lET X BE RATIONAL AND Y BE IRRATIONAL.iS XY NECESSARILY IRRATIONAL?GIVE REASONS AND JUSTIFY WITH EXAMPLE?

Asked by raoss1967 | 6th May, 2015, 12:48: PM

Expert Answer:

Yes. The product of one rational and one irrational number is an irrational number.
It can be proved by method of contradiction.
Let X be a rational number such that X equals p over q comma space q not equal to 0
Y is an irrational number such that XY=R , a rational number, where R equals a over b comma space b not equal to 0
So we get
p over q Y equals a over b rightwards double arrow Y equals a over b cross times q over p equals fraction numerator a q over denominator b p end fraction comma space a space r a t i o n a l space n u m b e r
The above conclusion is a contradiction, since Y is an irrational number.
Hence, our initial assumption that XY is rational is not true. Hence, XY is irrational. Proved.
 
For example, suppose X=3 and Y=√2 . XY=3√2 which is an irrational number.

Answered by satyajit samal | 7th May, 2015, 04:48: PM