Chapter 1.3, Problem 35WE

To determine

To explain:

The integers set is a field or not.

Answer to Problem 35WE

The integers set is not a field.

Explanation of Solution

Definition of property of reciprocal:

A number $x$ and its inverse or multiplicative inverse is the number that gives $1$ when multiplied by $x$ .

The product of a number and its reciprocal number is therefore $1$ . This property is referred to as a reciprocal property.

Thus, from the above definition of the property of reciprocals for any real number $a$ except $0$ , there is a unique real Number $\frac{1}{a}$ such that $a\cdot \frac{1}{a}=1$ .

  $2\cdot \frac{1}{2}=1$

The number $\frac{1}{2}$ is not an integer.

We know that to satisfy the property of reciprocals an integer number is to be multiplied with its reciprocal. For all numbers except $1$ , the reciprocal is a fraction.

Since for all numbers except $1$ the reciprocal is a fraction, the integers set is not a field.