Question 1 - Think and Reflect (Page 58) - Decimal Expansion of Real Numbers - Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)

Transcript

Question 1 - Think and Reflect (Page 58) The decimal expansion of 𝑝/𝑞 will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why? A terminating decimal is simply a fraction that can be written with a denominator that is a power of 10 (like 10,100,1000 ). For example: 0.15=15/100. The only prime factors of 10 are 2 and 𝟓 (10=2×5). Therefore, any power of 10 (like 100=2^2×5^2 ) is also built exclusively out of 2s and 5s . If our denominator (𝒒) is built strictly out of 2s and 5s Then, we can easily multiply the top and bottom by whatever is missing to perfectly build a power of 10 on the bottom. Example: 𝟏/𝟓=𝟏/𝟓 ×𝟐/𝟐=𝟐/𝟏𝟎=𝟎.𝟐 If your denominator has any other prime factor (like 3, 7, or 11 etc) It is impossible to multiply it by a whole number to get 10,100 , or 1000 . Because it can never give power of 10 as denominator Example: 𝟏/𝟑 or 𝟏/𝟕