Q2 of 17 Page 181

Prove that the product of any irrational number and non-zero rational number is an irrational number.

Let us take an irrational number ‘k’ and a non-zero rational number ‘’ where (a,b≠ 0).

Assume that the product of an irrational number and non-zero rational number is an rational number.


Therefore, let


where c/d is another rational number.





⇒


We can say that (bc) and (ad) are also integers with (ad ≠0).


So, bc/ad is a fraction with integer values in the numerator and denominator (denominator not zero) making it a rational number.


This is a contradiction to the fact that ‘k’ as an irrational number.


So, the assumption is wrong.


Therefore, the product of any irrational number and non-zero rational number is an irrational number.


More from this chapter

All 17 →
1

Using the common form of rational numbers, prove that the sum, difference, product and quotient of any two rational numbers is again a rational number.

 3

Give an example of two different irrational numbers whose product is a rational number.

 1

Find the distance between the two points on the number line, denoted by each pair of numbers given below:

i) 1, –5



2

Find the midpoint of each pair of points in the first problem.