Question

If the product of two given real numbers is a non-zero rational number, show that the numbers are either both rational or irrational

Answer 1

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Answer 2

Answer:

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Step-by-step explanation:

Let us take two real numbers, say a and b.

Now, ab = c

Where, c is a non-zero rational number.

There can be three cases:

(1) When both a and b are rational

(2) When both a and b are irrational.

(3) When a is rational and b is irrational.

(4) When a in irrational and b is rational. As we know that product of two rational numbers is always rational, condition (1) gives the product as a rational number

let us take two irrational numbers, say

√2

and

√2x3√/2=3x2=6

As, we can see that the product of the above two irrational numbers is a rational number, so condition (2) also gives the product as a rational number.

Now, as product of a rational number and an irrational number is always an irrational number, so condition (3) and (4)

never gives the product as a rational number.

Hence, product of two real numbers is always a non-zero rational number, only if both the numbers are either rational or

irrational.