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© Copyright 2000, Jim Loy
Before you studied algebra, you knew plenty of algebra. While you spent many years learning addition and subtraction and fractions, you were learning other things.
Let's look at 7 cupcakes and 7 apples. What do they have
in common, from a mathematical point of view? 7 of course, number is one of the
obvious and important features of each collection (set). Well, does it matter
what order I count the apples? No, number (7 in this case) is so basic a
feature that it never changes, no matter how we do the counting. This is a
basic rule of algebra: It doesn't matter what order you count things.
Skipping to flowers, so we won't get too hungry,
addition is a more interesting way to count. Here we see 7+4 flowers. We can
just count all of the flowers, to get the sum (11). But we know some
arithmetic; we know that 7+4=11. So we leap to the answer. Does it matter in
which order we add (4+7 instead of 7+4)? Of course not, we know from experience
that 4+7=7+4. We always get the same answer, no matter what order we add.
Besides, addition is really counting, and we already decided that it doesn't
matter in what order we count. Well, that is the commutative law of algebra.
You knew the law; you didn't want to know the name.
We have the same law (commutative law) in
multiplication. Well, multiplication is just repeated addition (and so is
another form of counting). You actually knew that, although you may not have
actually thought about it. Here we have 4x3 trees, which is the same as 3x4
trees, which is the same as 12 trees. It doesn't matter whether you count the
trees or do the multiplication short cut. You already knew that multiplication
was commutative.
Subtraction and division are inverse (opposite) operations. Subtraction is the inverse of addition and division is the inverse of multiplication. You probably knew that. And you know that, in these cases, order does matter: 4-3 is not the same as 3-4, and 4/3 is not the same as 3/4. Subtraction works like this: "What number do I add to 4 to get 7?" There is an unknown ("what number") in that sentence. And we can solve for it; the answer is 3. That was algebra, mathematics using unknowns. You were doing algebra whenever you subtracted, same with division.
You also knew all of the other basic laws of algebra (except maybe the distributive law, below). Let me list them:
Didn't you know these already? You just didn't know their names. That is the hard part. But I guess we have to be able to identify them? By the way, factoring an expression like ab+ac=a(b+c) is the distributive law backwards.