Millions And Billions

(Reposted from 2010. I was thinking of this problem in relation to the Green New Deal, and grand government projects in general. I don't think people advocating for these programs have much of an idea of how big a billion is, and absolutely no idea how big a trillion is.  It's just a Big Number. I just have this worry that 93 trillion seems smaller to them than 730 billion, because 93 and 730 are ideas they can get their heads around, but billions and trillions are just Big Numbers. Of course  93 Big Numbers is less than 730 Big Numbers, so we can clearly afford it.)

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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.