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Playing by the Rules 

In order to play the game of algebra properly, you have to play by the rules. The rules define how the numbers are calculated -- how they're added, subtracted, multiplied, and divided.

The rules are given in general form, just as they are in tennis, or golf, or other games, because they have to apply to all the numbers, not just selected ones. If you had to say how each and every number individually had to be handled, you'd be exhausted before you even got started.

Mathematicians are nothing if not abstractionists. Or generalists. It's their nature to generalize concepts. And they like to pack as many ideas and as much information as they can into as few words and symbols as possible. So the rules are given in general form with a lot said and little spoken. It's up to you to fill in the details. You have to become aware of the content, and nobody can tell you how to do that.

One of the rules says how numbers can be added, how to go about it. It orders the operation. It says that, given any two numbers, you can add either one of them to the other and get the same answer. In symbols, the rule is simply:

x + y = y + x.

Since the letters stand for numbers, generally, this means, for example, that 2 + 5 is the same as 5 + 2, as if you didn't already know! You can start with 2 and add 5, or start with 5 and add 2. Big deal!

A similar rule tells you how you can multiply any two numbers. So, as you probably already guessed, it defines the order of multiplication for any two numbers. That is:

xy = yx.

So of course, 2 times 5, for example, is the same as 5 times 2. Hey, don't smirk! In other places the rule doesn't work!

Besides these so-called commutative laws, or properties, you also have a rule that says how numbers can be added and multiplied in combination, like multiplying the sum of 3 + 2 by 5 to get 25. The rule shows what can be done. This distributive law is as follows:

(x + y)z = xz + yz.

It says that you get the result in two ways:

1. You can first add the numbers to be added, namely x and y, and multiply the result by the multiplier, z.
2. You can first multiply each of the numbers to be added and then add the results of the separate multiplications.

So, if x = 3, y = 2, and z = 5, you can add 3 and 2 to get 5 and multiply this by 5 to get 25 -- that's the left side of the equality. Or you can multiply 2 by 5 to get 10 and multiply 3 by 5 to get 15 and add 10 and 15 to get 25 -- that's the right side. And you see the same result.

There are also rules defining the use of two special numbers, namely 1 and 0.

One of the rules -- the rule of identity for multiplication -- says you can multiply any number by the number, 1, without changing the value. That is:

1x = x.

But when adding, 1 does not act like 1 in multiplication. You can not say that x + 1 = x. The operation is forbidden.

So again big deal!

I know it's obvious, but the thing is, it's obvious only after you know the rule -- even if you've never before seen it written out.

What isn't so obvious is that any operation specified by a rule is legitimate, but anything else is absolutely forbidden. There are no exceptions.

The other rule is obvious, too, if you already know your numbers. It says: When zero is added to any number, you get the original number as a result. Adding nothing doesn't change anything. That is, x + 0 is identical to x, or

x + 0 = x.

The rule for zero in multiplication is straightforward: Any number multiplied by 0 reduces to 0.

0x = 0.

That is, if you multiply x by 0, you get 0. So, for example, 4 times 0 equals 0.  If you multiply something by nothing, what happens is nothing!

On the other hand, zero can't be used as a divisor. For example, 5/0 isn't just hard to define -- it's meaningless. It doesn't fulfill a rule. So this operation, too, is forbidden.

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In case you were wondering, here's where information packing shows up big, because the rules apply to more complex forms of the same thing. For instance, take the expression:

x + y + z = z + x + y.

If this is true, it should satisfy the simple commutation rule of addition.

To understand why it does, think of x + y, for instance, as just another one of the numbers in the big bag of numbers. You can call this the number w. In other words, you would have:

w = x + y.

Then, for instance, for:

x + y + z = z + x + y

you can substitute for x + y and write

w + z = z + w,

which satisfies the rule.

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