Write a proof by contradiction for S as a free response. Original Question Prompt in Description.

Question

Let S be the statement: The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number x is 1/x.)

Q: Which of the following is a negation for S?

A: There is an irrational number whose reciprocal is not irrational.

Answer 1

Proof by contradiction:

given x is irrational. Suppose the reciprocal of x

is rational.

Then reciprocal of x is a/b, where a and b are integers.

Specifically, 1/x = a/b

cross multiplying b = ax

b/a = x

so then x is the quotient of two integers, which

makes it rational, a contradiction.

The original statement is: For every irrational number, the reciprocal is irrational.

The NEGATION of the statement is: There exists an irrational number such that the reciprocal

is rational