Thursday, June 2, 2011

Is π any good beyond 39 decimal places?

Visualization of H atom energy level
probability densities

It has been said that π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.

Back of the envelope time:

The observabe universe has a diameter of about 93 billion light years or 8.79873*10^26m. The circumference of the observable universe (assuming that it is spherical in shape) is therefore π*8.79873*10^26m or about 2.76420*10^27m. An error in π of +/-1.0*10^-39 would result in an error in the circumference of something on the order of +/-1.0*10^-12m (very roughly). The average orbital radius of H is about 5.29177*10^-11m.

So, Is π any good beyond 39 decimal places (even though it has an infinite decimal expansion)?
Answer: Probably not other than for theoretical and computational interests.

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