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.