Tuesday, August 10, 2010

Millions And Billions

This post sets up a point in a later post, but is interesting in its own right.

When measuring vocabulary in IQ tests, it has long been observed that higher education effects disappear by the time individuals are in their 50's - and they aren't that dramatic after a few years out of college anyway. While education does expose one to more words, and creates contexts in which these words are used, much of it is the specialised vocabulary of each discipline. That is a fine thing, for everyone must mean highly similar things in academic discussion when they say neurotic or microeconomics, or little more can be learned.

Something similar happens with numbers. We are not born knowing what a million is, or even a thousand. To all of us, numbers are no more meaningful than the hrair of Watership Down, where everything beyond four is simply "many." We can visualize a dozen somethings and understand it pretty early in life, but we have to work up to an intuitive meaning of a hundred by fooling around with sevens and dozens as we go through childhood. By adulthood, we have enough experience with setting up chairs, making change, and estimating distances that the "hundred" idea is well-embedded. We don't really get to a thousand with that except in more specialised settings which vary from person to person. But we do get a pretty good idea of a thousand because we know that it's ten of those hundred-thingies. Nearly everyone deals with thousands and ten-thousands in some setting and knows what they mean in context. Because we learn to manipulate the symbols of mathematics as children, we are often able to work with these numbers even before we get an intuitive handle on them. 4000rpms has a meaning with engines - it has a sound, it has a feel. From such islands of knowledge we can work in both directions to solidify our understanding of numbers.

I had an advanced studies course in Concepts of Mathematics one summer in high school. Mr. Hulser, quite sure that all of us were going on to take many more advanced math courses, put some stress on our working with large numbers for its own sake. He stated that "research" had shown that people didn't really have much idea what a million was, and though they knew in the abstract that a billion was a thousand millions, the actual working estimates they used showed that they experienced a billion as about ten million. A billion was just a "big million." I don't know if there was ever any real research behind his statement or whether it was just his observation that he tried to give a little more authority to, but it has always seemed plausible to me, given the way numbers get used in conversation and in newspapers. Hulser was not talking about the common man when he gave this estimate, BTW. He was talking about those educated people who happened to be in other fields. They didn't really get big numbers. But we, the budding mathematicians, the chosen ones, must do better than that. We needed to develop that intuitive sense that a million was not a hundred thousands, but a thousand of them, and billions not ten million. He thought best we might hope for by the end of college was an idea of a billion that was only one order of magnitude short instead of two. But it was important nonetheless, because sometimes in solving an equation we might have to intuit where solutions might lie by envisioning the graph.

He stated that confusing millions with billions was a terrible error, and had caused much mischief in the world. The innumeracy of otherwise educated people appalled him - though he left that said only by implication, perhaps not wanting to make us any more arrogant than we already were.

I think he was very much onto something. As we go through adulthood we encounter big numbers all the time, and have to make some sense of there being 15,000 people in a town in 1980 and 19,000 in 2000; of salaries, world population, budgets, odometer readings. We gradually build up a storehouse of these, to gain some control (beyond the mere manipulation of the symbols) of big numbers. And we do this along much the same lines as we do vocabulary - the brighter ones keep acquiring more understanding and more control, so that even a number such as 11.2 million might have some sense to it. And we know that a billion is much more than that, even if we can't make a very precise picture in our heads. Much, much more.

It turns out to be one of those rough measures of intelligence. Anyone might mishear or misspeak on occasion, but consistently messing up million and billion as concepts, just because they sound alike, is a sign, not only of a person who doesn't work with large numbers much, but of one who does not understand the large numbers that flow past him in his life.

7 comments:

Gringo said...

I am reminded of two public figures and big numbers. Al Gore saying that the temperature at the middle of the earth was several million degrees.

With more fondness,here is Senator Everett Dirksen: "A billion here, a billion there, and pretty soon you're talking about real money."

One had number sense, the other was senseless about numbers. Among other things.

If Al Gore wanted to shill, perhaps he should have done super-detergent infomercials on TV. Odds are he would have known more about the detergent than he does about math, science and AGW.

jlbussey said...

"...but of one who does not understand the large numbers that flow past him in his life."

Sounds like a close approximation of our political classes. They through the numbers around but they really have zero grasp of their meaning.

jlbussey said...

And yes, I do know that I meant "throw" not "through" - I may need a little more caffeine this morning.

(I work with numbers for a living, but in the other direction. Parts per thousand, parts per million, parts per billion. Even for me it need reinforcement on occasion.)

Paul_In_Houston said...

I love the metric (SI) system for visualizing large numbers.

Here in Houston, the downtown streets were laid out 16 to the mile, making their spacing about 100 meters (close enough for government work, anyway).

So, a meter (a bit over a yard) is a thousand millimeters (thickness of a paper clip).

A square meter is thus a MILLION square millimeters, and a cubic meter would make up a BILLION cubic millimeters.

Or, linearly, a kilometer (a thousand meters) would be about 10 city blocks in length.

If I can visualize that, I am looking at a MILLION millimeters.

In other words, 10 blocks vs the thickness of a paper clip gives you a ratio of a million to one.
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Gringo said...

Estimation is one aspect of number sense that is very useful. This is also good for those who deal w a lot of numbers with calculators or computers. Should this be a billion or a hundred thousand? Does the answer make sense?

While "new math" got a lot of bad press back in the day, I greatly enjoyed "new math." Before "new math," math was easy but boring. Proofs turned me on to math.

Proofs involving the various distributive, communicative, and associative properties of numbers also indirectly assisted me in mental calculation and in estimating. When you realize that 21=20+1 and can use that in multiplication, mental calculation and estimating becomes that much easier.

Texan99 said...

I also like the joke, "Don't tell Obama what comes after a trillion."

Texan99 said...

The only way I can make sense of huge numbers is to change the units. So a trillion inches means little to me until I see that it's one-sixth of the distance between the Earth and the Sun. The distance around the world is 1.5 billion inches. Coast-to-coast U.S. is perhaps a 125 million inches; a million inches is the distance to town: 16 miles.

A million seconds is not quite 12 days. A billion seconds is 32 years. A trillion seconds is more or less Neanderthals. A quadrillion seconds is halfway back to the dinosaurs.