Science Forums 2

[CraigD]

Originally Posted by Boof-head

Ah cha, as Ramunandran would have said, what if we don't expand or calculate pi?

I suspect that most programmers with a math bent – who tend to spend a lot of time ignoring proven impossibility and just trying to code it anyway – have on at least one occasions seriously considered this question.

The real numbers, which consist of the rational and the irrational numbers, taunt one to find a simple scheme to represent digitally. For the other common numbers systems - the naturals, integers, rationals, and complex - numbers can be simply and compactly represented with pairs of numbers of the more elementary system –

,

,



– but not so with the reals.

Exactly representing an irrational real number is no big challenge – just write it using recognized symbols, , etc. The big deal ensues when you start doing arithmetic with them.

For example, the simple series

which has an approximate rational value 671.46295, has the exact rational value:






As long as the number of operations in a calculation are small, on a human scale, it’s not impractical to represent irrational numbers exactly. If, however, a calculation has billions or more of operations, as they commonly do when performed using a computer, the representations can become impractically unwieldy – intractable, to use a good term – with even simple operations such as comparisons requiring nearly the equivalent of repeating the entire preceding calculation. For some fairly modest calculations, the amount of data storage required for systematic schemes to exact represent a rational number can exceed the amount possible given the limited amount of matter and energy in the visible universe!

Intuitively, one can visualize this problem being that real and complex numbers can simply contain too much information to be represented without making shrewd, very context-dependent decisions about the representation scheme to be used. Trying to formalize “shrewd context-dependent decisions” can be a nightmare task.