Re: Irrational numbers

[tobyink]

in reply to Irrational numbers

Certain irrational numbers have a compact but useful representation; for example the square-root of two can be represented as bless(\'2 ** 0.5', 'Surd'). A suitably smart implementation of the Surd class could figure out that root-2 times 3-root-2 equals 6 without resorting to floating point arithmetic.

The problem with π is that it's not just irrational; it's transcendental. There's no convenient, accurate representation of it. You've just got to store an approximation as a float or string or whatever, and it will only be accurate to a certain number of decimal places.

[grondilu]

I'm afraid you missed the point I was trying to make. Pi could be represented as a lazy yet infinite list of rationals converging towards pi. So could any irrational numbers, including those who are transcendental.

Representing real numbers seems like a very natural use case of lazy lists.

[tobyink]

Let's say you want pi to a million digits. Using a lazy list populated by the Machin's formula will take you several hours of calculation. But a hard-coded decimal representation of a million digits will take less than a megabyte of memory/disk space, even if inefficiently stored as one digit per byte and can be retrieved virtually instantly.

In practice, there's little reason to need that sort of precision. A 64-bit float has about 20 decimal digits of precision, which is enough to calculate the circumference of the Earth to less than a hair's breadth. Apparently 61 digits of pi is enough to calculate the circumference of the observable universe to within one Planck length.

In practice, floats are simply good enough for most purposes.