Arithmetic was initially limited to the study of the properties of natural integers, relative integers and rational numbers (in the form of fractions), and to the properties of operations on these numbers. The traditional arithmetic operations are addition, division, multiplication, and subtraction. This discipline was then broadened by the inclusion of the study of other numbers like reals (in the form of unlimited decimal expansion), or even more advanced concepts, like exponentiation or square root. Arithmetic is a way of formally representing - in other words, "coding" - numbers (as a list of digits, for example); and (thanks to this representation) define the basic operations: addition, multiplication, etc.
Many integers have special properties. These properties are the subject of number theory. Among these particular numbers, the prime numbers are arguably the most important.
Prime numbers edit
This is the case for so-called prime numbers. These are the natural numbers having only two distinct positive divisors, namely 1 and themselves. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. The integer 1 is not prime because it does not have two distinct positive divisors, but only one, namely himself. There are infinitely many prime numbers. By completing a grid of size 10 × 10 with the first 100 non-zero natural integers, and crossing out those which are not prime, we obtain the prime numbers belonging to {1, ..., 100} by a process called a sieve of Eratosthenes, named after the Greek scholar who invented it.
Even and odd numbers edit
Natural numbers can be divided into two categories: even and odd.
An even {\ displaystyle n} n integer is a multiple of 2 and can therefore be written {\ displaystyle n = 2 \, k} n = 2 \, k, with {\ displaystyle k \ in \ mathbb {N} } k \ in \ N. An odd number {\ displaystyle n} n is not a multiple of 2 and can be written {\ displaystyle n = 2 \, k + 1} n = 2 \, k + 1, with {\ displaystyle k \ in \ mathbb {N}} k \ in \ N.
We show that any integer is either even or odd, and this for a unique {\ displaystyle k} k: we denote {\ displaystyle \ forall n \ in \ mathbb {N} \ quad \ exists! K \ in \ mathbb {N } \ quad \ left (n = 2 \, k \ lor n = 2 \, k + 1 \ right)} {\ displaystyle \ forall n \ in \ mathbb {N} \ quad \ exists! k \ in \ mathbb { N} \ quad \ left (n = 2 \, k \ lor n = 2 \, k + 1 \ right)}.
The first six even integers are 0, 2, 4, 6, 8 and 10. The first six odd integers are 1, 3, 5, 7, 9 and 11
