Astudent memorized the statement the product of any rational number and any irrational number is irrational. label this statement as true or false, and explain your reasoning.

a) true. for example, 3 ·5 is irrational, 7 · 2 3 is irrational, and (π)·(−4) is irrational.

b) false. suppose an irrational times a rational = rational, then irrational = rational rational . this means irrational = rational, which is a contradiction. thus an irrational times a rational = rational.

c) true. suppose an irrational times a rational = rational, then irrational = rational rational . this means irrational = rational, which is a contradiction. thus an irrational times a rational = irrational.

d) false. the rational number cannot be zero. suppose an irrational times a nonzero rational = rational, then irrational = rational nonzero rational . this means irrational = rational, which is a contradiction. thus an irrational times a nonzero rational = irrational.