CBSE Class 8 Mathematics Rational Numbers Notes Set 13

Rational Numbers and Their Standard Form

A rational number is a number of the form \( \frac{p}{q} \) where p and q are integers and q ≠ 0. Natural numbers, whole numbers and integers are all rational (for example, \( 5 = \frac{5}{1} \)). A rational number is in standard form when q is positive and p and q have no common factor other than 1; for example \( \frac{-12}{18} \) in standard form is \( \frac{-2}{3} \).

Addition and Subtraction

To add rationals with the same denominator, add the numerators and keep the denominator. If the denominators differ, replace each fraction by an equivalent one with the LCM of the denominators and then add. For example \( \frac{5}{12}+\frac{7}{8} \): the LCM is 24, so the sum is \( \frac{10}{24}+\frac{21}{24} = \frac{31}{24} \).
To subtract, add the additive inverse: \( a - b = a + (-b) \).

Properties of Addition

Subtraction of rationals is not commutative and not associative.

Multiplication and Its Properties

To multiply two rationals, multiply the numerators and multiply the denominators: \( \frac{a}{b}\times\frac{c}{d} = \frac{ac}{bd} \). Cancel common factors first to keep the numbers small.

Reciprocal and Division

The reciprocal (multiplicative inverse) of \( \frac{a}{b} \) is \( \frac{b}{a} \), provided a ≠ 0, because \( \frac{a}{b}\times\frac{b}{a} = 1 \). Zero has no reciprocal. Only 1 and −1 are equal to their own reciprocals. Dividing by a rational number means multiplying by its reciprocal: \( x \div y = x\times\frac{1}{y} \).

Rational Numbers on the Number Line and Between Two Rationals

To mark \( \frac{p}{q} \) with q > 0 on the number line, divide each unit length into q equal parts and count p parts from 0, to the right for positive p and to the left for negative p. Numbers get larger as you move right.
There are infinitely many rational numbers between any two rationals. Method 1: write both numbers with the same, large denominator and list the numerators between them. For example between \( \frac{3}{5} \) and \( \frac{3}{4} \) use denominator 160: \( \frac{97}{160}, \frac{98}{160}, \dots, \frac{106}{160} \). Method 2: the mean \( \frac{x+y}{2} \) lies between x and y; repeat the process to find more.

Laws of Exponents for Rational Bases

For a non-zero rational a and integers m and n: \( a^m\times a^n = a^{m+n} \); \( a^m\div a^n = a^{m-n} \); \( (a^m)^n = a^{mn} \); \( a^m\times b^m = (ab)^m \); \( a^0 = 1 \); \( a^{-n} = \frac{1}{a^n} \); \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \).

Worked Examples

Illustration 1. Find the value of \( \frac{5}{12}+\frac{7}{8} \).
Answer:
LCM of 12 and 8 is 24.
\( \frac{10}{24}+\frac{21}{24} = \frac{31}{24} \).

Illustration 2. Subtract \( \frac{3}{7} \) from \( \frac{-4}{11} \).
Answer:
\( \frac{-4}{11}-\frac{3}{7} = \frac{-28-33}{77} = \frac{-61}{77} \).

Illustration 3. Simplify \( \frac{4}{7}\times\left(\frac{2}{3}+\frac{3}{4}\right) \) using the distributive property.
Answer:
\( \frac{4}{7}\times\frac{2}{3}+\frac{4}{7}\times\frac{3}{4} = \frac{8}{21}+\frac{3}{7} = \frac{8+9}{21} = \frac{17}{21} \).

Illustration 4. Simplify \( \left(\frac{3}{7}\right)^{-2}\times\left(\frac{7}{6}\right)^{-3} \).
Answer:
\( \left(\frac{7}{3}\right)^2\times\left(\frac{6}{7}\right)^3 = \frac{49}{9}\times\frac{216}{343} = \frac{24}{7} \).