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Chapter 5: Problem 47

Use two numbers to show that the integers are not closed with respect to division.

Short Answer

Expert verified
Two integers, for example 10 and 3, can be used to demonstrate that integers are not closed under division since 10 divided by 3 is \( \frac{10}{3} \) or approximately 3.33 which is not an integer.

Step by step solution

01

Choose Two Integers

The first thing to do is to pick two integers. The numbers 10 and 3 are selected as they will clearly illustrate the concept.
02

Perform Division

Now perform the division operation on the chosen integers. So divide 10 by 3. This operation yields a quotient of \( \frac{10}{3} \) or approximately 3.33 when expressed in decimal notation.
03

Explain Why Integers are not Closed under Division

Since the result of dividing 10 by 3 (i.e., approximately 3.33) is not an integer, it can hence be concluded that the set of integers is not closed under the operation of division. This is because the operation of division on two integers has yielded a result that is not in the same set (i.e., not an integer).

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