# Is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis.

a rational.

b. irrational.

step-by-step explanation:

a. the 2 numbers are rational so we can write them as a/b and c/d where a, b c and d are integers (not = 0.)

a/b * c/d = ac / bd.

by the closure property of integers , the product of 2 integers must also be an integer so we can write ac/bd = e/f where e and f are integers so the product is rational.

thus the product of 2 rational numbers is rational,

b. the product of a nonzero rational and an irrational is irrational.

let a/b be the rational number and x the irrational.

let us assume that a rational times an irrational number gives a rational number.

so we assume multiplying a/b * x = m/n where m and n are integers.

making x the subject my multiplying both sides by b/a we get:

x = mb/an which is rational. because m, b , a and n are integers

but we assumed x is irrational but we have proved it rational. so that is our contradiction.

so our assumption was wrong and rational * irrational must be irrational.

c. the rational cannot be zero because when we multiply by zero the answer is zero. it gives a different result to b.