Math
Math |
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In this page, WTM is going to travel back in time and do some really "old-fashioned" mathematics. Specifically we will look at how the ancient Egyptians multiplied quantities and dealt with fractions. To make things more convenient for our discussion, we will use our own numbering system and symbols. This is a piece of good news/bad news. First, it facilitates our presentation: you will understand the numbers right away and I can write things with my computer keyboard. However, such does not how "what it really looked like to the ancient Egyptian's eyes and mind" and we can't truly appreciate why he did things that way (which seems more difficult to us with our modern notation). But it's the best I can do for now.
With that preface, step into our time travel machine, strap on your seat belts, and let's go!
We will begin with the idea of multiplication, as it is the easier of the two topics. Let's say we wish to multiply 13 by 28. We write things this way:
1 28 2 56 4 112 8 224Notice that in the two columns each number is the double of the number above it (that is, after the first row). That's the first step -- just multiply by 2! Everybody should know their "2 times table", right?
Next, we make use of a very important fact from number theory: Every positive integer (number) can be uniquely expressed as a sum of powers of 2. Our first column consists of just that, the powers of 2. So from among those numbers we should be able to find some that add up to 13. That would be:
1 28 2 56 4 112 8 224 13Now we will highlight the numbers in the second column that correspond to 1, 4, and 8, then add them:
1 28 2 56 4 112 8 224 13 364Therefore, we can say
That this is not a cute parlor trick that only works in special situations can be shown by using the famous Distributive Property of Multiplication over Addition. Note:
13 × 28 = (1 + 4 + 8) × 28 = (1 × 28) + (4 × 28) + (8 × 28) = 28 + 112 + 224 = 364
By way of reviewing another famous number property, the Commutative Property of Multiplication, we will do it all over again, reversing the factors to 28 × 13.
1 13 2 26 4 52 8 104 16 208 28 364You see? It works, just like it's supposed to. Now practice this idea with some problems of your own choosing. Some working advice: instead of using color to highlight the numbers you need to add, you could mark then with a star(*), an "x", or some other convenient mark; or you could cross out the ones you DO NOT need. Do whatever seems helpful to you.
How about some fractions now, Egyptian style. (Get ready for something really different here.)
One thing you must know right off: the Egyptians didn't write their fractions like we do, even allowing for the different way that they symbolized their numbers. This means they didn't write things like
1 5 4 8 12 ---, ---, ---, ---, or ----. 2 6 7 9 17No sir, no numerator-over-denominator for them. In fact, they had no numerator at all. Well, that's not quite true; they did have a numerator of sorts: 1. That's it, just 1. So all their fractions are what we nowadays call by the name unit fractions, because 1 means "unit". Some "unit fractions", therefore, are:
1 1 1 1 1 ---, ---, ----, ----, or -----. 3 8 14 26 150Now, as I said, they didn't write any numerator-over-denominator, you should be wondering now just how did they express their fractions. It's quite simple, actually. They wrote the denominator, then placed an oval or "mouth" above it. If I recall my Egyptian numbers, "4" was made with four sticks, like "||||". So, one fourth (1/4) would be
Sure, it looks strange to us to call that a fraction. But it worked for their time. (Actually, I kinda like it.) And remember, they didn't know about things like calculus or computers, or sending spaceships to the moon. As their day-to-day needs were somewhat simpler than ours, their mathematics could certainly be written differently as well.
That still doesn't account for non-unit fractions, though, does it? True enough, and here's where the real fun begins. If they wanted to use a fraction such as 3/4, they would express it as the sum of two unit fractions. In our modern notation, we would write it like this:
3 1 1 --- = --- + --- 4 2 4Now that certainly does the job, doesn't it? This presents us with an interesting problem: given any particular non-unit fraction, how might an Egyption student of that time express it as the sum of two (or more) unit fractions? Let's take 2/7 as a example.
You might think that should be rather easy. Just do this:
1 1 2 --- + --- = --- 7 7 7To which I must say, "Sorry, Charlie!" Those Egyptians had another strange rule to follow: there could be no repeated fractions in the expression. To be a properly written Egyptian fraction, each denominator must be distinct (different). So in our example, we need to write
1 1 2
--- + ---- = ---
4 28 7
See? We have two different unit fractions and still have the sum of 2/7. That's the Egyptian way. (Nowhere did I say it was going to be easier, just interesting.)
But how does one get from the 2/7 fraction to the unit fraction form? The Egyptians had their way, perhaps rather complicated to our way of thinking, and they also used prepared tables with many simple cases already worked out in advance. What I will show here are two ways that can be done more easily with our modern notation. They will give you some excellent practice in the concept of fractions and factors, while at the same time learning something new.
Step 1: Divide the denominator by the numerator, old- fashioned style. 7 ÷ 2 = 3 r 1 or 3½ or 3.5 Step 2: Round UP the answer to Step 1. It is 4. That will be the denominator of our first unit fraction: 1/4. Step 3: Our situation looks like this now: 2 1 1 --- = --- + --- 7 4 ? So to find the fraction "1/?", we need to subtract 2 1 1 --- - --- = --- 7 4 ? 8 7 1 ---- - ---- = --- 28 28 ? 1 1 ---- = --- 28 ?It appears we were lucky this time, as the result of Step 3 gave us a unit fraction right away: 1/28. So we have done it! 2/7 = 1/4 + 1/28. Case closed. If our difference had not been a unit fraction after Step 3, we would have merely repeated the entire procedure with the difference(s) obtained until we had all unit fractions. [You see, it is another fact from number theory that: Every proper fraction has an Egyptian representation as a sum of distinct unit fractions. (Hurd, p. 593)] This is a simplification of a method due to Fibonacci, a famous mathematician of the 13th century.
To see if you have caught on to this idea, try Fibonacci's process on the fraction 3/7. If done correctly, you should obtain 1/3 + 1/11 + 1/231 as your result.
Now, to extend this idea just a bit, look again at the result for 2/7. If we multiply all the fractions in the representation by 2, we have this interesting result:
4 2 2 --- = --- + ---- 7 4 28 4 1 1 --- = --- + ---- 7 2 14See? We have an Egyptian fraction form for 4/7. With very little extra effort on our part. Sort of like something "new" from something "old". This doesn't work all the time, but sometimes it does.
3 1 1 1 --- = --- + --- + ---- 7 4 6 84This is somewhat nicer that the other style, as the denominator values are smaller. To get this result, we must first utilize the concept of equivalent fractions. For 3/7 some equivalent fractions are
6 9 12 15 18 ---- = ---- = ---- = ---- = ---- ... 14 21 28 35 42As you might guess from looking at the solution, we need the fraction whose denominator is 84, obtained by multiplying both parts of the fraction by 12.
36 ---- 84We now "decompose" the number 36 into three numbers, all of which are factors of 84. So next we need the factors of 84. I like the T-chart listing; it helps me to "keep things straight".
84 ----------- 1 | 84 2 | 42 3 | 28 4 | 21 6 | 14 7 | 12The highlighted numbers in the chart have a sum of 36, so...
36 21 + 14 + 1 ---- = ------------- 84 84 36 21 14 1 ---- = ---- + ---- + ---- 84 84 84 84 3 1 1 1 --- = --- + --- + ---- 7 4 6 84[A little review problem is in order here since all our denominators are even numbers; can you show me an Egyptian expression for 6/7?]
Return to our T-chart for 84 once again. The highlighted numbers show another way to make 36.
84 ----------- 1 | 84 2 | 42 3 | 28 4 | 21 6 | 14 7 | 12So that should provide us with yet another way to express 3/7:
36 28 + 7 + 1 ---- = ------------- 84 84 36 28 7 1 ---- = ---- + ---- + ---- 84 84 84 84 3 1 1 1 --- = --- + ---- + ---- 7 3 12 84Now how about a "really big show"? Observe: the red numbers make a six term way to make 36, right?
84 ----------- 1 | 84 2 | 42 3 | 28 4 | 21 6 | 14 7 | 12This allows us to make a really big expression for our fraction.
36 14 + 12 + 4 + 3 + 2 + 1 ---- = ------------------------- 84 84 36 14 12 4 3 2 1 ---- = ---- + ---- + ---- + ---- + ---- + ---- 84 84 84 84 84 84 84 3 1 1 1 1 1 1 --- = --- + --- + ---- + ---- + ---- + ---- 7 6 7 21 28 42 84Here's a homework problem for you. Can you find a way to form 3/7 with 4 unit fractions? With 5 unit fractions? With 7 of them?
[NOTE: Hurd called this method the "convenient denominator" way. This means that a multiple of the original denominator was sought that would "conveniently" possess some factors that could be used to form a sum equalling the numerator. Actually, it's pretty easy to do, don't you agree now?]
Now, doesn't that look a little familiar to you? Like a unit fraction used in our work above. This suggests that we could also verify our solutions using that key. First, find the decimal form of the original fraction, such as 3/7; it is 0.428571428 (in a 10-digit display model). Then taking our first solution for this fraction (1/3 + 1/11 + 1/231), we can use this key sequence to check it out:
Since the display shows the same result, we can be reasonably confident that our work was correct. Wouldn't an ancient Egyptian scribe be impressed by that! How fast it is these days!
Here is a little item, not strictly related to the topic of this page, but found in one of the websites below. To see a pretty result, use the reciprocal key of your calculator to find the sum of the reciprocals of these numbers:
I found many websites that discuss Egyptian fractions; here are some of them:
Here is another nice website about Egyptian numeration in general: Egyptian Mathematics.
Finally, the magazine reference used was: Spencer P. Hurd, "Egyptian Fractions: Ahmes to Fibonacci to Today". MATHEMATICS TEACHER. October 1991. pp. 561-568
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