ICSE Class 8 Maths Numbers Chapter 06 Rational Numbers

Rational Numbers

Any number which can be expressed in the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0, is called a rational number. The set of rational numbers is denoted by Q.

\(Q = \left\{\frac{p}{q} : p, q \in I \text{ and } q \neq 0\right\}\)

Obviously, fractions are rational numbers. For example, \(\frac{2}{3}, \frac{5}{7}, \frac{-6}{11}, \frac{-23}{35}\) are rational numbers.

The natural numbers and integers are also rational numbers because \(7 = \frac{7}{1}, -25 = \frac{-25}{1}\), and so on.

Any terminating or recurring decimal can be written as a rational number. For example, \(1.79 = \frac{179}{100}\) and \(0.2 = \frac{2}{9}\) are rational numbers.

Numbers such as \(\sqrt{2}, \sqrt{3}\) and \(\pi\) cannot be expressed as rational numbers. They are called irrationals.

Properties Of Rational Numbers

1. The sum of two rational numbers is always a rational number.

Example: \(\frac{2}{7}\) and \(\frac{13}{9}\) are rational numbers.

Their sum = \(\frac{2}{7} + \frac{13}{9} = \frac{18 + 91}{63} = \frac{109}{63}\) is a rational number.

2. Two rational numbers can be added in any order.

Example: \(\frac{3}{5} + \frac{11}{17} = \frac{11}{17} + \frac{3}{5}\) because \(\frac{3}{5} + \frac{11}{17} = \frac{51 + 55}{85} = \frac{106}{85}\).

\(\frac{11}{17} + \frac{3}{5} = \frac{55 + 51}{85} = \frac{106}{85}\)

3. While adding three rational numbers, they can be grouped in any order.

Example: Consider three rational numbers \(\frac{1}{2}, \frac{3}{7}\) and \(\frac{5}{9}\).

\(\left(\frac{1}{2} + \frac{3}{7}\right) + \frac{5}{9} = \frac{(7 + 6)}{14} + \frac{5}{9} = \frac{13}{14} + \frac{5}{9} = \frac{117 + 70}{126} = \frac{187}{126}\)

\(\frac{1}{2} + \left(\frac{3}{7} + \frac{5}{9}\right) = \frac{1}{2} + \frac{(27 + 35)}{63} = \frac{1}{2} + \frac{62}{63} = \frac{63 + 124}{126} = \frac{187}{126}\)

\(\therefore \left(\frac{1}{2} + \frac{3}{7}\right) + \frac{5}{9} = \frac{1}{2} + \left(\frac{3}{7} + \frac{5}{9}\right)\)

4. 0 is a rational number such that the sum of any rational number and 0 is the rational number itself.

Example: \(\frac{7}{9} + 0 = \frac{7}{9}, \quad 0 + \frac{7}{9} = \frac{7}{9}\)

\(\therefore \frac{7}{9} + 0 = 0 + \frac{7}{9} = \frac{7}{9}\)

5. For every rational number \(\frac{p}{q}\), there is a rational number \(\frac{-p}{q}\) such that

\(\frac{p}{q} + \left(\frac{-p}{q}\right) = 0 = \left(\frac{-p}{q}\right) + \frac{p}{q}\)

Example: \(\frac{3}{5} + \frac{-3}{5} = 0 = \frac{-3}{5} + \frac{3}{5}\)

6. The difference of two rational numbers is also a rational number.

Example: \(\frac{9}{11}\) and \(\frac{3}{5}\) are rational numbers.

\(\frac{9}{11} - \frac{3}{5} = \frac{45 - 33}{55} = \frac{12}{55}\) is also a rational number.

7. The product of two rational numbers is always a rational number.

Example: \(\frac{2}{9}\) and \(\frac{11}{13}\) are rational numbers and their product = \(\frac{2}{9} \times \frac{11}{13} = \frac{22}{117}\) is also a rational number.

8. Two rational numbers can be multiplied in any order.

Example: \(\frac{3}{17}\) and \(\frac{15}{19}\) are two rational numbers. \(\frac{3}{17} \times \frac{15}{19} = \frac{45}{323}, \frac{15}{19} \times \frac{3}{17} = \frac{45}{323}\)

So, \(\frac{3}{17} \times \frac{15}{19} = \frac{15}{19} \times \frac{3}{17}\)

9. While multiplying three (or more) rational numbers, they can be grouped in any order.

Example: Consider the rationals \(\frac{1}{2}, \frac{3}{5}\) and \(\frac{-7}{11}\).

\(\left(\frac{1}{2} \times \frac{3}{5}\right) \times \left(\frac{-7}{11}\right) = \frac{3}{10} \times \left(\frac{-7}{11}\right) = \frac{-21}{110}\)

and \(\frac{1}{2} \times \left(\frac{3}{5} \times \left(\frac{-7}{11}\right)\right) = \frac{1}{2} \times \left(\frac{-21}{55}\right) = \frac{-21}{110}\)

\(\therefore \left(\frac{1}{2} \times \frac{3}{5}\right) \times \left(\frac{-7}{11}\right) = \frac{1}{2} \times \left(\frac{3}{5} \times \left(\frac{-7}{11}\right)\right)\)

10. For any rational number \(\frac{p}{q}\), we have a rational number 1 such that

\(\frac{p}{q} \times 1 = 1 \times \frac{p}{q} = \frac{p}{q}\)

Example: Consider the rational \(\frac{11}{13}\).

\(\frac{11}{13} \times 1 = \frac{11}{13}, \quad 1 \times \frac{11}{13} = \frac{11}{13}\)

\(\therefore \frac{11}{13} \times 1 = 1 \times \frac{11}{13} = \frac{11}{13}\)

11. For any rational number \(\frac{p}{q}\), there is a rational number \(\frac{q}{p}\) such that

\(\frac{p}{q} \times \frac{q}{p} = \frac{q}{p} \times \frac{p}{q} = 1\)

Example: \(\frac{3}{5}\) and \(\frac{5}{3}\) are rationals such that \(\frac{3}{5} \times \frac{5}{3} = \frac{5}{3} \times \frac{3}{5} = 1\).

12. If \(\frac{p}{q}\) and \(\frac{r}{s}\) be two rationals such that \(\frac{r}{s} \neq 0\) then \(\frac{p}{q} \div \frac{r}{s}\) is also a rational number.

Example: \(\frac{2}{7}\) and \(\frac{3}{19}\) are two rationals.

\(\frac{2}{7} \div \frac{3}{19} = \frac{2}{7} \times \frac{19}{3} = \frac{38}{21}\) is also a rational number.

13. If x and y be two rational numbers such that x < y then \(\frac{x + y}{2}\) is a rational number between x and y, that is, \(x < \frac{x + y}{2} < y\).

Example: A rational number between \(\frac{1}{7}\) and \(\frac{1}{3}\) is \(\frac{\frac{1}{7} + \frac{1}{3}}{2} = \frac{10}{42} = \frac{5}{21}\)

\(\therefore\) the rational \(\frac{5}{21}\) lies between \(\frac{1}{7} \left(= \frac{3}{21}\right)\) and \(\frac{1}{3} \left(= \frac{7}{21}\right)\).

Irrational Numbers

A number \(\sqrt{a}\) (square root of a) is called an irrational number if a is positive and a is not the square of a rational number.

Examples: (i) \(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{\frac{7}{5}}, \frac{\sqrt{6}}{3}\) are positive irrational numbers.

Similarly, \(-\sqrt{2}, -\sqrt{3}, -\frac{\sqrt{6}}{5}\), etc., are negative irrational numbers.

But numbers such as \(\sqrt{4}, \sqrt{9}, \sqrt{\frac{16}{25}}\) are not irrational because \(4 = 2^2, 9 = 3^2, \frac{16}{25} = \left(\frac{4}{5}\right)^2\), that is, 4, 9, \(\frac{16}{25}\) are squares of rational numbers.

(ii) The solutions of the equation \(x^2 = k\) are irrational numbers if k is not a perfect square.

Real Numbers

The set of real numbers is the collection of rational numbers and irrational numbers.

A number line diagram is shown displaying real numbers from -6 to 6, with markers at \(\frac{1}{2}, \sqrt{2}, \pi,\) and 5.85.

Properties Of Real Numbers

1. The sum, difference and product of two real numbers are real numbers.

2. The division of a real number by a nonzero real number gives a real number.

3. Every real number has a negative real number. 0 is its own negative number.

4. The sum, difference, product and quotient of a rational number and an irrational number are irrational.

Example: 2 is rational and \(\sqrt{3}\) is irrational, but \(2 + \sqrt{3}, 2 - \sqrt{3}, 2\sqrt{3}\) and \(\frac{2}{\sqrt{3}}\) are all irrationals.

5. The sum, difference, product and quotient of two irrational numbers need not be irrational.

Example: \((2 + \sqrt{3}) + (2 - \sqrt{3}) = 4\), which is not irrational.

\((\sqrt{3} + 5) - (\sqrt{3} - 5) = 10\), which is not irrational.

\((3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7\), which is not irrational.

\(\frac{9 + 3\sqrt{3}}{3 + \sqrt{3}} = \frac{3(3 + \sqrt{3})}{3 + \sqrt{3}} = 3\), which is not irrational.

6. Given two real numbers a and b, \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}, \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) and \((\sqrt{a})^2 = \sqrt{a} \times \sqrt{a} = \sqrt{a} \times a = a\).

Rationalization

3 + \(\sqrt{5}\) is an irrational number. Let us multiply it by another irrational number 3 - \(\sqrt{5}\).

\((3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4\), which is a rational number.

This process is called rationalization. We say that 3 - \(\sqrt{5}\) is the rationalizing factor of 3 + \(\sqrt{5}\). Similarly, 3 + \(\sqrt{5}\) is the rationalizing factor of 3 - \(\sqrt{5}\).

Note: If a + \(\sqrt{b}\) is an irrational number then a - \(\sqrt{b}\) is the rationalizing factor of a + \(\sqrt{b}\). Similarly, a + \(\sqrt{b}\) is the rationalizing factor of a - \(\sqrt{b}\). a + \(\sqrt{b}\) and a - \(\sqrt{b}\) are said to be conjugate to each other.

Example: Rationalize the denominator of \(\frac{5}{6 - \sqrt{3}}\)

Solution: The given expression = \(\frac{5}{6 - \sqrt{3}} = \frac{5}{6 - \sqrt{3}} \times \frac{6 + \sqrt{3}}{6 + \sqrt{3}} = \frac{5(6 + \sqrt{3})}{(6 + \sqrt{3})(6 - \sqrt{3})}\)

= \(\frac{30 + 5\sqrt{3}}{6^2 - (\sqrt{3})^2} = \frac{30 + 5\sqrt{3}}{36 - 3} = \frac{30 + 5\sqrt{3}}{33} = \frac{10}{11} + \frac{5}{33}\sqrt{3}\)

Teacher's Note

When calculating quantities like recipe portions or splitting bills fairly, we use rational numbers daily - understanding their properties helps ensure accurate real-world calculations.

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