Episode 138: January 18, 2013
Real World Math
by Jason Marshall
Now that we’ve successfully completed our journey through the wonderful world of working with mixed fractions, it’s time to spend a few weeks stretching out our brains in an entirely different direction. Which is why we’re about to embark on a trip back through time to the birth of numbers. Long time math fans may remember our first foray into this history way back when the podcast was just getting started. But we only covered the tip of the iceberg then, so let’s once again jump into our time machine and head back to begin our journey through a brief history of numbers. How far back? Way, way back…
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The Birth of Arithmetic
Nobody knows precisely when our story begins, but we know that for some reason around 30,000 years ago people started making tiny notches in bones. Researchers who found these bones have speculated that the marks carved in them were probably used to keep track of things like the number of sheep in a field or the number of days since the last harvest. In other words, these so called “tally-bones” represent the origin of counting.
But this system for keeping track of numbers isn’t so great (which is why we aren’t all using it!). Why do I say that? Well, imagine you’re an ancient tally-bone carver and your job is to make one notch on your bone for every full moon since the last harvest. After four full moons, your bone has four notches:
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Which, perhaps, means that it’s time to plant your crops again and restart your tallying. For this purpose the system seems to work pretty well. But what if your job was to keep track of the number of weeks that have passed instead of the number of months? After those same four months, your tally-bone would contain a lot of notches—almost 20 of them:
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All of those notches means that it’s a lot tougher to tell at a glance how long it’s been since the harvest. Even worse, what if your job is actually to keep track of the number of days that have passed. Then your tally-bone would contain over 100 notches:
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What a mess! Even if you don’t run out of room on your tally-bone, every time you want to know how many days have passed you have to start from the beginning and count each and every mark. That’s obviously not too smart…which led people to start looking for a better way to do things.
About 5,000 years ago the ancient Egyptians developed a hieroglyphic system that was definitely an improvement. As we’ll soon see and appreciate, the key to this system is the idea that it’s much more efficient to use many symbols to represent numbers rather than just the single symbol used by the tally-bone carvers. The first five symbols in the ancient Egyptian numeral system are:
Notice that the symbol for the number 1 is basically a notch—exactly like the one our tally-bone carving friends used. And, not coincidentally, it looks a lot like our numeral “1”. But that’s where the similarity to the tally-bone system ends since the Egyptians also had symbols for the numbers 10 (shaped like a yoke used to plow fields), 100 (a twisted length of rope), 1000 (a rather cheerful looking flower), 10,000 (a finger), 100,000 (a frog), and 1,000,000 (a happy human with arms raised to the sky). But what about the numbers 2, 3, and everything else that doesn’t have a unique hieroglyph?
It’s more efficient to use many symbols to represent numbers rather than a single symbol.
All numbers that don’t have their own symbols are written by adding two or more symbols together—basic arithmetic! For example, the ancient Egyptians would write the number 2 by writing the symbol for the number 1 two times. But, you might be thinking, that’s exactly like the tally-bone system! How is this more efficient? Well, the efficiency comes from using those extra symbols we’ve seen for higher powers of 10. So although the ancient Egyptians did have to write an extra hieroglyph for each number between 2 and 9, when they got to 10 they returned to writing only a single symbol. Then, to represent a number like 36, all you have to do is write down 3 hieroglyphs that each represent 10 next to 6 hieroglyphs that each represent 1, which represents a total of (3 x 10) + (6 x 1) = 36.
While this system is certainly more efficient than notch-carving, it’s definitely not without its flaws (which, again, is why we’re not all using it today). After all, you still have to do quite a bit of work to write down a number like 3,358—try it and see! Even worse, imagine having to do addition and subtraction of large numbers using this ancient Egyptian system. Just to get a taste for how “fun” this can be, I encourage you to try your hand at solving an arithmetic problem Egyptian style—perhaps 1,927 + 2,743. If you do it, you’ll definitely gain an appreciation for the beauty of the decimal system that we use every day. Aren’t you glad you live in the modern world with our modern numeral system? I know I am!
So what comes next in our brief (and certainly incomplete) tour through the history of numbers? Have any other clever people come up with clever methods of counting? There’s one other system that I bet you’re already at least somewhat familiar with: the Roman numeral system. How does that work? Well, unfortunately we’re all out of time for today. Which means that our journey into the world of ancient Rome, ancient Romans, and their numerals will have to wait until next time.
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