Humans, and probably most

animals, have an

intuitive perception of

numbers up to about

6. (See the 1956 article

The Magical Number Seven Plus or Minus Two by

George A. Miller). These can be thought of as "class-0 numbers", the "intuitive" numbers.

Up to about a

million (1000000 or 10

^{6}), the number can be directly

perceived. For example, you can see 20,000 people in a stadium and get a good feel for how big 20,000 is. These are the class-1 or "perceivable" numbers.

Above that the numbers are perceived more as ideas that differ from one person to another, resulting in the symptoms of

innumeracy. For example, the

marketing department of the

Ideal Toy Corporation thought that

Rubik's Cube would sell better if they claimed it had "over 3 billion combinations" than if they told the

truth, which is "over 43 quintillion combinations" (4.3×10

^{19}).

It is common for people to think of large numbers in terms of how many

digits they have. In extreme cases, a number with twice as many digits seems only twice as large.

Computers can

store and

manipulate large numbers easily, until the number of digits approaches the

memory capacity of the computer. Depending on the task, the limit is around 10

^{106} or a bit higher. Any number up to this size can be written down

exactly. We can call these "class-2 numbers", they are "exactly representable".

Beyond that, things get tricky. To manipulate class-3 and higher numbers, you need to use some form of

logarithms. If you aren't careful you'll run into an

exponent paradox. For example, which of these do you think is larger:

7

^{777} or 1000

^{10001000}
The first one is much larger (about 10

^{(3.177×10695974)}) than the second (10

^{(3.0×103000)}).

Another example of the exponent paradox appears with class-4 numbers, like
10

^{1010100}. This number is so large that it appears to be equal to its

square.

The concept of number "classes" presented here was inspired by the "levels of perceptual reality" treatment given by

Douglas Hofstadter in his May 1982
"Metamagical Themas" column for

Scientific American. (article title "On Number Numbness").

*For more about large numbers find my web pages on the topic. To avoid dead URLs I suggest using a Google search for "large numbers dyadic mrob".
*

Copyright © 2001-2002 Robert Munafo. Robert
Munafo is mrob27.