## 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.