Last week, in Number Sense 009, we defined what we meant by larger than and smaller than on a number line, and used those definitions to prove that two numbers on different marks are different numbers. This week we are going to take a look at subtraction. We took our first look at subtraction in Number Sense 006, demonstrating how it worked on a number line, and then, in Number Sense 007, saw how number lines showed how “Fact Families” for addition and subtraction worked.
One of the things we saw was that we could subtract a large number (6) from a small number (4)
but, when we did this, we ran into uncharted territory on our number line. Before we move on to what these new numbers are, I'd like to take a brief detour to introduce an idea that we will use, partly to explain this phenomenon, and partly because we will see it again later.
The idea is that an operation, such as addition, subtraction, multiplication or division, can be “open” or “closed” for a type of number.
For example, suppose we group our whole numbers into odd numbers and even numbers. On a number line, it would look like this:
An operation (such as addition) is “closed” in a group of numbers if, when you do that operation on members of the group, you get another member of the group.
Think of a party in a very swanky hotel, and everyone who is anyone is there (bunch of snobs). So, the group is “people at the party” and the operation is “name dropping.” When you eavesdrop on any conversation between two people at the party, and someone “drops a name” then the operation “name dropping” is closed if the person mentioned is at the party. Since everyone who is anyone is at this party, we could close the doors and never hear a name of someone (who is nobody) outside the group. So name dropping in this particular snobby crowd is closed. Only people in the in-crowd get mentioned, and the entire in-crowd is at the party.
Let's look at even numbers and addition.
We take two even numbers and add them 2 + 4 = 6. The sum is another even number. Let's try again: 4 + 6 = 10. The sum is another even number!
We can do this as much as we like, the sum of two even numbers will always be another even number. If we had a definition for even numbers, we could prove it, but we haven't defined even numbers yet. But that's not the point. The point is that even numbers are closed under addition.
On the other hand, odd numbers are not closed under addition. A single example will show this: 1 + 3 = not an odd number!
If we look at two Odd number lines, and add 1 and 3, we can see they add up to a number that is not labeled on the odd number line. (Now, we do know that numbers that are not odd are even, and we can peek at our even number line to find the answer is 4, but the point is that we have to go elsewhere to find the sum.)
Now, the same thing happened when we tried to subtract. We ran into uncharted territory.
Subtraction is not closed for whole numbers. So we need to expand our idea of number, add some more labels to our number line so we know what those new numbers are.
Imagine going backwards for a bit. Suppose we started only knowing odd numbers. We would quickly discover that we couldn't add if all we had were odd numbers, because none of our sums would be odd. To solve this problem, we would have to expand our idea of numbers to include even numbers. Then addition would be closed under our new odd and even numbers, and we could get answers to our addition questions. And, since we like to make life simple for ourselves, we would call this expanded group “whole numbers” instead of “odd and even numbers” because “whole numbers” is easier to say.
So, we have discovered that subtraction isn't closed for whole numbers, so we have to expand our idea of numbers to include those unlabeled tick marks to the left of zero. The conventional expansion is this:
We call them “negative numbers” and we get them by subtracting one from zero to get -1, then subtracting one from -1 to get -2, then subtracting 1 from -2 to get -3, and so on.
And, instead of calling this larger group of numbers the “whole numbers and negative numbers” we will simply call them “integers.”
Have fun in the comments : )