NCERT Class 7 Maths: Rational Numbers Guide

Welcome to the exciting world of rational numbers! In earlier classes, you learned about natural numbers, whole numbers, fractions, and integers. But what happens when we need to express values like a depth of 750 meters below sea level as a fraction in kilometers? Simple integers or positive fractions aren't enough. That is where rational numbers step in!

In this complete guide to rational numbers class 7 ncert, we will master the definition of rational numbers, learn how to represent them on a number line, convert them into standard form, and easily perform mathematical operations like addition, subtraction, multiplication, and division. With your YoLearn AI Tutor by your side, you will build a solid base for algebra and real numbers in higher classes. Let's jump in!

What is a Rational Number?

A rational number is defined as a number that can be written in the form p/q, where p and q are integers and q is not equal to 0. The word 'rational' comes from 'ratio'. A simple ratio like 3:5 can be written as 3/5, which is a rational number. Here, p is called the numerator and q is called the denominator.

Why can q never be zero? Because dividing any number by zero is mathematically undefined!

Every integer is also a rational number. For instance, the integer -7 can be written as -7/1, and the integer 0 can be written as 0/1. Therefore, rational numbers act as a grand category that beautifully includes all integers, whole numbers, and fractions.

Key Terminology and Definitions

Rational Number
Any number that can be expressed in the form p/q, where p and q are integers and q is not equal to 0.
Positive Rational Number
A rational number whose numerator and denominator both have the same sign (both positive, like 3/4, or both negative, like -5/-8).
Negative Rational Number
A rational number whose numerator and denominator have opposite signs (e.g., -3/7 or 4/-9).
Standard Form
A rational number is in standard form if its denominator is positive and the highest common factor (HCF) of the numerator and denominator is 1.

How to Convert a Rational Number into Standard Form

  1. Step 1: Check the Denominator's Sign — If the denominator is negative, multiply both the numerator and the denominator by -1 to make the denominator positive. For example, convert 5/-8 to -5/8.
  2. Step 2: Find the HCF — Calculate the Highest Common Factor (HCF) of the absolute values of the numerator and the denominator. For example, for -18/45, find the HCF of 18 and 45, which is 9.
  3. Step 3: Divide and Simplify — Divide both the numerator and the denominator by this HCF. Dividing -18 by 9 gives -2, and dividing 45 by 9 gives 5. The standard form is -2/5.

Step-by-Step Worked Examples

  • Example 1: Express -36/24 in standard form. • Step 1: Denominator is positive 24. No sign change needed. • Step 2: HCF of 36 and 24 is 12. • Step 3: Divide numerator and denominator by 12. (-36 ÷ 12) / (24 ÷ 12) = -3/2. • Final Answer: -3/2.
  • Example 2: Find three rational numbers between -1 and 0. • Step 1: Let us write -1 and 0 as rational numbers with a denominator of 4. -1 = -4/4 and 0 = 0/4. • Step 2: Look at the numerators between -4 and 0: they are -3, -2, -1. • Step 3: Write down the rational numbers: -3/4, -2/4 (which is -1/2), and -1/4. • Final Answer: -3/4, -1/2, and -1/4 are three rational numbers between -1 and 0.

Avoid These Common Mistakes in Exams

  1. The Negative Denominator Trap: In standard form, never leave a minus sign in the denominator. Always bring it up to the numerator (e.g., transform 2/-3 to -2/3).
  2. The Zero Confusion: Remember that 0 is a rational number (0/1), but any number divided by 0 (like 5/0) is NOT a rational number.
  3. Cross-Multiplication Signs: When comparing negative rational numbers, pay close attention to signs. For instance, -15 is smaller than -14, so -15/20 < -14/20. Don't rush!

Practice Questions with Solutions

  • Q: Reduce -44/72 to its standard form. A: Step 1: Check the denominator. 72 is positive, so the sign stays as is. Step 2: Find the HCF of 44 and 72. The factors of 44 are 1, 2, 4, 11, 22, 44. The factors of 72 include 4. The HCF of 44 and 72 is 4. Step 3: Divide both the numerator and denominator by 4. (-44 ÷ 4) / (72 ÷ 4) = -11/18. Final answer: -11/18
  • Q: Find the sum: 5/3 + (-11/3). A: Step 1: Since the denominators are the same, we simply add the numerators. Step 2: Sum of numerators = 5 + (-11) = -6. Step 3: Keep the denominator same: -6/3. Step 4: Simplify to standard form: -6/3 = -2. Final answer: -2
  • Q: Which is greater: -3/4 or -2/3? A: Step 1: Find the common denominator by taking the LCM of 4 and 3, which is 12. Step 2: Convert both numbers to equivalent rational numbers with denominator 12. -3/4 = (-3 3) / (4 3) = -9/12 -2/3 = (-2 4) / (3 4) = -8/12 Step 3: Compare the numerators: since -8 is greater than -9, we have -8/12 > -9/12. Final answer: -2/3 is greater than -3/4
  • Q: Find the product of -6/5 and 9/11. A: Step 1: Multiply the numerators: -6 9 = -54. Step 2: Multiply the denominators: 5 11 = 55. Step 3: Combine them: -54/55. There are no common factors other than 1. Final answer: -54/55

Frequently Asked Questions

Are fractions and rational numbers the same thing?

All fractions are rational numbers, but not all rational numbers are fractions. Fractions only use positive integers for both numerator and denominator, while rational numbers can use negative integers as well.

Is 0 a rational number?

Yes, 0 is a rational number. It can be written in the p/q form as 0/1 or 0/5, where the denominator is not zero.

How do you find rational numbers between two given rational numbers?

Convert both rational numbers to equivalent rational numbers with a larger common denominator. This allows you to find multiple integers between their numerators.