True or false the reciprocal of any rational number is irrat

True or false the reciprocal of any rational number is irrational

Solution

The statement is true

If it\'s a rational number, then by definition it can be written in the form of a/b, where \"a\" and \"b\" are both integers. The only way it can be zero is if a = 0, so since it\'s non-zero, a 0.

The reciprocal of a/b is just b/a. We\'re not dividing by zero, and we have one integer written over another integer. Therefore, by definition, b/a is a rational number too.

Another way of proving this, as some other people did, is to say that a rational number times its reciprocal is 1. A non-zero rational number times an irrational number will always give you back an irrational number. If a non-zero rational number\'s reciprocal was irrational, then multiplying the two together would give you an irrational number. But the product has to be 1, which is not irrational. Therefore, the reciprocal has to be rational.