Rational and Irrational Numbers

Rational Numbers

Any number that can be expressed in the form $\\frac{p}{{q}}$ where p, q are integers and q $\\neq$ 0 is called a rational number. p is called the numerator and q the denominator of the rational number.

Any integer can be expressed as a rational number. For example, 5 can be written as $\\frac{5}{1}$ in the rational form.

The numbers $\\frac{4}{-5}$ , $\\frac{6}{4}$ , $\\frac{-11}{-9}$ , $\\frac{8}{1}$ , $\\frac{-7}{-1}$ , $\\frac{0}{-3}$ ,… are all rational numbers

All fractions are rational numbers. All integers are also rational numbers. There are many rational numbers which are neither fractions nor integers.

Rational numbers can also be represented on the number line.

Example: Which of the following is (are) rational number(s)?

2, $\\frac{1}{3}$ , $\\sqrt{2}$ , -9, $\\frac{-5}{12}}$

It can be seen that apart from $\\sqrt{2}$ , all other numbers can be written in the form $\\frac{p}{q}}$ .

So, except $\\sqrt{2}$ , all the given numbers are rational numbers.

Irrational Numbers

All numbers which are not rational are called irrational numbers, which means all the numbers that cannot be put in the form $\\frac{p}{q}}$ where p, q are integers and q $\\neq$ 0 are called ‘irrational numbers’.

Any rational number will be a terminating decimal or a repeating decimal. For example, $\\frac{7}{8}}$ = 0.875(terminating decimal)

$\\frac{5}{12}$ = 0.41666…(repeating decimal)

An irrational number is a decimal that is neither terminating nor repeating.

The square roots of 2, 3, 5, 6, 7, 8 are all irrational numbers.

Example: Which of the following is NOT irrational?

$\\sqrt{2}$
$\\sqrt{7}$
$\\sqrt{16}$
$\\sqrt{8-4}$

$\\sqrt{2}$ and $\\sqrt{7}$ are irrational numbers.

$\\sqrt{16}$ = 4, which is a rational number.

$\\sqrt{8-4}$ = $\\sqrt{4}$ = 2, which is also a rational number.

So, $\\sqrt{16}$ and $\\sqrt{8-4}$ are NOT irrational numbers.

Check Point

1. Is 0.31 311 3111… an irrational number?

2. Identify the rational numbers: 21π, $\\sqrt{83}$ , $\\frac{10}{7}$ .

3. Which of the following is an irrational number?

4. Is 0 a rational number?

5. Which list contains only rational numbers?

Answer key

Yes
$\\frac{10}{7}$
π
0 is a rational number because we can write it as $\\frac{0}{1}$ , $\\frac{0}{-4}$ , which implies that 0 is a rational number.
(A)

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