Question 15 - End-of-Chapter Exercises - Chapter 3 Class 9 - The World of Numbers (Ganita Manjari I)

Transcript

Question 15 Show that the rational number ((𝑎+𝑏))/(2 ) lies between the rational numbers a and b. This is asking us to prove that the average of two numbers will always sit exactly between them on the number line. We can prove this using a little bit of algebral Let 𝑎 and 𝑏 be two different rational numbers. One of them has to be smaller than the other, so let's assume 𝒂<𝒃. Now, taking our starting inequality: 𝑎<𝑏 Adding 𝒂 to both sides of the inequality: 𝒂+𝒂<𝒂+𝒃 2𝑎<𝑎+𝑏■(@) Dividing both sides by 2 𝒂<(𝒂 + 𝒃)/𝟐 This proves the average is strictly greater than the smaller number). Now, again with our starting inequality: 𝑎<𝑏 This time, adding 𝒃 to both sides 𝒂+𝒃<𝒃+𝒃 ■(@𝑎+𝑏<2𝑏) Divide both sides by 2 : (𝒂 + 𝒃)/𝟐<𝒃 This proves the average is strictly less than the larger number From (1) and (2), we can write 𝒂<(𝒂+𝒃)/𝟐<𝒃 Thus, (𝑎 + 𝑏)/2 always lies strictly between 𝑎 and 𝑏. Hence proved