Proofs

[INF3RN0666]

Hi

Can anyone help me prove if a and b are rational numbers, then a(b^2) - 5 is a rational number.

[krtxmrtz]

If a and b are rational, then they can be written as the quotient of 2 integer numbers, for instance:

a = p/q
b = v/w

where p,q,v,w belong to Z (the set of integer numbers)

So, then

ab² - 5 = (p/q)(v/w)² - 5 = (pv² - 5qw²) / (qw²)

which in its turn is the quotient of integer numbers since only sums, substractions and products are involved in both the numerator and the denominator.

[superbovine]

Not to be picky would it still be rational if q=w=0? or q=0 or w=0?

[Logophobic]

Neither q nor w can be zero, because a and b are given to be rational.

[krtxmrtz]

If a=0 and b=0 then

ab² - 5 = - 5

which is rational.

And 0 itself is rational. It can be expressed as a quotient:

0 = 0 / A

where A is any integer number different from 0.