# Why aren't there real numbers?

## Types of numbers: From natural to complex numbers

**Different types of numbers are defined in mathematics. There are natural numbers, whole numbers, negative numbers and so-called complex numbers. In this section we will introduce you to the different types of numbers in more detail.**

**Natural numbers**

In principle, everything I can count is called a natural number: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and so on. You can continue counting here as you like. At this point there is usually a question: Is the zero also a natural number? The answer: It depends on how you define natural numbers. In the case of a definition with a zero, the zero is included. If you define the natural numbers without zero, they do not belong to them. You can find more information on this topic in the article Natural Numbers.

Show:**Negative numbers**

Negative numbers can be recognized by a minus sign in front of the number, e.g. -5 or -23 or -8.23. The easiest way to understand this is with a bank account. If I have 1000 euros, I have +1000 euros in the account. However, if I borrowed money from the bank, e.g. 1000 euros, then I have -1000 euros. So I have to give the bank 1000 euros to get the 0 euros on my account and not have any debts. You can find more information on this topic in the article Negative Numbers.

**Whole numbers**

The whole numbers are an extension of the natural numbers. They include not only 1, 2, 3, 4 etc. but also negative numbers such as -3, -2, -1. The 0 is also counted. The whole numbers are thus: .... -3, -2, -1, 0, 1, 2, 3 .... You can find more information on this topic in the article Whole Numbers.

**Rational numbers**

A rational number is a (real) number that can be represented as the ratio of two whole numbers. It includes all numbers that can be represented as a fraction that contains whole numbers in both the numerator and the denominator. Examples: 8/3, 3/4, 232/579. **Every integer and every natural number is a rational number**. Note: We have a separate section for fractions, which goes into this area in more detail. You can learn more about rational numbers in our article Rational Numbers.

**Irrational numbers**

Rational numbers can be represented as a fraction, irrational numbers cannot. For example, if you take the root of the number 2, you get about 1.4142. However, this number is imprecise, because the root of 2 has an infinite number of places after the decimal point. This also applies to the circle number π (pronounced: pi), for which the value 3.14 is usually used as an approximation in schools. In practice you break off after a certain place after the comma and thus get a finite decimal number (point number). You can find more about this type of number in our article Irrational Numbers.

**Real numbers **

The set of real numbers is the union of the rational numbers and the irrational numbers. The definitions for these two types of numbers can be found above. You can find more about this type of number in our article Real Numbers.

**Complex numbers**

As a rule, one only deals with complex numbers in college or university, but not in school. The complex numbers expand the range of numbers. This is achieved by introducing a new number i or j (depending on which one would prefer to use) with the property i^{2} = - 1. This number i is called an imaginary unit. Complex numbers are usually represented in the form a + b · j. Example: 4 + 3y or also 2 + 5y. The first number is the real part and the number with the "j" is the imaginary part. They can be plotted in the complex Gaussian plane. The axes are not labeled with x-y as usual from school, but with real-imaginary. The complex numbers are an extensive topic. More on this also in the complex numbers basics.

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