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.