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 106), 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" (4.3×1019).

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 10106 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:

7777 or 100010001000

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 101010100. 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.

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