## 4.5.1.1 Natural Numbers

A natural number is a countable number.

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In this image, there are five images of "the industry". Note that:

• You cannot have a negative number of industries
• You cannot have half an industry

The set of numbers, $$\{0, 1, 2, 3, ...\}$$ is represented by $$\mathbb{N}^0$$ or $$\mathbb{N}_{0}$$

Sometimes, 0 is not included. This set, $$\{1, 2, 3, 4, ...\}$$ is represented by $$\mathbb{N}^*$$, $$\mathbb{N}^+$$, $$\mathbb{N}_{1}$$ or $$\mathbb{N}_{>0}$$

## 4.5.1.2 Integer Numbers

An integer number is a number that represents a whole number of things.

$$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$$

## 4.5.1.3 Rational Numbers

A rational number is a number which can be represented as a fraction of integers.

Number Rational? Why
$$\pi$$ No $$\pi$$ cannot be written as a fraction
$$\sqrt{2}$$ No $$\sqrt{2}$$ cannot be written as a fraction
$$7$$ Yes 7 can be written as $$7\over1$$. All integer numbers are therefore also rational numbers
$$\sqrt{-1}$$ No $$\sqrt{-1} = i$$, and complex numbers cannot be written with integers.

## 4.5.1.5 Real Numbers

Real numbers are numbers which can be represented as an integer number, a rational or an irrational number.

One may consider a real number as a "real world quantity". The set of numbers is represented as $$\mathbb{R}$$

Number Rational? Why
$$\pi$$ Yes $$\pi$$ can be found in the real world. Tasty.
$$\sqrt{2}$$ Yes It is possible to cut a length $$\sqrt{2}$$.
$$7$$ Yes You can count seven objects.
$$\sqrt{-1}$$ No You cannot count a complex number.

## 4.5.1.6 Ordinal Numbers

An ordinal number is a number that can be used to describe a position within a set.

In the set $$S = \{"a", "b", "c", "d"\}$$, "a" would be the 1st letter in the one indexed set.

## 4.5.1.7 Counting and measurement

Natural numbers are for counting. Real numbers can be used for measurement.