What are natural numbers? How do define them? How can they be visualized "graphically"?

1- Peano defines numbers like this:

0 is a natural number

...

For every natural number *n*, *S*(*n*) is a natural number.

...

The set of natural numbers is then defined as: *N* = {0, *S*(0), *S*(*S*(0)), …}

*NOTE: S(n) is basically n+1*

2- S(n) has interesting properties:

and

3- It is possible to define natural numbers using only sets:

* s*(*a*) = *a* ∪ { *a* }

Visually it could be: 0 = [] "zero is a box"

and: S( [a] ) = [ a [a] ]

So: [] -> S([]) = [[]] -> S(S([])) = [ [] [[]] ] -> ...

4- Another way to visualize natural numbers is to use a string:

Z for zero

S for S(.), and S(Z) = SZ is one

So N = { Z, SZ , SSZ , SSSZ , ... }

This basically is the unary system, only that a special symbol is used ("Z" here) to mark the beginning of the string representing a number. (this offers advantages when using a Turing Machine or lambda calculus to define arithmetic operators).

5- Since each natural number can be expressed in a unique way by its prime factors (up to sorting), it is possible to define a diagram for each natural number, as they do here; these diagrams are also described here:

They are nice and prime numbers can be recognized because they are circles! But the diagrams tend to be large an less interesting for natural numbers with large factors.

6- Gödel numbering: a way to encode a list of numbers into a single (LARGE) one, so that it is always possible to re-compute back the original list of numbers.

The idea is:

where p_n is some prime.

Example: the list (1,2,3) can be encoded as the number 2^{1} * 3^{2 }* 5^{3 }= 2*9*125 = 750

Since 2, 3 and 5 are the first 3 primes. To de-code the list, simply use the fundamental theorem of arithmetic, and find the prime factors of 750:

750 = 2^{1} * 3^{2 }* 5^{3}

then be sure that the primes are sorted from smaller to larger, and read the exponents: (1,2,3) *et voila'* :)

## Sunday, January 27, 2013

### Numbers? Natural :P

By at 22:20

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