Tuesday, June 16, 2009

Number theory, base, and Sumerian freaks.

So, numbers. Number theory is kind of interesting because numbers are something everyone knows, but almost no one really thinks about. They're the foundation of civilization as much as peanut butter toast, and are just as ancient.

Numbers are said to be the oldest, most abstract idea the human race has ever had. They may be the FIRST abstract idea we ever had. We've used them for so long, and learn them at such a young age, that I bet everyone is thinking "abstract? What's abstract about three?" well, quite a lot, actually.

It took the human race quite a long time to understand that this

(those are herd animals of early neolithic tribes; really) is the same as this.

And from there, it was really quite a huge leap to decide that those things are both the same as this:

Think about it. What does a 3 on a piece of paper REALLY have to do with three herd animals? Nothing, except in your mind. You could make any squiggle, and assign it the value of those three herd animals and call it three or Fred or George or cement, and it would be just as accurate as 3 is. VERY abstract. You can use the counting words for all kinds of wild stuff: three hours (time), three meters (distance), three goats (concrete stuff), three planets (astronomy), you name it. Invent numbers, and you can control (or pretend to control) the world.

What's really interesting is, numbers have been invented more than once. You'd think this would be a major big deal like the wheel and only get invented once. But no. People all over the world have developed different systems of varying complexity. Tally sticks - bits of wood, ivory, clay, whatever, with little hatch marks scratched into them - date back to the stone age.

This one is from Africa and is twenty five thousand years old. Sometimes, with these really old sticks, we can figure out what they were counting. Usually days, phases of the moon (of course there are piles of sticks we've no idea what they were counting). There are ancient caves with drawings of animals that have hatch marks next to them, for all the world like the 'kills' graphic on the sides of planes:

Except the ones in caves are thousands of years old and about bears and stuff.

The more things change, the more they stay the same.

Before you decide this is all hopelessly primitive and we've left it behind in our new, high-tech, computerized world, guess again. Anyone do this, counting rows while knitting?

I do it all the time. Guess what? Goes back to those twenty-five thousand year old tally sticks. You may be counting something different, but the method is exactly the same.

To go further with this, we now have to get technical. Unfortunately.

BASES (also known as radix)

There are almost as many number systems as there are languages - the two seem to go hand in hand. One of the ways they are classified is by base. It's the number of unique digits (including zero, if they have it), before the higher numbers are expressed as combinations of those unique numbers. For instance, the number fourteen is ten-and-four, linguistically; four-ten, fourteen. The technical, modern world runs - mostly - on base ten (I am not starting on binary and hexidecimal, at least not today). We've got the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. All other numbers are expressed as variations of those ten. Get it? (I hope so, 'cause I'm trying to keep this from getting complicated.)

Right. Well. Obviously base ten - the decimal system, deci from Latin or Greek or whatever meaning ten - comes from counting on our fingers. Simple, easy, count your herd of goats on your fingers and from there gradually move forward to the Grand Unification Theory, counting on your fingers the whole way. The vast majority of the planet ran on base ten, through human history. Chinese, Egyptians both ancient and modern, Greeks, Hebrews, Romans, Indian and Indus civilizations, they ran on base ten.

Occasionally there were civilizations who ran on base five - the fingers of one hand. The Carib and Arawak tribes in the Caribbean, some Oceania islanders, the Khmer of SE Asia, and assorted African tribes used base five, which makes sense and works just fine as long as you're counting goats and not moving into really complex math.

Then there were the civs that ran on base twenty - all your fingers and toes. The thought of their multiplication tables makes my head hurt, but using your fingers and toes to count on still makes sense. From linguistic studies, it's thought the Celts used base twenty (from whom we have the holdover word, score, as a special word for twenty). The Basques still use base twenty. Some parts of Scandinavia, Inuits, and the Ainu of Japan all ran on twenty. But the most notable folks to use base twenty were the Maya, who had an extremely complex mathematical and callendrical system they worked out using their numerical system.

Then it gets weird.

The Sumerians used base sixty. And because we ultimately inherited their writing and numbering systems, we have vestiges of that damned base sixty system floating around us, all day every day. Buy a dozen donuts? Special words for eleven and twelve? Twenty-four hours in a day, sixty minutes, sixty seconds, 360 degrees in a circle... all the Sumerians' fault.

Why? Well, that's for tomorrow.

Did I mention they were the world's first accountants? This is a bill of sale.


Caryn O'Keefe said...

Always wondered why 360 degrees in a circle. Now I'm wondering why base 60? You'd have to borrow two other people's fingers and toes.

amy said...

I remember reading John Holt--How Children Learn, maybe?--talking about the complexity of numbers and children learning them. I think he was talking about counting: We count one, two, three, but for a child to grasp that what we point to when we say "two" isn't actually "two" without the "one" before it...not an easy thing. I've tried to keep that in mind when dealing with numbers with young kids.

Anyway. Very interesting topic!

Emily said...

I never heard of base twenty before, and base 60?!?! Whoa.

I can hardly wait for the next installment. And, given how math & I get along together, that's quite a statement!

janet said...

This is awesome, thanks.

A friend and I were wondering this weekend when the use of zero as an actual number came up - and discussing why you can't divide by zero. So this is right up my alley.

Amy Lane said...

Damn, girl--you ARE good. And if it makes you feel better about the dumbassery that IS public education, at least once a year, as kids are writing, I hear that EXACT question (or a variation thereof: "How did the word 'three' come to mean 'three'?" Which is good--it means that there are prehumans out there ready to move into abstract thought. Nice post!

Galad said...

Fascinating! Can't wait to read tomorrow's installment.

elleninindy said...

i read an interesting piece recently that contended that kids in asian countries do better in math because their counting systems/words don't include oddities like eleven and twelve.

eleven translates as ten one, twelve as ten two, thirteen as ten three, etc.

also read elsewhere that all chinese number words have just one syllable, a boon to processing speed.

and then there's the geek/cool factor. on malcolm gladwell's blog, a chinese student attending a u.s. university points out that math stars are cool where he's from, but that none of his friends there have much in the way of muscles. working the brain is cooler there than working the bod.

there's also the question of what percentage of each population actually is tested for the often-cited results.

but all in all, american teaching of/learning of math sucks. i freely (though shame-facedly) admit that math is my greatest area of ignorance.

TinkingBell said...

yan, tan, tethera (one, two, many!)

I still remember vividly the moment when my only just 2 year old son was sitting on his bed, looking at me and holding up one hand and going '5' and the other '5' and then saying - 'Mummy, I have 10 fingers!' and being really excited. (why yes - he is a bit advanced - he is now 4 - reads, has discovered zero, understands the zero secret - that adding it to a number stays the same while multiplying it always gives zero- has discovered the secret of the tens, understands that you can make the same number by adding different numbers) His kindergarten teacher is tearing her hair out - heh heh!

Shoveling Ferret said...

Tied in to the accounting and abstract symbols for numbers theme, there's been quite a lot of ink spilled suggesting that writing (in Mesopotamia at least) came about primarily as an expansion of accounting systems in tracking trade of items. Some of the proto-writing stuff from Mesopotamia, like bullae and tokens are essentially numbers and tokens to track things like sheep being sent from place to place and work like a packing list or receipt. Or as a way to control inventory. Doors or containers could be sealed with clay and hatch marks for numbers along with a sign indicating contents.
And the vast majority of early "documents" are primarily economic in nature. It's only later that you see recording of myths, king lists, laws, literature, etc.
It's not quite that clear cut in Egypt, but some of the more recent finds suggest the possibility of a token system roughly similar to that in Mesopotamia.

Louiz said...

I'm convinced kids counting games are holdovers of old counting systems. I had a surprise when I heard my dad count to 20 in Welsh - it was the "picking" counting game we used as a kid.

And anyone who's watched a kid discover numbers knows they're abstract. Kathryn went 5 fingers, oh, five toys, oh, numeral 5.

And I use tally marks to count rows when I run out of row counters (doesn't happen very often though)

Ginger_nut said...

actually - the 360 degrees in a circle is because the sumarians thought that that is how many days it took for the sun to go around earth and return to the same place (ie a year). Apart from the fact they thought the Earth stood still, they weren't far wrong - only 5.25 days off :)

and if you had learnt base 5 or 20 all your life you would probably think that trying to do higher math in base 10 is really insane!

of course the one curious detail you missed out is why the US is so keen to hold onto the imperial system when the rest of the civilised (hehe) world has moved onto the decimal system. When I took high school science classes in Oregon 10 years ago I was amazed at how confused Americans got about the decimal system... meanwhile I was trying to convert all my language into imperial (30cm = a foot = 12 inches... at sea level water freezes at 0C = 32F and boils at 100C = ?? F... 1 gallon = 3.8 litres etc etc... and don't even get me started on why Americans insist on cutting pies into forths instead of quarters when you use the words quarts and quarters so often :)

PS - I'm not trying to pick a fight, I find all this really interesting and just want to share some oddities I have noticed between the two countries I have lived in, with really similar cultures but great bit gaping distances in some areas :)

Ginger_nut said...

actually... i just remembered counting in "men at arms" (discworld, pratchett)


Donna Lee said...

I always wondered about the 24 hours in a day and the dozen. Base 60 blows my mind.

Barbara said...

I'm glad I don't live in your brain, Julie, but I sure do love what comes out of it. Keep it up! More numbers, more Brussels Sprouts, more of everything. Geeks are too cool.

NeedleTart said...

Rabbi recently told us that counting people is rude so to be sure that we have a minyan (10 adults to make prayer "official") we count, not one, not two, not three, and so on.
A long time ago I read the "Cross Time Engineer". Can't remember why he was thrown back to the early middle ages but he introduced base 12 to make commerce easier. Much easier to divide into halves, quarters, and thirds than 10.

Roxie said...

Ah, Julie, you blow my mind every time! Teach us how to multiply and divide on an abacus? I remember going into a shop in Chinatown when I was a tyke, and the gentleman at the counter made those little beads just fly!! I whined till I got a little toy abacus, but didn't understand much beyond adding.