The Realm of Integers (Types of Numbers #3)

To kick off the tower of numbers, so to speak, the last article in this series discussed the “basic” numbers – mainly focusing on the positive whole numbers but also including zero. Now, we haven’t quite explored every aspect of the number zero yet, and we ran into problems with subtraction within the so-called “natural numbers”. The goal of our discussion now is to solve our subtraction problem.

Overview of Previous Layer of Numbers (Natural Numbers)

The so-called natural numbers are often denoted by \mathbb{N} by mathematicians. For my purposes, I will say that \mathbb{N} = \{ 0, 1, 2, 3, \dots \} is the full list of positive whole numbers along with zero. (Sidenote: The font here is called “blackboard bold” and, as its name suggests, began to exist because mathematicians needed a way to write a letter in boldface with chalk on a blackboard.) With positive numbers in \mathbb{N} like 1, 2, 3, and so on, we know how to add and multiply already. We also know that adding by zero is the same as “doing nothing”. We have also learned in our schooling how to multiply by zero, but this isn’t actually very obvious, so we will pretend we don’t know how to do that and talk about multiplying by zero later on.

What the Natural Numbers are Missing

Notice what we know and don’t know how to do “easily” with natural numbers: we know how to multiply any two positive numbers, and we know how to add any two numbers. Isn’t is sort of odd that subtraction is missing completely? Plus, we don’t quite know how to multiply by zero. At least, it isn’t obvious how to multiply by zero. As simple as the natural numbers are, it is annoying that such a simple idea as subtraction just doesn’t work here. There is, for example, no natural number that is equal to 2 - 5. We know that 5 - 2 = 3, but we do not have the foggiest idea yet of what 2 - 5 might be. Our goal now will be to reduce the number of things we don’t know how to do.

How Integers Fill in the Gap

The solution, which will might sound weird, is just to define more numbers! In other words, just because 2 - 5 isn’t a natural number doesn’t mean it isn’t a number at all. We will call it an integer (more specifically a negative number) when all is said and done. This new world of “integers” will eventually be called \mathbb{Z} (which comes from the German world zahlen which means “numbers”).

But how can you just invent new numbers? We don’t even know how to add these new numbers. How can I add a “new number” to and “old number”? That doesn’t make a lot of sense… at least not yet. This is a fair criticism, and therefore in order to call these things genuine numbers we have to learn how to combine them with the old numbers. To do this, we need some kind of motivation for making sense of these new numbers in light of the old ones.

The way that negative numbers initially came to make sense to people was through monetary calculations. People knew that when you add a debt, you lose money, and when you subtract a debt, you gain money. This is the exact opposite of how money normally works, since if you add money to an account you now have more money, not less. Over time, people began to realize that it is possible to think of debt as a “different kind of number” – or at least they would pretend it was in order to make their calculations faster. The fundamental rule for making these “debt numbers” work is the following:

If I add a debt of x dollars to my account and then add x dollars to my account, I have neither gained nor lost any money. That is, the net affect on my account was to add 0 dollars.

Let’s write -x as the symbol to stand in for a debt of x dollars. Then what this sentence tells us is that x + (-x) = 0. This should look familiar, as in the world of natural numbers we already know that x - x = 0. So, what we have really done is this:

We have discovered a way to think about an equation we already knew about from a different angle.

Instead of thinking about subtraction (as in the equation x - x = 0) we are now thinking about addition with our “new” negative numbers, as in x + (-x) = 0. This is the definition of adding negative numbers! For example now, we can say that 2 - 5 = 2 + (-5). Since we can determine that (2 - 5) + 3 = 0, we can now actually conclude that in our new number system, 2 - 5 = -3!

To a reader who already has studied negative numbers before, this is relatively straightforward. But remember, we are trying to think about this as if we had never come up with the idea of negative numbers before. I am, of course, moving quite a lot quicker than a teacher would when they teach negative numbers for the first time. I can move faster since this isn’t a classroom, and also because I know most people reading this are probably old enough that they learned about negative numbers sometime in the past. The point I want to make, and which will be made much more explicitly soon, is the concept of extending one world of numbers into another world of numbers.

Making Multiplication Work

Now, the idea of multiplication is defined in the context of positive whole numbers. Multiplication is repeated addition. It makes sense how to calculate 3 \times 5, for instance, because I know what it means to add 3 to itself 5 times. But what can we say now about 0 and negative numbers? What do 2 \times 0, or -3 \times 2, or -7 \times -4 even mean? The idea of repeated addition no longer works as clearly in these situations. So how do we make sense of such multiplications?

Here lies the single most important thing to remember when extending a realm of numbers to a new realm of numbers. Our number one goal must always be this:

Make all the rules that work in the smaller realm of numbers also work in the bigger realm of numbers.

This concept is what we will use to make sense of multiplying by zero and by negative numbers. We have three problems we now have to solve.

Problem 1: Zero times any number

For any positive whole numbers, we have the distributive property a(b+c) = ab + ac. Using our main principle, we would like this to still be true if we make some of these numbers equal to zero instead of a positive number. To do this, we need to mess around with different situations until we learn something useful.

You might first try making a = 0, and then you’d end up with the equation 0 * (b+c) = 0*b + 0*c, but we still don’t really know what any of the three terms are, and we can’t really cancel anything out, so this wasn’t a very useful attempt to understand how multiplying by zero works.

Perhaps eventually you decide to try b = c = 0, so then our equation is a*(0+0) = a*0 + a*0 by the distributive property. Well, this is actually quite helpful since we already know that 0 + 0 = 0. If we use this fact we know, then we have a new equation a*0 = a*0 + a*0. Since a*0 is just a number, we can subtract it from both sides of the equation (since we are allowed to subtract now!) and therefore a*0 = a*0 - a*0. Since any number minus itself is equal to 0, we conclude that a*0 = 0. Since multiplication is commutative (that is, a*b = b*a) we also know now that 0 * a = 0.

This should be very interesting if you’ve never seen it before. Notice what I have done – I used formulas like a*(b+c) = a*b + a*c that are true for positive integers, I then assumed that it would also be true for other numbers too, and I used that to figure out how to correctly multiply by zero.

We will now use the same idea – along with our new knowledge that 0 * a = 0, to discover how to correctly multiply by negative numbers.

Problem 2: Negative number times positive number

We now want to know how to multiply (-a) * b. To the curious reader, I encourage you now: try to use the distributive law to figure out how to multiply the negative number -a by the positive number b.

The idea is exactly the same as with Problem 1. We are going to make use of the distributive law a(b+c) = ab + ac. If we assume that this law is also true when some of the numbers are negative, one of the things we would learn is that a(b + (-c)) = ab + a(-c). The clever step we will use to discover the solution is by setting b = c. When we do this, we learn that a*(b + (-b)) = a*b + a*(-b). Now, since b + (-b) = 0, we now have learned that a*0 = a*b + a*(-b), and by our solution to the first problem, we now have also learned that 0 = a*b + a*(-b). We are allowed to subtract from both sides, and therefore by subtracting a*b from both sides we arrive at

a*(-b) = -(a*b).

Ok, so now we know how to multiply one positive number by one negative number. But before we move on, there will be an extremely useful observation we can make from this. If I decide to look at the situation when b = 1, then I conclude that a * (-1) = -a. If you’ve learned about negative numbers before, then this should feel quite obvious. Really, all of this would feel obvious. But remember, someone had to figure all this out for the first time for themselves. It wouldn’t have been obvious to them. By going through all this work, we retrace the steps of brilliant mathematicians of the ancient past.

This innocuous equation a * (-1) = -a will help us greatly simplify our last equation.

Problem 3: Negative number times negative number

The last thing we need to learn how to do is how to multiply (-a) * (-b). We can massively simplify this task by using the commutative property ab = ba along with the rule a * (-1) = -a that we just developed. Using these,

(-a)*(-b) = a * (-1) * b * (-1) = (-1) * (-1) * ab.

Therefore, all we really need to know is how to multiply -1 * -1, then we know everything else. Without walking through all the same steps again, I will show most of the works in one main string of equations, which the reader should slow down to make sure they understand:

0 = -1 * 0 = -1 * (1 - 1) = -1 * (1 + (-1)) = (-1 * 1) + (-1 * -1) = -1 + (-1) * (-1).

Since 0 = -1 + (-1) * (-1), we conclude that (-1)*(-1) = 1. Therefore, (-a)(-b) = ab.

Conclusion

We have now learned how to add, subtract, and multiply together any two integers, any two numbers in the realm of \mathbb{Z}. We have seen how \mathbb{Z} extends the familiar positive whole numbers into a bigger world that, while it is more complicated in some ways, is actually simpler in some other ways. Next time, we will do the same thing with division that we have just done with subtraction, expanding \mathbb{Z} into a new bigger world where division is (almost) always allowed.

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