Definition of Rational Numbers

Rational numbers are numbers that can be reduced to a ratio of two integers a and b, where b cannot be 0

{{a, b\in\mathbb{Z}:\frac{a}{b} with b\neq 0}

They are called Rational because of the fact that they are a ratio or quotient of two integers. 'Ratio' is the root of rational and that's what it means in this sense, the ability to be represented as a ratio.

The total group of all Rational Numbers is denoted by \mathbb{Q}. The Natural Numbers \mathbb{N} and Integers \mathbb{Z} are both subsets of the Rational Numbers. This means that every Natural number and every Integer is also a Rational number

\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}

where \subseteq means 'is a subset of'.

Every Rational number can be represented as a ratio of two integers. Rational numbers have the property of having a finite or infinitely repeating equivalent decimal value.

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Examples of Rational Numbers and their Equivalent Decimal Values

\frac{a}{b}=\frac{1}{2}=0.5

\frac{a}{b}=\frac{1}{3}=0.\overline{33}

\frac{a}{b}=-\frac{2}{3}=-0.\overline{66}

\frac{a}{b}=\frac{5}{4}=1.25

\frac{a}{b}=\frac{2}{1}=2

Notice that the integer 2 is also the fraction \frac{2}{1} or \frac{4}{2} or \frac{60}{30}. All of these are just alternate forms of the number 2.

Rational Number Diagram

Rational Number Diagram

More about Rational Numbers

Any Natural number or Integer n is the ratio of that number n to 1

n=4=\frac{4}{1}

The Rational Numbers are not just the numbers known as fractions, although that is ultimately a property that each one shares.

We can imagine a fraction that includes an irrational number, which would not be a Rational number because it includes a number that is non-repeating and non-ending.

\frac{\pi}{2} is an example of a fraction that is not Rational

Rational numbers can come in many forms: Natural numbers, integers, decimals, ratios, fractions or quotients; any form of number that can be reduced to a ratio of integers.

Number Line Representation of Rational Numbers

The Rational Numbers 'fill-up' some of the space between the Integers on the Number Line. We can divide any region from one integer to another integer into any amount of subdivisions we want.

We can subdivide the region between 0 and 1 into fifth's by marking off 4 points a, b, c, d that will make 5 equal spaces in between

0\to 1

Divided Number Line

the 4 points a, b, c, d in between will be at

a=\frac{1}{5}, b=\frac{2}{5}, c=\frac{3}{5}, d=\frac{4}{5}

making 5 equally spaced segments between 0 and 1 of

0\to\frac{1}{5}

\frac{1}{5}\to\frac{2}{5}

\frac{2}{5}\to\frac{3}{5}

\frac{3}{5}\to\frac{4}{5}

\frac{4}{5}\to\frac{5}{5}

Number Line in Fifths

How to Divide a Line into Equal Segments

Equipartitioned Line

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Patrick