Question

1. If pq = r, p, q and r being real numbers, then
show that either p and q both are rational or p
and g both are irrational.​

Answer 1

We know " The real numbers include all the rational numbers, such as the integer −5 , 3 ,0 , 1, -2 ..and the fraction 12 , 34 .... and all the irrational numbers ( that have non terminating and non recursive decimal digits . )

So,

We have

pq = r , And p , q and r are real numbers .

So when we take p and as rational number ,

So,

pq = r is real numbers ( As we know all rationa numbers are real numbers )

And we know multiplication of any two rational number gives us a rational number . As :

p = 2 , q = 34

So,

r = 2 ( 34 )

r = 32 that is in form of pq , and q≠ 0 , so that is a rational number and we know that any rational number is a real number , so

Any value of p and q as rational number gives r = real number.

Now

We take p and q both are irrational number .

we know that multiplication of two irrational number gives us a irrational number , And any irrational number is a real number . As

p = 2√ , q = 3√ so,

r = ( 2√ ) ( 3√ )

r = 6√ that is irrational number , and we know all irrational number can be real number , so

And if we use same irrational number than they gives us rational number, As

P = 2√ , q = 2√ So,

r = ( 2√ ) ( 2√ )

r = 4√

r = 2

r is a real number when p and q are irrational number.