In Our Brains, a Billion Doesn’t Count for Much

This year’s proposed national budget is over a trillion dollars; the Milky Way galaxy contains 200 billion stars; chemical reactions take place in femtoseconds (quadrillionths of a second); life has evolved over a period of 4 billion years.

What are we to make of such numbers?

The unsettling answer is, not much.

In fact, there’s good evidence that our brains may not be engineered to cope with extremely large or small numbers--which may have something to do with our state of denial about the national debt, and why we can’t seem to discriminate between millions for children and billions for savings and loan bailouts.

Scientists are well acquainted with the human inability to absorb large and small numbers. UC Berkeley geologist Raymond Jeanloz, for example, likes to impress his students with the power of large numbers by drawing a line designating zero on one end of the blackboard, and another marking a trillion on the other.

Then he asks a volunteer to draw a line where a billion would fall. Most people put it about a third of the way between zero and a trillion, he said. Actually, it falls very near the chalk line that marks the zero.

Compared to a trillion, a billion is peanuts. That’s because a trillion is a thousand times a billion, and multiplication is a powerful engine for growth.

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Take the legend of the mathematician who invented chess. The king liked the game so well, the story goes, that he offered the mathematician any prize she wanted. She asked only for two grains of wheat to be placed on the first square of the chessboard, four on the second, eight on the third, and so forth, doubling the number of grains for each square on the board.

How much grain did she win?

More than had been produced in the entire history of the world. That’s the power of doubling.

A more recent (and chilling) example comes from University of Colorado physicist Albert Bartlett, who studies the mathematics of overpopulation.

Bartlett asks us to imagine a colony of bacteria that live in a Coke bottle. They double their population once every minute. They start growing at 11 a.m. and the bottle is jampacked with bacteria by noon.

What time would it be, Bartlett asks, when even the most farsighted bacteria saw an overpopulation problem on the horizon? Certainly not before 11:58, he answers, because at that point the bottle would only be one-quarter full (two doublings away from full).

Even if the bacteria explored new territory and discovered three new empty Coke bottles, it would still take only two minutes for them to fill those bottles to the brim, Bartlett says.

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In fact, everything that grows exponentially doubles sooner or later. Compound interest at 7% doubles your money in 10 years. The human population is expected to double in 50 to 60 years. But it’s unlikely that people will get too perturbed about it any time soon--in part because our brains can’t perceive the magnitude of this growth. Rather, brains appear to be calibrated like the magnitude scales that measure the power of earthquakes.

As most people know, a quake that registers 8 on the scale is vastly more powerful than one that registers 7--a lot more than the difference between the numbers 7 and 8 would suggest. Magnitude scales work like the chessboard example--where the energy of the quake multiplies like the grains of wheat, but the number on the scale only increases by a single unit.

So if earthquake power grew like the grains of wheat, the sixth square would contain 2 times 2 times 2 times 2 times 2 times 2 (or 64) grains, but the scale would measure only 6. This “logarithmic scale” is required to measure earthquakes because they range over such a broad spectrum of strengths. Using the chessboard example, in order to have a different square for each, the chessboard would have to be bigger than the world.

The same is true of neurons. The human eye can see over a range of a million different shades of brightness--but we don’t perceive the brightest thing we can see as a million times brighter than the dimmest. There simply isn’t enough room in the brain. So sight works on a scale similar to earthquake measurements.

This innate human blindness to the growth of big numbers is a good thing to remember next time someone tells you that the world will always find another Coke bottle.