If a and b are rational numbers, then ab is a rational number.?

Question

True or False?

Why?

Answer 1

If a is rational, we can write it as a fraction

a=a1/a2,

and similarly

b=b1/b2.

Then we know how to multiply fractions, so

ab=(a1b1)/(a2b2)

You can obviously see that this is a fraction, so ab is rational too.

Answer 2

True.

Suppose a and b are rational numbers.

Then we can write a = m/n and b = p/q, where m, n, p, and q are integers and n and q are nonzero.

Then ab = (mp) / (nq).

The product of integers is an integer, and the product of nonzero integers is nonzero, so ab is also rational, because it is the quotient of two integers, the second of which is nonzero.