Natural numbers are probably the most common numbers that you use in everyday life. Natural numbers are in general used for counting such as “there will be *seven* people at the party” and for ordering such as “look, Usain Bolt is *first* again”.

The set of natural numbers contains numbers 1, 2, 3, 4, 5, etc. We denote the set of natural numbers as *ℕ*. Sometimes we include the number zero to a set of natural numbers. You can also see the symbol *ℕ _{0}*. It explicitly mark that we’re talking about set of natural numbers, zero included. On the other hand you can also see symbol

*ℕ*and this explicitly says that we are talking about positive numbers, i. e. without zero.

^{ + }## Properties of Natural Numbers

- The set of all natural numbers is infinite. The proof is quite easy. Think of any natural number. Now you add one – bam! You have a new natural number. Thus we can find for any natural number
*x*a number*x + 1*which is different and greater than*x*. - Natural numbers are closed under addition. It means that if you add any two natural numbers you will always get another natural number. Try it! E. g. 5 + 6 = 11 and all of them are natural numbers.
- The same goes for multiplication. We can multiply any two natural numbers and we always get another natural number. E. g.
*2 · 7 = 14*. - Natural numbers are not closed under subtraction. E. g.
*4 − 9 = − 5*. The numbers*4*and*9*are both natural numbers but the result of subtraction is not. The number*− 5*is not a natural number. - The same goes for division:
*5 / 2 = 2.5*and the number*2.5*is not natural.

## Graphical Representation of Natural Numbers

We can show natural numbers using a simple graph with a single axis:

We see the *x*-axis with ten marked points. Every point represents a natural number. As you can see, it’s the set *ℕ ^{ + }* since the zero point is not highlighted. Of course we cannot show all natural numbers since there infinite amount of them.