Is the Set of Counting Numbers Closed Under Multiplication and/or Division?

Before we got into our first Math Mystery, The Checkerboard Problem, we were discussing when the set of counting numbers was closed. We decided, so far, that the set of counting numbers

• is closed under addition, but 
• is NOT closed under subtraction. 

At the end of the post, I left off with a question about the operations multiplication and division.

Multiplication

The set of counting numbers IS closed under multiplication. The reason for this is because you can write a complete TRUE equation using nothing but counting numbers. Look for as long as you like, you will not find a counterexample to this rule! (If you need to review the definition of "counterexample," go back to the original post on closed sets.)

Division

On the other hand, there are many counterexamples for the operation of division. Let's take a look at one:

While 1 and 2 are both counting numbers, the answer to make the equation true, 0.5 is NOT a counting number!

Since we have found an equation that begins with counting numbers, but can not be completed with them, that means that the set of counting numbers is not closed under division.

To sum it all up, looking only at the four basic operations (addition, subtraction, multiplication and division), here is what we can say about the set of counting numbers.

The set of counting numbers is

• closed under addition.

• NOT closed under subtraction.

• closed under multiplication.

• NOT closed under division.

MORAL OF THE STORY: 

We NEED more numbers if we are going to do more than add and multiply!

 ...stay tuned to see which types of numbers we'll be talking about next!