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Example 8 - Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)
Last updated at May 11, 2026 by Teachoo
Decimal Expansion of Real Numbers
Class 9
Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)
- The Human Need to Count
- Exercise Set 3.1
- The Revolution of Śhūnya
- Integers
- Exercise Set 3.2
- Rational Numbers
- Exercise Set 3.3
- Rational Numbers on the Number Line
- Absolute Value of a Rational Number
- Rational Numbers between two Rational numbers
- Exercise Set 3.4
- Irrational Numbers
- Construction of Length √n on Number line
- The Story of Pi (π) and Madhava’s Infinite Series
- Real Numbers & Imaginary Numbers
-
Decimal Expansion of Real Numbers
- Magic of Cyclic Numbers
- Constructing a Square Root Spiral
- Exercise Set 3.5
- End-of-Chapter Exercises
Transcript
Example 8 Convert 2.357 ̅ into the form 𝑝/𝑞. Let x = 2.35777….. Since this is a general repeating decimal, so we look at both non-repeating (after decimal points) and repeating digits Here, ‘35’ is non-repeating (2 digits) ‘7’ repeats (1 digit) Since two digits are non-repeating (i.e. after decimal point) Multiplying equation (1) by 102 i.e. 100 100x = 100 × (2.35777…) 100x = 235.7777… Since one digit is repeating Multiplying equation (2) by 101 i.e. 10 10 × 100x = 10 × (235.7777…) 1000x = 2357.777… Doing (2) – (1) 1000x – 100x = 2357.7777… – 235.7777… Since both have same digits after decimal, we can cancel them 900x = 2357 – 235 900x = 2122 x = 2122/900 x = 𝟏𝟎𝟔𝟏/𝟒𝟓𝟎 Thus, 2.357 ̅ = 𝟏𝟎𝟔𝟏/𝟒𝟓𝟎
