The set of the natural numbers, order and representation

We use natural numbers to count things. For instance, $$1, 6$$ and $$100$$ are natural numbers since we can say: $$1$$ book, $$6$$ shoes, $$100$$ people. There is no consensus concerning $$0$$ as a natural number because it came later, and in fact, we do not use it to count things (we don't tend to say "There are $$0$$ seats”). However, we will think of $$0$$ as a natural number.

Decimal numbers are not natural numbers (like $$6.1$$ or $$0.3$$) neither negative numbers (like $$-1$$ or $$-6$$) since they don’t serve to count objects or people.

Thus, the natural numbers are: $$0, 1, 2, 3, \ldots$$ and all coming thereafter. To represent the set of the naturals we are going to use the $$n$$ of natural, but it is written as follows in order to distinguish it from the letters usually write: $$\mathbb{N}$$. You might think of it as a box where all the natural numbers are.

Order and representation of the natural numbers

The natural numbers are ordered: $$0$$ is less than $$1$$, $$1$$ is less than $$2$$, etc... Instead of writing it like this and to save time and space, in mathematics this is written with the symbol $$ < $$. For example, to say: "$$3$$ is less than $$7$$" we write: $$3<7$$

Likewise, to say "is greater than" we use the symbol $$>$$. For example: "$$5$$ is greater than $$1$$" is written: $$5>1$$

The natural numbers can be represented in a line arranged from lowest to highest. To do so, we should identify a point on the line to determine the number zero. Then the natural numbers are written on the right of zero from low to the high, each one at the same distance: