Rational and Irrational Numbers

In the article Classification of Numbers we have already defined Rational Numbers and Irrational Numbers. We also touched upon a few fundamental properties of Rational and Irrational numbers. In this article we shall extend our discussion of the same and explain in detail some more properties of Rational and Irrational Numbers.

Rational Numbers

In general the set of rational numbers is denoted as $\mathbb{Q}$ .

$\therefore$ $\mathbb{Q} = \dfrac{a}{b}$ ; $a,b \in Z$ and $b \neq 0$

The condition $b \neq 0$ is a necessary condition for $\frac{a}{b}$ to be rational number, as division by zero is not defined. $\frac{5}{11}, -\frac{4}{9}, \frac{3}{19}$ , etc. are rational numbers.
Just as, corresponding to any integer $n$ , there is it’s negative integer $-n$ ; similarly corresponding to every rational number $\frac{a}{b}$ there is it’s negative rational number $-\frac{a}{b}$ .
Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal if and only if $ad=bc$ i.e., $\frac{a}{b} = \frac{c}{d}$ or $a \times d = b \times c$ . Also, $\frac{a}{b} > \frac{c}{d} \Rightarrow a \times d > b \times c$ and $\frac{a}{b} < \frac{c}{d} \Rightarrow a \times d < b \times c$
In between any two rational numbers $a$ and $b$ , there exists another rational number $\frac{a+b}{2}$ . If $a>b$ then $a > \frac{a+b}{2} >b$ and if $a<b$ then $a < \frac{a+b}{2} < b$

Example: Find three rational numbers between 3 and 5.

Solution:

$\because$ $3 < 5 \Rightarrow 3< \dfrac{3+5}{2} <5$ (inserting one rational number between 3 and 5)

or, $3 < 4 < 5$

or, $3 < \dfrac{3+4}{2} < 4 < \dfrac{4+5}{2} < 5$

or, $3 < \dfrac{7}{2} < 4 < \dfrac{9}{2} < 5$

$\therefore$ $\dfrac{7}{2}, 4, \dfrac{9}{2}$ are three rational numbers between 3 and 5.

Properties of Rational Numbers

The sum of two or more rational numbers is always a rational number. For example: i) If $a$ and $b$ are any two rational numbers then $a+b$ is also a rational number. ii) $\because$ $\frac{3}{5} \in \mathbb{Q}$ and $\frac{4}{7} \in \mathbb{Q}$ then $\frac{3}{5} + \frac{4}{7} = \frac{21+20}{35} = \frac{41}{45}$ , which is also a rational number.
The difference of two rational numbers is always a rational number. For example: i) If $a$ and $b$ are any two rational numbers then each of $a-b$ and $b-a$ is also a rational number ii) $\because$ $\frac{3}{5} \in \mathbb{Q}$ and $\frac{4}{7} \in \mathbb{Q}$ then $\frac{3}{5} - \frac{4}{7} = \frac{21-20}{35} = \frac{1}{35}$ , which is also a rational number. Also, $\frac{4}{7} - \frac{3}{5} = \frac{20-21}{35} = -\frac{1}{35}$ , which is also a rational number.
The product of two or more rational numbers is always a rational number. For example: If $a$ and $b$ are two rational numbers, then $ab \in \mathbb{Q}$
The division of a rational number by a non-zero rational number is always a rational number. For example: If $a$ and $b$ are two rational numbers and $b \neq 0$ , then $\frac{a}{b}$ is always a rational number

Note: Since the sum of two rational numbers is always a rational number; we say the set of rational numbers is closed for addition.

In the same way the set of rational numbers is closed for:

Subtraction
Multiplication
Division (if divisor not equal to zero)

Decimal representation of Rational Numbers

Every rational number can be expressed either as a terminating decimal or as a non-terminating decimal.

Examine the following rational numbers:

$\dfrac{1}{8} = 0.125$

$\dfrac{1}{25} = 0.04$

In each example given above the division is exact. The quotients of such divisions are called terminating decimals.

Now examine the following rational numbers:

$\dfrac{3}{7} = 0.42857$

$\dfrac{18}{23} = 0.7826086$

In each example above the division never ends, no matter how long it continues. The quotients of such divisions are called non-terminating decimals.

Now examine the following divisions:

$\dfrac{4}{9} = 0.444444444$

$\dfrac{11}{30} = 0.3666666666$

These non terminating decimals in which a digit or a set of digits repeats continually, is called a recurring or a periodic or a circulating decimal. The repeating digit or the set of repeating digits is called the period of the recurring decimal.

Note: If the denominator of a rational number can be expressed as the power either of 2 or of 5 or of 2 and 5 both, the rational number is convertible into a terminating decimal. Otherwise, the rational number is convertible to a recurring decimal.

Irrational Numbers

Any real number that cannot be expressed as a ratio of integers, i.e., any real number that cannot be expressed as simple fraction is called an irrational number.

The square roots, cube roots, etc of natural numbers are irrational numbers, if their exact values cannot be obtained. $\sqrt{2}$ is irrational since exact value of it cannot be obtained.

A non- terminating and non-recurring decimal is an irrational number.For example, 0.424344445

The number $\pi = 3.1415926535$ is also an irrational number.

Example: Identify the number as rational or irrational $\sqrt{12}$

Solution:

$\sqrt{12} = \sqrt{2 \times 2 \times 3}= 2 \sqrt{3}$ , which is the product of a rational number 2 and an irrational number $\sqrt{3}$

Example: Find two irrational numbers between 2 and 3.

Solution:

If a and b are two positive numbers such that ab is not a perfect square then :

i ) A rational number between $a$ and $b = \dfrac{a+b}{2}$

ii) An irrational number between $a$ and $b = \sqrt{ab}$

$\because$ 2 and 3 are rational numbers and $2 \times 3 = 6$ is not a perfect square

$\therefore$ one irrational number between 2 and 3 $= \sqrt{2 \times 3} = \sqrt{6}$

An irrational number between 2 and $\sqrt{6}= \sqrt{2 \times \sqrt{6}}$

$\therefore$ required rational and irrational numbers are:

$\sqrt{6}$

$\sqrt{2 \times \sqrt{6}}$

Example: Insert a rational and an irrational number between 3 and 4.

Solution: Since, 3 and 4 are positive rational numbers and $3 \times 4 = 12$ is not a perfect square, therefore:

i) A rational number between 3 and 4 $= \dfrac{3+4}{2} = 3.5$

ii) An irrational number between 3 and 4 $= \sqrt{3 \times 4}= 2\sqrt{3}$

More about irrational numbers

The sum of two irrational numbers may or may not be irrational.
The difference of two irrational numbers may or may not be irrational.
The product of two irrational numbers may or may not be irrational.
The negative of an irrational number is always irrational.
The sum of a rational and an irrational number is always irrational.
The product of a non-zero rational number and an irrational number is always irrational.

Exercise

Insert three rational numbers between:
1. 4 and 4.5
2. 5 and -2
Represent as a decimal number:
1. $\dfrac{8}{13}$
2. $\dfrac{4}{9}$
Find which of the following rational numbers have terminating decimal representation:
1. $\dfrac{32}{45}$
2. $\dfrac{17}{40}$
Find the decimal representation of $\dfrac{1}{7}$ and $\dfrac{2}{7}$ . Deduce from the decimal representation of $\dfrac{1}{7}$ , without actual calculation, the decimal representation of $\dfrac{3}{7}, \dfrac{4}{7}, \dfrac{6}{7}$ .
Prove that $ \sqrt{m} $ is not a rational number, if $m$ is not a perfect square.
Compare : $\sqrt{4}$ and $\sqrt{3}$
Write in ascending order:
1. $5\sqrt{3}, 4\sqrt{5}, 3\sqrt{7}$
2. $3\sqrt{2}, 2\sqrt{3}$
Write a pair of irrational numbers whose difference is irrational.
Write a pair of irrational numbers whose product is rational.
Insert two rational numbers and two irrational numbers between $\sqrt{3}$ and $\sqrt{7}$

Rational Numbers

Properties of Rational Numbers

Decimal representation of Rational Numbers

Irrational Numbers

More about irrational numbers

Exercise

Comments

Leave a Reply Cancel reply