From Andrew
secondary
student

I have always wondered; is it possible to find the value of an irrational number, such as (phi) at it's nth decimal place???

You would plug the decimal place into the formula and the value would be given at the specified decimal place.

When we look at the expression { (sqr 5 + 1)/2 } we are in a sense visualising the number in it's entirety, so the formula may include elements of the above expression in some form.

Any thoughts on how this can be done?

Much appreciated

Andrew

There is a recent paper:

Bailey, D. H., Borwein, J. M., Borwein, P. B.. Plouffe, S., The quest for pi. Math. Intelligencer 19 (1997), no. 1, 50--57.

where something amazing is proved: If you write pi in base 16, so that its digits are called "hexadecimals" (ranging among 0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F) rather than decimals, there is an algorithm that does exactly what you talk about: You just plug in a hexadecimal place and it computes that hexadecimal without computing all the preceding ones.

On the other hand, I think that there are also numbers for which it is known that this cannot be done: finding the next decimal would involve more work than finding all previous decimals (I think this has to do with Chaitin's work on "algorithmic randomness" but I could not find a precise enough statement).

You can also construct irrational numbers with just this purpose in mind: The number is irrational but any decimal is easy to compute; for instance take x = 0.110100010000000100 ... where the n-th decimal is a 1 if n is a power of 2 and a 0 otherwise. Then x is irrational, but it is easy to decide the value of the 1023-th decimal, or the 1024-th decimal, or even the million-th decimal.

(Note: About your comment "When we look at the expression { (sqr 5 + 1)/2 } we are in a sense visualising the number in it's entirety", I wish that my students would understand that rather than automatically convert to 1.6180.)