Rational Numbers

What Are Rational Numbers?

Rational numbers are numbers that can be obtained by dividing one integer by another.

Let me explain with a typical example.

$3 \hspace{0.2em}$ and $\hspace{0.2em} 4 \hspace{0.2em}$ are two integers (numbers with no fractional or decimal part). So, if we divide $\hspace{0.2em} 3 \hspace{0.2em}$ by $\hspace{0.2em} 4 \hspace{0.2em}$ , we get a rational number. That means $\hspace{0.2em} 3/4 \hspace{0.2em}$ is a rational number. So is $\hspace{0.2em} 4/3 \hspace{0.2em}$ .

So, if you see a fraction whose top and bottom numbers are both integers, that’s a rational number.

But rational numbers don’t always come in the form of fractions. They can be integers or decimals too. As long as a number can be expressed as a division of two integers (a common fraction), it is a rational number.

Here are a few examples.

Integer/Decimal	In Fraction Form
$\hspace{0.2em} 2 \hspace{0.2em}$	$\hspace{0.2em} 2/1 \hspace{0.2em}$
$\hspace{0.2em} -5 \hspace{0.2em}$	$\hspace{0.2em} -5/1 \hspace{0.2em}$
$\hspace{0.2em} 0.6 \hspace{0.2em}$	$\hspace{0.2em} 3/5 \hspace{0.2em}$
$\hspace{0.2em} 1.\overline{3} \hspace{0.2em}$	$\hspace{0.2em} 4/3 \hspace{0.2em}$

So now we are in a position to look at the more technical definition of rational numbers.

A rational number is a number that can be written in the form of a fraction, $\hspace{0.2em} \frac{p}{q} \hspace{0.2em}$ , such that $\hspace{0.2em} p \hspace{0.2em}$ and $\hspace{0.2em} q \hspace{0.2em}$ are integers and $\hspace{0.2em} q \neq 0 \hspace{0.2em}$ .

Why can’t $\hspace{0.2em} q \hspace{0.2em}$ (the bottom number) be zero, you ask?

Division by zero is not defined. And so, we cannot have zero at the bottom of a fraction.

How to Identify Rational Numbers?

While it’s not always possible to just look at a number and say whether or not it is rational, there are a few rules that can help you do that easily in most cases.

Any fraction with integers as its numerator and denominator (also known as a common fraction) is, by definition, rational.
All integers are rational, since they can be written as fractions (with integers at the top and bottom).
All terminating decimals are rational numbers.
All recurring (or repeating) decimals are rational numbers.
Non-terminating non-recurring decimals are not rational numbers. They are irrational.

Real numbers that are not rational are known as irrational numbers. For example, $\hspace{0.2em} \sqrt{2} \hspace{0.2em}$ , $\hspace{0.2em} \sqrt{3} \hspace{0.2em}$ , $\hspace{0.2em} \sqrt[3]{4} \hspace{0.2em}$ , etc.

In their decimal forms, irrational numbers are non-terminating and non-repeating. For example, $\hspace{0.2em} 1.2315263128... \hspace{0.2em}$

Fun Fact — Some of the coolest and most famous numbers in math are irrational. For example, $\hspace{0.2em} \pi \hspace{0.2em}$ (pi), $\hspace{0.2em} e \hspace{0.2em}$ (Euler’s number), or $\hspace{0.2em} \phi \hspace{0.2em}$ (the golden ratio). And it’s perfectly fine if you haven’t heard of them all.

Properties of Rational Numbers

Like integers, rational numbers also have certain properties related to common mathematical operations. Let’s look at some of them.

Closure Property

Rational numbers are closed under addition, subtraction, and multiplication.

That’s just a fancy way of saying — if you add, subtract, or multiply two rational numbers, the result is also rational.

Also, rational numbers are not closed under division. That’s because 0 is rational but division by zero is not defined.

Commutative Property

Rational numbers are commutative under addition and multiplication.

That means when adding or multiplying two rational numbers, the order doesn’t matter. So the numbers can commute or change places. For example –

\frac{1}{8} + \frac{5}{8} \, = \, \frac{5}{8} + \frac{1}{8}

and

\frac{1}{2} \times \frac{3}{4} \, = \, \frac{3}{4} \times \frac{1}{2}

However, this idea cannot generally be extended to subtraction or division. So –

\frac{5}{3} - \frac{1}{3} \, \neq \, \frac{1}{3} - \frac{5}{3}

and

\frac{1}{3} \div \frac{5}{4} \, \neq \, \frac{5}{4} - \frac{1}{3}

Associative Property

Rational numbers are associative under addition and multiplication.

That means when adding, or multiplying, three rational numbers, how we group (or associate) the numbers together does not matter. For example,

\left ( \frac{1}{10} + \frac{2}{10} \right ) + \frac{6}{10} \, = \, \frac{1}{10} + \left ( \frac{2}{10} + \frac{6}{10} \right )

and

\left ( \frac{1}{2} \times \frac{2}{3} \right ) \times \frac{3}{4} \, = \, \frac{1}{2} \times \left ( \frac{2}{3} \times \frac{3}{4} \right )

Again, this cannot be said for subtraction or division.

\left ( \frac{1}{2} \div \frac{2}{3} \right ) \div \frac{3}{4} \, \neq \, \frac{1}{2} \div \left ( \frac{2}{3} \div \frac{3}{4} \right )

Distributive Property

Rational numbers follow the distributive property of multiplication over addition and subtraction.

Let me explain. Say, a factor is being multiplied by the sum of two numbers.

The distributive property says we will get the same result whether you take the sum first and then multiply by the factor, or multiply the factor by each number being added and then add the products. So,

{\color{Red} \frac{1}{2}} \times \left ( \frac{1}{3} + \frac{1}{4} \right) \, = \, {\color{Red} \frac{1}{2}} \times \frac{1}{3} + {\color{Red} \frac{1}{2}} \times \frac{1}{4}

Now, in place of addition, we could have had subtraction, and that would be fine too.

{\color{Red} \frac{1}{2}} \times \left ( \frac{1}{3} - \frac{1}{4} \right) \, = \, {\color{Red} \frac{1}{2}} \times \frac{1}{3} - {\color{Red} \frac{1}{2}} \times \frac{1}{4}

But we cannot distribute division like this.

{\color{Red} \frac{1}{2}} \div \left ( \frac{1}{3} + \frac{1}{4} \right) \, \neq \, {\color{Red} \frac{1}{2}} \div \frac{1}{3} + {\color{Red} \frac{1}{2}} \div \frac{1}{4}

Additive Identity

The additive identity for rational numbers is $\hspace{0.2em} 0 \hspace{0.2em}$ .

That means adding zero to the number will not change it in any way.

\frac{1}{2} + {\color{Red} 0} \, = \, \frac{1}{2}

Additive Inverse

The additive inverse of a rational number is the negative of that number.

In other words, if you add the negative of a number to the number itself, you get $\hspace{0.2em} 0 \hspace{0.2em}$ .

For example, the additive inverse of $\hspace{0.2em} 2/5 \hspace{0.2em}$ is $\hspace{0.2em} -2/5 \hspace{0.2em}$

\frac{1}{2} + \left ( {\color{Red} - \frac{1}{2}} \right ) \, = \, 0

Similarly, the additive inverse of $\hspace{0.2em} -4 \hspace{0.2em}$ is $\hspace{0.2em} -(-4) \hspace{0.2em}$ or $\hspace{0.2em} 4 \hspace{0.2em}$ .

Multiplicative Identity

The multiplicative identity if a rational number is $\hspace{0.2em} 1 \hspace{0.2em}$ .

Meaning, multiplying by $\hspace{0.2em} 1 \hspace{0.2em}$ will does change the number.

\frac{1}{2} \times {\color{Red} 1} \, = \, \frac{1}{2}

Multiplicative Inverse

The multiplicative inverse of a rational number is its reciprocal.

So, if you multiply a rational number by its reciprocal, you get $\hspace{0.2em} 1 \hspace{0.2em}$ .

For example,

\frac{1}{2} \times {\color{Red} \frac{2}{1}} \, = \, 1

And that brings us to the end of this tutorial on rational numbers. Until next time.