Quick Revision Notes: Class 7 Mathematics

Chapter: Rational Numbers

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1. What is a Rational Number? (परिमेय संख्या क्या है?)

• Definition: A rational number is a number that can be expressed in the form p/q, where p and q are integers, and q ≠ 0.
• Examples: 3/4, -5/7, 0, 6 (since 6 = 6/1), -15.
• Note: All integers and fractions are rational numbers.

2. Positive and Negative Rational Numbers

• Positive Rational Numbers: Both the numerator and denominator have the same sign (both positive or both negative).
  • Example: 3/5 or (-3)/(-5).
• Negative Rational Numbers: Either the numerator or the denominator is negative.
  • Example: -3/5 or 3/(-5).

3. Rational Numbers on a Number Line

• Just like integers and fractions, rational numbers can also be represented on a number line.
• Positive rational numbers lie to the right of zero (0).
• Negative rational numbers lie to the left of zero (0).

4. Rational Number in Standard Form (Lowest Form)

• A rational number p/q is said to be in the standard form if:
  1. q is positive.
  2. The greatest common divisor (HCF) of p and q is 1 (i.e., they have no common factor other than 1).
• Example: To reduce -45/30 to standard form, divide both by their HCF (15): (-45 ÷ 15) / (30 ÷ 15) = -3/2.

5. Equivalent Rational Numbers

• By multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, we get equivalent rational numbers.
• Example: 1/2 = 2/4 = 3/6 = 4/8

6. Comparison of Rational Numbers

• Rule 1: Every positive rational number is greater than zero and every negative rational number.
• Rule 2: Every negative rational number is less than zero.
• Rule 3: To compare two rational numbers with different denominators:
  1. Make the denominators positive.
  2. Find the LCM of the denominators.
  3. Convert both rational numbers to equivalent rational numbers with the LCM as the common denominator.
  4. Compare the numerators. The one with the greater numerator is greater.

7. Operations on Rational Numbers (संक्रियाएँ)

A. Addition and Subtraction

• Same Denominator: Add or subtract the numerators while keeping the denominator the same.
  • a/c + b/c = (a + b) / c
• Different Denominator: Take the LCM of the denominators, convert them to like fractions, and then add or subtract.
  • Example: 1/2 + 1/3 = (3 + 2) / 6 = 5/6 (LCM of 2 and 3 is 6)

B. Multiplication

• Multiply the numerators together and the denominators together.
  • (a/b) × (c/d) = (a × c) / (b × d)

C. Division

• To divide one rational number by another, multiply the first rational number by the reciprocal (व्युत्क्रम) of the second rational number.
  • (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)

8. Reciprocal (Multiplicative Inverse / व्युत्क्रम)

• The reciprocal of a non-zero rational number a/b is b/a.
• Example: The reciprocal of 2/3 is 3/2.
• Note: The product of a rational number and its reciprocal is always 1. (a/b × b/a = 1)