## Really Big Numbers

10 July 2010

Much is made in popular mathematics writing of the human impulse to contemplate infinity, and even more is made of how counterintuitive the infinite can be.  Cantor’s Hotel, the fact that there are as many counting numbers as there are fractions, and so forth.  But you don’t have to go all the way to infinity to get confused; math is confusing enough “near” infinity, i.e. at really big numbers.  Consider this quotation from distinguished mathematician Ronald Graham.

The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.﻿

I have heard it said (though I don’t remember right now who said it) that humans intuitively perceive numbers much as a person standing in a large meadow perceives distance markers placed, say, at 1-foot intervals.  We see 2 as significantly more than 1, and 10 is a lot more than that.  But it’s hard to compare a million and a billion; they’re both essentially on the horizon.  Indeed, in some ways 3 and 10 can feel further apart than, say, a billion and a trillion.  It’s something like asking a small child whether two stars in the sky are closer together or further apart than, say, her house and her school.  The intuition that comes standard on people is a local thing.  And almost all numbers, like almost all places, are really far away.

Here’s an interesting construction that almost impossible to believe at first, because all the interesting stuff is happening way far out down the number line.