Prove or disprove: The product of an irrational number and a rational number is irrational.

Answer 1

Suppose not...

Suppose the product of an irrational and rational is RATIONAL...

Then by contradiction x * y = z for irrational x and rationals y and z

Since y and z are rational , y = a/b and z = m/n for integers a,b,m,n with a,b and m all not zero

Substitution:

x ( a/b) = m/n

Multiplying by b/a:

x = (m/n)(b/a) = (mb)/(an)

By closure property of integer multiplication , mb and an are integers

so then x is the quotient of two integers, which makes it rational, a contradiction