## Monday, January 28, 2013

### Fractions

To non-mathematicians, math must seem like “just numbers.” You know, the three “R’s,” readin’, ‘ritin’, and ‘rithmatic. Certainly most people are exposed to math beyond simple Arithmetic … if you assume Arithmetic is simple. (Actually, some of the most complex and difficult mathematics that exists works with Arithmetic. It is called “Number Theory,” but we’ll save that for another day.)

Most students were introduced in High School to Algebra where letters substitute for numbers, Geometry where there weren’t any numbers at all, and Trigonometry, which, in my day, meant using a lot of tables to get back to “numbers.” Some have had the joy of Calculus. As the saying goes, “no one flunks the Calculus, they just flunk the Algebra.” That refers to the fact that there’s a lot of Algebra in the Calculus. (You may have noticed we just say “Algebra,” but we say “The Calculus.” That’s just a funny little habit mathematicians have.)

Actually, my experience teaching adults mathematics is that the problems start in Arithmetic. The place where the problems begin is with fractions. If you really understand fractions, then the leap to Algebra isn’t all that hard.

(I would also review “powers,” that’s exponents and logarithms … that’s exponents too. Fractions and exponents were where people who had been out of school for a few years needed a good review.)

So, let’s start with a little review. We will begin with the rules for multiplying fractions. First a reminder of terminology: the part of a fraction on the top is called the “numerator” and the part on the bottom is called the “denominator.”

To multiply two fractions, you just multiply the numerators and then you multiply the denominators and you have the new fraction that is the “product.” For example, 1/3 * 3/4 = 1 * 3 over 3 * 4 or 3/12. You would reduce that to 1/4 … the simplest form.

(I’ll have to write my fractions this way since this word processor doesn’t let me write them above and below a line like I would prefer. Mathematicians and word processors sometimes have to arm wrestle, and — in this case — the word processor won.)

In words you could pronounce the multiply sign as “of.” So the problem is spoken as one-third of three-quarters. Well, one-third of three-quarters is a quarter. Sure, if it was left-over pizza, that’s how you would divide it among three people. Note that somehow, almost mysteriously, multiplication of fractions seems more like division. Well, it is. There is this thing called the "reciprocal." But let’s not think about that right now. We’ve got some other fish to fry. (Hmmmm, fish tacos, why not fish pizza? This is more than a math class. This is a cooking class. And now we’re really cooking!)

It is a little harder adding of two fractions. To add two fractions, they must have the same denominator. Adding 3/8 + 1/8 is easy since they both have 8 for a denominator. The rule is that you add the numerator, and then write that over the “common” denominator.

3/8 + 1/8 = 3 + 1 over the 8, which equals 4/8. Of course, now you should do some simplification where you convert the answer to the “lowest denominator.” For example, 4/8 = 2/4 = 1/2. Now you are at the lowest denominator.

Let’s do another. 3/8 + 5/8. That’s 8/8 = 4/4 = 2/2 = 1. Remember, when the numerator = denominator, that’s one. Are these rules coming back to you?

Now, in math, just as in real life, once you figure out the easy cases, they throw the hard ones at you. What if the denominators are different? Well, they you convert the fractions until they have the same denominator … a “common” denominator.

Let’s add 3/8 + 1/4. You probably notice, with this simple problem, that the “4” in the second fraction is a factor of the “8” in the first fraction. So what we could do is multiply the 4 in the denominator by “2” and we would get “8.” But we must do this without changing the fractions value. You may recall the trick was to multiply both the denominator AND the numerator by 2. So you get 2/2 * 1/4 = 2/8.

Basically, 2/8 is the same as 1/4. What is really happening is that you are multiplying the fraction by another fraction that has the same number in the numerator and the denominator. Recall the rule that any number divided by itself is 1, and 1 times any number is still that number. So, effectively, you are just multiplying the original fraction by one and that doesn’t change anything. I didn’t lose you, did I? Read it again. Get out a pencil and try for yourself. Now you’ve got it.

Ah ha, just reminded you that a fraction is also a division problem where the numerator is the “dividend” and the denominator is the “divisor.” (The denominator “goes-inta” the numerator, as we use to say in the sticks.)

Remember the pizza example. Multiplying by a fraction is really a form of division. (You may even recall that to divide fractions, you invert and multiply. Slippery little devils aren’t they. Oh yes, it’s all coming back now!)

So, 1/4 is 1 divided by 4 and 2/8 is 2 divided by 8. Now back to our problem.

3/8 + 1/4 = 3/8 + 2/8 = 5/8 and that’s the “final answer.”

Well, that was easy. What if the denominators don’t have any common factors? Suppose it was 3/4 + 1/3?

Well, then you just multiply the first denominator by the second denominator and you multiply the second denominator by the first. That’s actually the general rule that will always work. Of course, to keep from changing the fraction, you have to multiply the numerator by the same amount. Get out your pencil and follow along:

So you multiply the 3/4 by the denominator of the second fraction: 3.

So you get 3/3 * 3/4 = 9/12.

Now multiply the second fraction by the denominator from the first: 4.

That’s 4/4 * 1/3 = 4/12.

Now they’ll both have the same denominator. Works every time. Now you can add:

3/4 + 1/3 = 9/12 + 4/12 = 13/12.

That’s actually the lowest form, but some would convert that to a mixed fraction of 1-1/12. (Spoken as one and one-twelfth.)

Now we’re ready for the jump to Algebra. Algebra is just generalized Arithmetic where you work the calculations with letters that stand for “variables.” Basically, a variable can be (almost) any number. The results will be true if you then substitute actual values for the variables. That’s the start of Algebra.

So, if you had to add a/b + c/d, you don’t know what values b and d will have, so you take the basic rule I went over above and multiply the first fraction by the denominator of the second and vice-versa.

That would be

(and I’ll use the Algebra shortcut that you don’t need the multiply sign (*), you can just write the letters next to each other to signify multiplication)

d/d * a/b = ad/bd and b/b * c/d = cb/bd

(Notice the order of multiplication doesn’t matter: b * d = d * b. That’s called “commutativity.” Another rule of algebra.)

So the addition becomes ad/bd + cb/bd = (ad + cb) / bd. (I added the parentheses because this editor won’t let me write the fractions as above and below the line like I would prefer.)

I’ll try:
bd

Sometimes we used Algebra to create "formulas" that can then be used to simply solve problems. That last formula is the solution for adding two fractions, no matter what the denominators are. It is the first fraction numerator (a) times the second denominator (d), plus the second fraction numerator (c) times the first fraction denominator (b). Then all that is divided by the product of the two denominators. So this formula is the "general solution" for adding two fractions. Program it into your computer, and you can forget all the Arithmetic you ever knew.

So now you may have a better understanding of how Algebra can be considered generalized Arithmetic. Also notice that it is true that most students begin to have trouble with math at fractions. If you really know and understand fractions well, then you have a great advantage when you start studying Algebra. And good Algebra skills make Calculus easy — just a tip from the old master. Now wasn’t that fun? Who says math is hard?!!?