Using the Closure Property to Divide Whole Numbers & Integers

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

A mathematical operation like division can be performed on two numbers. In this lesson, we explore what happens when we divide whole numbers and when we divide integers. In particular, we look at the property of closure under division for these two sets of numbers.

Closure Property

A nice way to organize numbers is to place them in a spreadsheet. Accountants often find spreadsheets very useful.

A spreadsheet.
A spreadsheet.

We will use the idea of a spreadsheet to explore the property of closure under division.

Set of Whole Numbers and Closure Under Division

Whole numbers are all the positive numbers including zero but not including fractions and decimals. Some whole numbers are 0, 1, 2, 3, etc. Some numbers that are not whole numbers include -130, -4, 2/3, 3/2, .5, etc. Remember, fractions, decimals and negative numbers are not whole numbers!

To use the spreadsheet, we write some whole numbers in the first column and the same numbers in the first row. The numbers don't matter. The numbers selected are your choice, provided they are whole numbers. The symbol for division is written in the top-left corner. A spread sheet with the numbers 2, 3 and 4 looks like.

Spreadsheet of whole numbers.
Partial table whole numbers.

Take a number from the first column, divide by a number from the first row and enter the result in the spreadsheet. 2 divided by 2 is 1; 3 divided by 3 is 1 and 4 divided by 4 is 1. 2 divided by 3 is 2/3. Now, we fill in the rest of the spreadsheet.

Spreadsheet of whole numbers filled in.
Table whole numbers.

The whole numbers are closed under division if the numbers in the results are all whole numbers. Even if one entry is not a whole number, we don't have closure. There are fractions in our results! Fractions are not whole numbers! We see that whole numbers are not closed under division. Is it necessary to fill in the entire spreadsheet? No, as soon as we find one result is not a whole number we can stop. Do we have to try all the whole numbers? No, there are an infinite number of whole numbers. We try a few, and check the results. If we are not convinced, we try some more whole numbers.

What about the number zero? Good question! Zero was intentionally left out because 0 divided by any number is 0 but a number divided by itself is 1. Zero divided by zero is undefined. Is this a problem for determining closure? If we already have at least one case where the result of dividing a whole number by another is not a whole number then we don't need to be concerned about zero divided by zero. So far our accounting is quite good. Let's continue with another set.

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