by Damien Sullivan
5 Dec 2004
(Young Achilles is playing near a road, upon which comes the Tortoise.)
Tortoise: Come away from your playing, Young Achilles! For I, the Great and Polymathic Tortoise, have come to teach you fractions!
Achilles: Why do I need to learn fractions? I will grow up to be the greatest warrior of the Achaeans! People will do my fractions for me!
Tortoise: You may need fractions to cook in the field, to handle your money, and to know that other people are not cheating you if they cook or handle your money for you. Also, wily Odysseus will laugh at you if you don't know your fractions.
Achilles: Wily Odysseus will laugh at me anyway!
Tortoise: He would laugh less if you learned more.
Achilles: But he's wily and learns faster, so as I learn, he learns even more, and I can't catch up. It's a Zeno's Paradox of Ignorance. I may as well play.
Tortoise: THAT is so wrong that you MIGHT be right. You have made my brain hurt, which is a cruel thing to do to a poor old Tortoise of very little brain. But come, let us get to the business our Author intended us for, lest our Reader grow bored and go away, causing us to cease to exist.
Achilles: Very well.
Tortoise: In the fraction 2/3, what do we call the top part?
Achilles: The NUMERATOR.
Tortoise: Good, but why?
Achilles: How should I know?
Tortoise: Chiron your tutor has taught you a large vocabulary; list some words resembling numerator.
Achilles: Enumerate, enumeration... I can't think of more.
Tortoise: Those are enough. Note that those words deal with counting, and in fact all three share a root with the word NUMBER. The numerator is simply a number. Now what is the bottom?
Achilles: The DENOMINATOR. And you'll ask me "Why?", and I won't know, and then you'll ask me to list similar words, so DENOMINATION, as of a currency or a church (not that paper money or churches have been invented yet!) or NOMINATE, as in nominating someone to office, or a NOMINATION, having been nominated.
Tortoise: Very good. And all of those share a root (in Latin, which also hasn't been invented yet), with NAME. The denominator is simply the name of something, its type.
Achilles: So 2/3 is two things, which happen to be called, or of the type, thirds. Two-thirds. 3/5 would be three things, of type fifths, three-fifths. 7/4 would be 7 things of type fourths, seven-fourths.
Achilles: What about 2?
Tortoise: We can re-write that as 2/1, two things of type ones, only we don't say ones, we say WHOLES. Two wholes. Or just two.
(Annoying jingling approaches.)
Achilles: Ooh, the baklava man! I'll be back!
Tortoise: If you have two apples, and you add three apples, what do you have?
Achilles: Five apples, and what does this have to do with fractions?
Tortoise: Wait. And three oranges plus four oranges?
Achilles: Seven oranges.
Tortoise: Now if you have two apples and add three oranges, you have five apples, right?
Achilles: What?! No! You can't add apples and oranges!
Tortoise: Aha! So you already know the first rule of adding fractions: they must be of the same type.
Achilles: Which means they must have the same denominator? What about multiplication?
Tortoise: Yes, and don't worry about that yet. Now, if they ARE of the same type -- have the same denominator -- then you can just add their NUMBERS, or their numerators. So, 2/7 + 3/7 would be?
Achilles: Two-sevenths plus three-sevenths, so two plus three things, of type sevenths, or five-sevenths. 5/7. And I'll try some of my own: 2/3 + 2/3 should be two plus two, of type thirds, or 4/3. I bet you can subtract as well: 5/7 - 2/7 is five minus two, of type sevenths, or 3/7 again.
Tortoise: Perfect! How about 1/4 + 1/4?
Achilles: 2/4. Should I reduce that or something?
Tortoise: You CAN, but you never HAVE to, unless your teacher forces you to, or you have some other need. Sometimes you want to reduce, to have smaller numbers; other times you want to keep the current form. For example, 1/4 + 1/4 + 1/4. It would be silly to get 2/4 + 1/4, then reduce to 1/2 + 1/4, and not be able to keep on adding, when you could have gone on to get 3/4.
As a last question, how about 2/3 + 0/3?
Achilles: Two plus zero... 2/3. So we just add the numerators, across the top -- if the denominators are the same.
Tortoise: Yes. And it works for negative numbers, or even fractions, in the numerators. (-1)/3 + 2/3 = 1/3. (1/4)/3 + (2/4)/3 = (3/4)/3.
Achilles: I'll have to think about that last one. What about adding when the fractions are of different types?
Tortoise: Briefly, you simply convert the types of the fractions until they are the same type, and then add them. But to understand how we do that you will need multiplication, so we will do that first.
Tortoise: Now, recall that multiplication is fast addition. Two times one orange is one orange taken twice, or one orange plus one orange, for two oranges. 2*1 = 1 + 1 = 2. 2*3 = 3 + 3 = 6. 3*2 = 2 + 2 + 2 = 6. (Or we could make a rectangle out of dots, 2 dots along one side and 3 dots along the other, and count the dots.)
Achilles: All right.
Tortoise: So what do you think 2 * 3/5 might be?
Achilles: 3/5 taken twice, or 3/5 + 3/5, or 6/5?
Tortoise: Yes. 3 * 2/5?
Achilles: 2/5 + 2/5 + 2/5, or 6/5 again.
Tortoise: 4 * 1/2?
Achilles: 1/2 + 1/2 + 1/2 + 1/2, or 4/2. It seems that if I multiply by a whole, I can just multiply the numerator by the whole and be done.
Tortoise: Indeed. Now let us look at the bottom; this will take more thought. It might even be a good time for a break... Ready? We have been hauling denominators around all this time without asking what they are. What are these types, where do they come from? They are in fact our fractions. Divide an apple into two equal parts, take one of those parts, and you have one half the apple, or 1/2. Divide a pizza into 6 equal slices, take a slice, and you have 1/6.
Achilles: I know this.
Tortoise: But have you thought it all the way through? 2 divided by 5 is 2/5. What is 2 times 1/5?
Achilles: Also 2/5.
Tortoise: 3 divided by 4 is 3/4. 3 * 1/4?
Achilles: 3/4. Dividing by an integer n is the same as multiplying by 1/n, I see.
Tortoise: Now, take that slice of pizza -- 1/6 of the whole. Cut it in two equal pieces. Do you agree that if each of the 5 other slices had also been cut in two, we would have 12 new slices?
Achilles: Yes, and each is one-twelth, or 1/12.
Tortoise: We divided 1/6 by 2 to get 1/12. Or, (1/6)/2 = 1/12. But as you just said, we could have multiplied by 1/2 instead. So 1/6 * 1/2 = 1/12.
Achilles: We multiply fractions across the top and bottom, separately?
Tortoise: Yes. Or, we multiply the numbers (numerators), and then the types (denominators.) In yet other words: each type represents a division which happened at some point. We multiply all the numbers on the top, and divide that by all the numbers on the bottom, or divide by the product of all those numbers.
Tortoise: 2/3 * 4/5 * 2/7 = (2*4*2)/(3*5*7) = 16/105. Or it could be 16/3, then (16/3)/5, or 16/15, then (16/15)/7, or 16/105. But let me go back to a concrete example, a bigger pizza this time. Cut it into 4 equal pieces, and take two of those, so we have half the pizza, in 2 pieces -- 2/4.
Achilles: Got it.
Tortoise: Now imagine dividing each fourth into 3 equal pieces, so we have six-twelths altogether (6/12), and then take one piece from each set of 3, so we have taken 2/12. 2/4 divided by 3 is (2/4)/3, or 2/12. And 2/4 * 1/3 is 2/12, by the rule we've decided on.
Achilles: By the way, I've never seen notation like (2/4)/3 before.
Tortoise: Indeed. But it seems the only way of representing that I am dividing a fraction by another number in this medium. More common would be writing 2/4, and then a big line underneath that, and a 3 beneath the big line.
Tortoise: Let us go back to the 2 apples and 3 oranges. If we decided to think of them simply as "fruit" instead, then could we add them?
Achilles: Certainly. 5 pieces of fruit.
Tortoise: But we could be less generic. We might say that each apple is an apple-or-orange, and likewise for the oranges, and then say that we have 5 apple-or-oranges.
Achilles: That would be weird, and I'm wondering if that should be apples-or-oranges, but yes, you could do that.
Tortoise: Excellent. Now consider the coins of a country which doesn't exist yet: what is a dime plus a quarter?
Achilles: A dime is 10 cents, and a quarter is 25 cents, so 35 cents. I could do that before you waylaid me!
Tortoise: I thought you didn't know fractions then.
Tortoise: Those coins are fractions. Oh, never mind for now. But as we said before, we need things to be of the same type before we add them, but we can make them the same type, if we're clever. What is 3/4 * 2/2?
Tortoise: Is that the same number? The same amount of pizza, say?
Achilles: It's 3 fourth-size pieces of pizza, versus 6 eighth-size pieces, but if you started with the same pizza, then yes, it'd be the same amount -- just more and smaller slices.
Tortoise: Good. That should make sense. Two divided by two is 1, 2/2 = 1, so I really just multiplied 3/4 by 1.
Achilles: I see that.
Tortoise: Then you're ready for the trick. We want to add 1/2 + 1/3 --
Achilles: Oh! I just remembered Chiron talking about a cross-multiply rule. Can we use that?
Tortoise: We COULD, but instead we will DERIVE it. Instead of pushing symbols around on paper blindly, we will see what is happening (nothing) to the real mathematical entities. To continue, 1/2 + 1/3. Now, 1/2 * 3/3 is still 1/2 REALLY, because we are multiplying by 1, but if we carry through we get 3/6. And 1/3 * 2/2 is 2/6.
Achilles: But now they're the same type, and we can add them! 5/6! But is that right?
Tortoise: Well, imagine half a pizza, or half pie. Now add another third of the whole pizza or pie (not one third of the half you've already visualized, mind you!) Does the resulting total look close to 5/6?
Tortoise: Then that is good enough for now. I'm not trying to teach you to be a modern mathematician, but to be able to use numbers to think about the real world. If you can write things down and quickly get the same answer you would get by slowly measuring and cutting up and counting things out for real, then I'm happy. Well, I do also want you to see WHY, but that doesn't have to be a perfectly rigorous proof.
Achilles: So what's the rule?
Tortoise: Back to the coins. You know the coins have values in cents, but they are also fractions of a dollar. There are 10 dimes in a dollar, so a dime is 1/10 dollars. A quarter is 1/4 (one quarter!) And a cent is 1/100 of a dollar.
Achilles: So I turned the dime into ten cents, or 10/100 of a dollar, and the quarter into 25/100, and added to get 35/100 of a dollar, or 35 cents.
Tortoise: Yes. And going one more step, I was asking you to add 1/10 + 1/4 (of a dollar.) Which could turn into 1/10 * 10/10 + 1/4 * 25/25 = 10/100 + 25/100 = 35/100. But there other ways.
Achilles: Which are?
Tortoise: Same method, different numbers. Ignoring the dollars and cents for now, just think about 1/10 + 1/4. The first number which they both divide is 20, so that could be our target type instead (and smaller.) 1/10 * 2/2 + 1/4 * 5/5 = 2/20 + 5/20 = 7/20. And 7/20 * 5/5 = 35/100, so it's the same answer, really.
Achilles: Got it.
Tortoise: Now, the cross-multiply rule you learned gives a third path to the answer. You probably learned something like a/b + c/d = (a*d + b*c)/(c*d). Here, that would give us (1*4 + 10*1)/(10*4), or 14/40.
Achilles: Which is 7/20 again.
Tortoise: Yes, and here is why that works. 1/10 * 4/4 + 1/4 * 10/10 = 4/40 + 10/40 = 14/40, and --
Achilles: Wait! a/b + c/d = a/b * d/d + c/d * b/b = (a*d)/(b*d) + (c*b)/(d*b) = (a*d + c*b)/(b*d), since b*d = d*b. And c*b = b*c, so I've just derived the cross-multiply rule.
Tortoise: Excellent! And now you know fractions! Oh, except for one little thing...
Tortoise: One final thing you should know, and which will give you a glimpse of doing mathematics. You've seen (1/3)/2. What is 2/(1/3)?
Achilles: I can't just regroup the parentheses?
Tortoise: Of course not! This is division, not multiplication. That would be like saying (5-2) - 1 was the same as 5 - (2-1).
Achilles: Well, 2/3 was 2 things, each cut up into 3 equal parts, and then I take 2 of the 6 parts. 2/(1/3) is 2 things, each cut up into 1/3 part... that doesn't make any sense!
Tortoise: Not really. But what mathematicians do sometimes, when they have some symbols which give good answers some of the time, and don't make sense (in real world meanings) other times, is to make up meanings which allow the rules to keep on going. Such as the square root of negative numbers. And we will often do that by trying to think more deeply about what we're trying to do, or about how that is related to other things. Now, one sense of division is cutting things up. Another sense is of taking away pieces (7/2 is 3 remainder 1, meaning we were able to subtract 2 from seven 3 times, with 1 left over) but that won't help here either. A third thought is that division is the inverse, or reversal of multiplication.
Tortoise: That if we multiply 2 by 3, and add 1 back, we get 7 again. Or that if 10/2 = 5, then 5*2 = 10. If (1/3)/2 = 1/6, then 1/6 * 2 = 1/3.
Achilles: I think I get it. So we want 2/(1/3) to equal SOMETHING, such that that same something * 1/3 will give us back 2?
Tortoise: Exactly! Now what is the something?
Achilles: I'm not sure.
Tortoise: It's just simple algebra, child! Or guessing, but let's do algebra: SOMETHING * 1/3 = 2. Now multiply both sides by 3: SOMETHING * 1/3 * 3 = 2 * 3. 1/3 * 3 = 1, so we have SOMETHING = 2 * 3 = 6.
Achilles: So 2 divided by one-third gives 6? That still doesn't make real sense. Although I suppose if dividing by bigger and bigger numbers gives smaller and smaller answers (1/2, 1/3, 1/4, 1/5...) then it makes a bit of sense that dividing by a small number such as 1/3 should give bigger numbers.
Tortoise: Now you're thinking like a mathematician! Try some more?
Achilles: Well, (1/3)/2 was 1/6. What is 1/(3/2)? Something such that SOMETHING * 3/2 = 1. Multiply by 2, divide by 3, and I get SOMETHING = 2/3. Hmm, it just flips.
Tortoise: In fact, that is the rule. When dividing by a fraction, flip it over and multiply instead. It even works in funny cases: 2/3 = 2/(3/1) = 2 * 1/3 = 2/3. And now you're done with fractions. Apart from practice, perhaps.