This article describes the different types of numbers from natural to real numbers.

The natural numbers, also called positive integers, are the numbers \(0, 1, 2, 3, 4, 5, 6, 7, 8,....n+1\).

Natural numbers are infinitely, because every number n has a successor n + 1.

In addition, add two numbers \(a + \)b to a new number. Here, \(a\) and \(b\) are the summands and the result the sum.

In multiplication multiply two numbers \(a · b\) to a new number. \(a\) and \(b\) are called factors and the result is the product.

Natural numbers can be added and multiplied without restriction. Substractions and divisions are limited. \(7 - 5 = 2\) or \(8 / 2 = 4\) can be calculated with natural numbers. With \(5 -7\) the range of natural numbers is left. The result \(-2\) falls within the range of the integer numbers

The integers expand the range of natural numbers by the negative range

The range of integers numbers is \(-(n+1) ... -3, -2, -1, 0, 1, 2, 3, ...n+1\)

To get a set of numbers, which also includes every result of a division, we need to expand the integers. The rational numbers are used for this. These are formed by fractions of integers. The integers are also included in the rational numbers.

This allows the execution of all four basic operations. The following examples show operations with ratinal numbers.

When adding, the denominators of the two summands must be put on a common main denominator

$$\frac{a}{b}+\frac{m}{n}=\frac{\mathrm{an}}{\mathrm{bn}}+\frac{\mathrm{bm}}{\mathrm{bm}}=\frac{\mathrm{an}+\mathrm{bm}}{\mathrm{bn}}$$

$$\frac{1}{2}+\frac{3}{4}=\frac{4}{8}+\frac{6}{8}=\frac{10}{8}=1\frac{1}{4}$$

A subtraction can be performed via the addition.

$$\frac{a}{b}-\frac{m}{n}=\frac{a}{b}+\frac{\mathrm{-m}}{n}$$

In multiplication, the numerator is multiplied by the numerator and the denominator by the denominator.

$$\frac{a}{b}*\frac{m}{n}=\frac{\mathrm{am}}{\mathrm{bn}}$$

Division durch einen Bruch ist gleich der Multiplikation mit seinem Kehrwert.

$$\frac{a}{b}/\frac{m}{n}=\frac{a}{b}*\frac{n}{m}=\frac{\mathrm{an}}{\mathrm{bm}}$$

Rational numbers can be written as a decimal fraction. Decimal fractions are comma numbers. Before the comma is the whole part of the number, after the comma the decimal places. Rational numbers in decimal notation have the property that they either have limited decimal places or there is a period in the decimal places in which digit sequences repeat themselves.

Example of rational numbers in decimal notation

$$\frac{1}{2}=\mathrm{0.5}\phantom{\rule{80px}{0ex}}\frac{1}{4}=\mathrm{0.25}\phantom{\rule{80px}{0ex}}\frac{3}{8}=\mathrm{0.375}\phantom{\rule{80px}{0ex}}\frac{888}{100}=8.88$$

$$\frac{5}{12}=0.4166666=0.416\phantom{\rule{80px}{0ex}}\frac{7}{1111}=0.006300630063=0.0063$$

Numbers with an infinite number of decimal places without period are irrational numbers. Known irrational numbers are the number \(Pi = 3.14159265358 ...\) and the Euler number \(e\).

The rational numbers along with the irrational ones are the real numbers.