Appendix 1: Proofs in Mathematics | A1.2

Question 2

Given that the product of two rational numbers is rational, and suppose aa and bb are rationals, what can you conclude about abab?

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Solution

We will use the given property about rational numbers to find the nature of abab.

Step 1 — Identify the numbers

We are given two numbers. Let the first number be a. Let the second number be b. Both a and b are rational numbers.

Step 2 — Apply the given property

The problem states a key fact. The product of two rational numbers is rational. We have two rational numbers, a and b. Their product is ab. So, ab must also be a rational number.

Answer

(i) The product abab is rational.

More questions in A1.2

Q1

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A?

Q2

Given that the product of two rational numbers is rational, and suppose aa and bb are rationals, what can you conclude about abab?

Q3

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and 17\sqrt{17} is irrational, what can we conclude about the decimal expansion of 17\sqrt{17}?

Q4

Given that y=x2+6y = x^2 + 6 and x=−1x = -1, what can we conclude about the value of yy?

Q5

Given that ABCD is a parallelogram and ∠B=80∘\angle B = 80^\circ. What can you conclude about the other angles of the parallelogram?

Q6

Given that PQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

Q7

Given that p\sqrt{p} is irrational for all primes pp and also suppose that 3721 is a prime. Can you conclude that 3721\sqrt{3721} is an irrational number? Is your conclusion correct? Why or why not?

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