Chapter 8.6, Problem 53E

To determine

Whether the given statement is always, sometimes or never true.

Answer to Problem 53E

The given statement is never true.

Explanation of Solution

Given:

The given statement is:

An integer is an irrational number.

Calculation:

An integer can be zero, a positive number or a negative number.

For example:

  $\{-3,-2,-1,0,1,2,3,\mathrm{....}\}$

This can also be written as:

  $\{-\frac{3}{1},-\frac{2}{1},-\frac{1}{1},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1},\mathrm{....}\}$

A rational number can be expressed as a ratio such as $\frac{p}{q}$ , where p and q are integers and q is not equal to 0.

For example:

  $\{-\frac{3}{2},-\frac{4}{3},\frac{1}{4},\frac{0}{1},\frac{2}{13},\mathrm{....}\}$

Therefore, an integer is always a rational number.

Also, any real number that is not a rational number is an irrational number.

Definition of real number states that all numbers that are rational, irrational or integers are real numbers.

Therefore, an integer is never an irrational number.

The given statement is never true.