CBSE Notes Class 7 Maths Chapter 8 Rational Numbers

A rational number is any number that can be expressed in the form $\frac{p}{q}$ , where p and q are integers, and $q\ne 0$ . Rational numbers include both positive and negative numbers as well as zero. They can be represented on a number line, with positive values to the right of zero and negative values to the left. Every fraction is a rational number, though not all rational numbers are positive. Rational numbers follow properties like closure, commutativity, and associativity under addition, subtraction, and multiplication. They also have additive and multiplicative identities: 0 and 1, respectively.

1.0Definitions

1. Natural Numbers (N): Counting numbers like 1, 2, 3, 4, etc., are called natural numbers, represented as N = {1, 2, 3, 4, ...}.
2. Whole Numbers (W): Adding 0 to the set of natural numbers gives us whole numbers: W = {0, 1, 2, 3, 4, ...}. Every natural number is a whole number, but 0 is the only whole number not classified as a natural number.
3. Integers (I or Z): Integers include all positive counting numbers, their negatives, and zero: Z = { ..., -3, -2, -1, 0, 1, 2, 3, ... }.
4. Rational Numbers (Q) : A rational number can be written as \frac{p}{q}  , where p and q are integers, and q \neq 0. For example, -\frac{5}{8} is a rational number. Here, p is the numerator and q is the denominator.

• Positive Rational Numbers: If both numerator and denominator are either positive or negative, the rational number is positive, e.g., $\frac{3}{5},\frac{-5}{-9}$ .
• Negative Rational Numbers: If either the numerator or the denominator is negative, the rational number is negative, e.g., $-\frac{3}{5},\frac{-7}{4}$.

2.0Properties of Rational Numbers

1. Equivalent Rational Numbers: For $\frac{p}{q}$  and any non-zero integer m, $\frac{p}{q}=\frac{p\times m}{q\times m}$.
2. Reducing to Simplest Form: Simplify $\frac{p}{q}$ by dividing both p and q by their HCF. $\frac{p}{q}=\frac{p÷m}{q÷m}$.
3. Standard Form: A rational number $\frac{p}{q}$ is in standard form when p and q have no common factors other than 1 (i.e., they are co-prime), and q is positive. This form ensures the simplest representation of the number.

3.0Rational Numbers on a Number Line

Rational numbers can be placed on a number line just like integers. Positive rational numbers lie to the right of zero, while negative rational numbers are to the left. Each rational number corresponds to a unique point, allowing us to visualize their relative values easily.

4.0Comparison of Two Rational Numbers

To compare two rational numbers, $\frac{a}{b}$ and $\frac{c}{d}$ , follow these steps:

1. Make the Denominators Equal: Find a common denominator, usually the least common multiple (LCM) of b and d. Rewrite each fraction with this common denominator:

_{\frac{a}{b}=\frac{a \times d}{b \times d} \text { and } \frac{c}{d}=\frac{c \times b}{d \times b}}

2. Compare the Numerators: With the same denominator, compare the numerators. If a × d < c × b, then $\frac{a}{b}>\frac{c}{d}$ ; otherwise, $\frac{a}{b}<\frac{c}{d}$.

This method allows for direct comparison of the two rational numbers.

5.0Rules for Inserting Rational Numbers

Case I: Same Denominators

1. Compare the numerators if the denominators are the same.
2. If numerators differ significantly, insert numbers by increasing the numerator step-by-step while keeping the denominator constant.
3. If the difference between numerators is small, multiply both rational numbers by multiples of 10 to create space for more numbers.

Case II: Different Denominators

1. To find rational numbers between two with different denominators, make the denominators equal.
2. This is done by finding the LCM of the denominators and rewriting the numbers accordingly.

6.0Addition of Rational Numbers

Case I: Same Denominator To add $\frac{p}{q}$ and $\frac{r}{q}$ , add the numerators and keep the common denominator:

$\frac{p}{q}+\frac{r}{q}=\frac{p+r}{q}$

Case II: Different Denominators

1. Find the LCM of the denominators.
2. Rewrite the rational numbers with the LCM as their common denominator.
3. Add the resulting rational numbers.

7.0Subtraction of Rational Numbers

Subtraction is the opposite of addition. For rational numbers x and y, subtracting y from x means adding the negative of y:

x – y = x + (–y)

For two rational numbers $\frac{a}{b}\text{ and }\frac{c}{d}$:

$\frac{a}{b}-\frac{c}{d}=\frac{a}{b}+\left(-\frac{c}{d}\right)$

This involves adding the additive inverse of $\frac{c}{d}$ .

8.0Multiplication of Rational Numbers

To multiply two rational numbers, multiply their numerators and their denominators:

$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$

This gives the product of $\frac{a}{b}\text{ and }\frac{c}{d}$

9.0Reciprocal of Rational Numbers

A rational number y is the reciprocal of x if $x\times y=1$ . For example, the reciprocal of $\frac{7}{18}\text{ is }\frac{18}{7}$ because:$\frac{7}{18}\times \frac{18}{7}=1$

The reciprocal of $\frac{7}{18}$  can also be expressed as ${\left(\frac{7}{18}\right)}^{-1}$ .

10.0Division of Rational Numbers

Division is the inverse of multiplication. For integers a and b ($b\ne 0$) , division is defined as:

$a÷b=a\times \frac{1}{b}$

To divide a rational number x by a non-zero rational number y, multiply x by the reciprocal of y.

11.0Absolute Value of a Rational Number

The absolute value of a rational number is its distance from zero on the number line, regardless of its sign.  

For example:

$\left|\frac{1}{2}\right|=\frac{1}{2},\left|-\frac{1}{2}\right|=\frac{1}{2}$

The absolute value is denoted by vertical bars around the number: $\left|\frac{p}{q}\right|=\frac{|p|}{|q|}.$

12.0Sample Questions on Rational Numbers

1. What do we mean by rational numbers?

Ans: Numbers that can be expressed as the ratio of 2 integers, with the denominator not equal to zero is called Rational Numbers. (e.g., $\frac{2}{3},\frac{-5}{7}).$

2. How can rational numbers with the same denominator be added?

Ans: Add the numerators and keep the denominator the same:

$\frac{p}{q}+\frac{r}{q}=\frac{p+r}{q}$

3. What is the reciprocal of a rational number?

Ans: The reciprocal of $\frac{a}{b}\text{ is }\frac{b}{a}$ , as long as neither a nor b is zero.