Natural numbers

The natural numbers are used for counting objects like: $3$ eggs, $5$ students, $17$ trees.

The set of natural numbers is denoted by $\mathbb{N}$ (which is printed in so-called ``blackboard bold'')

Two natural numbers can be added or multiplied: $3 + 5 = 8$, $\quad 9 + 13 = 22$.

In other words, the sum and the product of natural numbers is a natural number.

There are subsets of $\mathbb{N}$ that are of particular interest, e.g.

• even numbers: ${2,4,6,8,\dots}$
• odd numbers: ${3,5,7,9,\dots}$
• squares: ${1,4,9,16,25,\dots}$

We say that a number $n \in \mathbb{N}$ is divisible by a number $d \in \mathbb{N}$, if there exists a $m \in \mathbb{N}$ such that $m \cdot d = n$.

A natural number different from $1$ that is divisible only by $1$ and itself is called a prime number.

• primes: ${2,3,5,7,11,13,\dots}$

Every natural number is a unique product of primes (up to permuting the factors):

$1050 = 2 \cdot 3 \cdot 5 \cdot 5 \cdot 7$, $\quad 1001 = 7 \cdot 11 \cdot 13$

There are infinintely many primes in $\mathbb{N}$.

The difference of natural numbers need not be a natural number: $2 - 5 \not\in \mathbb{N}$.