \frac{(n+1)\cdot n}{2}.$

Let’s do another example, before we move on to a theorem where there is actually some work to do. Even the Pythagoreans may have noticed the connection between sums of odd numbers and squares of numbers. For example, $2^2 = 1 + 3$ , but also $3^2 = 1 + 3 + 5$ , and $4^2 = 1 + 3 + 5 + 7$ and so on and on. Again, easy to prove with arithmetic, algebra or induction, but we can also do it with only one visual:

Squares from 1 to 5 visualised as sums of odd numbers

Notice that each odd number $2k + 1$ wraps around perfectly around $k^2$ , giving $(k+1)^2.$ Let’s combine these above two statements into a beautiful visual result of Pythagorean Arithmetic.

This result is one of those nuggets from number theory: easy to prove, easy to remember, and somehow it’s not immediately clear why it is true. For any natural number $n$ , we have

$(1 + 2 + 3 + \ldots + n)^2 = 1^3 + 2^3 + 3^3 + \ldots + n^3.$

For example, $(1 + 2 + 3 + 4)^2 = 10^2 = 100 = 1 + 8 + 27 + 64$ . Then, combined with the two previous statements, here is the visual you’ll need to ‘prove’ this using Pythagorean arithmetic:

It may require quite a bit of thinking to show the above theorem using this visual, and perhaps you’ll solve it without using the two statements we ‘proved’ above. In any case, you’ll find several interesting numerical facts and I hope you enjoy thinking this out using Pythagorean arithmetic as much as I did!

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