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!

What Makes a Set Closed?

Two of the questions I asked at the end of the last post on the set of Counting Numbers was

• What types of things CAN we do if we only have Counting Numbers?
• When do we need MORE numbers than just the Counting Numbers?

When mathematicians start asking these questions, they start talking about whether or not a set of numbers is closed. 

Let's begin with a magic bucket that holds ALL of the counting numbers in it:

When you reach in this bucket, you can grab out any counting number you need!

This bucket now represents the set of counting numbers. (If you need a review, now is a good time to go back to the last post all about the Counting Numbers!)

A Daze-y Definition

A set is just a collection of things. In math, we give certain sets names (like the set of counting numbers) to describe a collection that have some common characteristics. We use set brackets, { }, to hold them all together.

 *Reminder: The set of Counting numbers (also called Natural numbers) contains all non-decimal numbers greater than zero, or {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...}.

 Let's grab our magic bucket and do some math. Don't freak out! We'll start with addition. Here is the question we need to answer to decide whether or not the set of counting number is closed  under addition:

Can we create and answer ANY addition problem using only the numbers in our bucket?

I've got one! 1 + 1 = 2. All of the numbers are counting numbers. Also 3 + 5 = 8 works

Since I found two problems that work already (1 + 1 = 2 and 3 + 5 = 8), your job is to try to think of a counterexample.

Another Daze-y Definition

 A counterexample is one that breaks the rule that we think is true. If you can find even just one situation that breaks the rule, then we say the rule is NOT TRUE.

Can you think of an example that doesn't work?  Here are some tricks to try that usually reveal counter examples:

• Change the order of the numbers in the problem. (Don't always put bigger numbers first!)
• Use numbers more than once. (I did that in the 1 + 1 problem - will that always work?)
• Try using extreme numbers - maybe a really big number with a really small number - does it still work?? You can use a calculator if you need to! (I actually added a really basic one to the sidebar, if you don't have one of your own.)

OK. **Spoiler Alert** I am about to tell you what you should have found, so if you want to keep working on it, don't read this part yet. The fact is, no matter how hard you try, no matter what you do with your numbers, you will not find a counterexample for this. This is pretty cool, this means that the set of counting numbers is CLOSED under addition.

Now don't get too excited, that does not necessarily mean it is closed under every operation! We still don't know about subtraction, multiplication or division. Let's look at addition's evil twin: SUBTRACTION.

Can we create and answer ANY subtraction problem using only the numbers in our bucket?

Some equations that work 152 - 48 = 104, or, if you like smaller numbers 6 - 2 = 4.

 What about any counterexamples? Think about the tricks we can use to find the hidden counterexamples... Can you think of any? Did you already know it before we even got to subtraction?!

Here's ONE counterexample:

Remember: Zero is NOT a counting number! That means it is NOT in our bucket!! Uh-oh!

There are actually a whole bunch of counterexamples for this, so that means the set of counting numbers is NOT closed under subtraction.

Do you think you have an idea of what it means to be closed (in mathematics, anyway)? Here is my Daze-y definition for it.

Last Daze-y Definition in This Post

A set of numbers is CLOSED UNDER AN OPERATION when a complete equation (that means even the ANSWER) of that operation can be created using ONLY numbers from that set. Here is the picture that always pops into my mind when I think about CLOSED SETS:

I think, in closed sets I can keep going around and around to my "magic bucket" fill with the numbers in my set to make up my equations, but when a set is not closed, at some point, when I try to go back to the "bucket" I can't find what I am looking for (that's what causes the question mark)!

This is a pretty abstract concept to get, so before we move on, I think it is best if you practice on your own to see if this is something you really understand.

 There are two more operations that WE didn't play around with yet - multiplication and division. Why don't you use the calculator (if you need to), or a bunch of paper to figure out whether or not counting numbers are closed under those operations?

Pictures to ponder...

Picture 1: Multiplication

Is the set of counting numbers closed under MULTIPLICATION? (Notice, we are using the floating dot to represent multiplication, instead of an "x".)

Picture 2: Division

Is the set of counting numbers closed under DIVISION?

 

I can't wait to hear what you find out! I look forward to your responses (you can put them right in the comments section). If you decide that one of these operations is closed, tell us the counterexample you came up with!

Have fun!

The Counting Numbers... Naturally

The first numbers you ever began to discover were those that came naturally to you, the ones that allowed to realize that there were different amounts, or quantities, of things in the world. As your mind started to see these patterns emerge, someone in your life, probably a parent started to teach you about counting. If you were lucky, you were even taught how to do this in different languages. As you put together the patterns in your head that came naturally to you, with the numbers your loved ones taught you were used for counting, you probably became a counting monster!

I always find it fun to be around a child that just learned to count!

During this time you can learn about a lot of numbers and one fun activity is to see how high you can go, but (here's the really cool part) there is no end! You can go higher forever!

These numbers that you first learn as a child and then continue to use for the rest of your life have a name. They are called (this is a really great name if you ask me) the COUNTING NUMBERS. They are also known as the NATURAL NUMBERS. Both of these names make perfect sense to me because they are the numbers that we kind of already know naturally, but they are also the numbers we use for counting. So what numbers are they, exactly?


A Daze-y Definition

The set of Counting Numbers, or Natural Numbers, includes all of the non-decimal numbers (known as whole numbers - we'll get to that in a later post) greater than zero.  See the symbol and set below:

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...}

 It is very important to note that ZERO is NOT A COUNTING NUMBER. Zero is a very complicated number (even though it is all about nothing!), so it is something we will address in a later post. Just remember that the counting numbers begin with one and go up from there.

Questions to ponder... and then COMMENT on:

• What's the highest counting number you know?
• How many languages do you know the counting numbers in?
• What things can we do in life with counting numbers?
• What things in life need MORE numbers than just the counting numbers? (That seems crazy... I'm pretty sure I said these numbers would go on forever!)
• Do you remember who first taught you your counting numbers? (Have you thanked them yet???)
• Are you currently being tortured by a counting monster (aka - a child who just learned to count and, therefore must count EVERYTHING in sight)? 

Art created using the Drawing Online website.