Numerical Systems

From the outside, mathematics looks pretty organized. It is organized, rigorous, exact, and as beautiful as it is hard.

It is not simple to prove trivial things such as (1) 0*x = 0 or (2) a*b=b*a for some group. It seems impractical and purposeless to do such things, but it isn’t.

For many years things like (1) or (2) were used by anyone as physicists and engineers without any comprobation or certainty.

It was about the 19th century that mathematicians such as Cantor, Hilbert, Riemann et al. began to develop the foundations of mathematics from the ground up, i.e., from natural numbers up to complex numbers.

In ancient cultures, counting was something humans had to learn to survive the harshness of nature. Everything periodical was necessary, such as seasons, crops, and hours. The use of calendars was fundamental to the development of astronomy and beads for the beginnings of trade between one culture and another.

Natural Numbers $\mathbb{N}$

Natural numbers are those used for counting 1,2,3,4,5… but how do we define them?

We should try 1 and 1′ is the sucessor of 1 which is true etc.

We could say that 1′ = 2, 1” = 3 and so on

1. The successor of 1 is 1′.
2. Every number has one successor

These two axioms permit writing sequences like 1,2,3,4,1,2,34… Such sequences are not attractive to the formulation of a numerical system, so we add another axiom to avoid repetitions of the same sequence.

• No number has 1 as its successor

It seems obvious, but fortunately, we don’t have a numerical system yet, we need to have operations to handle those numbers.

The Arithmetic Operations

There are two fundamental operations on Natural Numbers, They are the sum (+) and multiplication (*).

One of the more important discovery in humanity was the number 0.

What do we say about (a) 1 + 0 = 1 or (b) 0*x = 0. We can prove (b) as follows:

$y*x + 0*x= (y+0)*x$

$y*x + 0*x= y*x$

$y*x+ 0*x= y*x +0$

$0*x= 0$

$\blacksquare$