Too Much Pi

I came across this article earlier this week:

How Many Decimals of Pi Do We Really Need?

It's some interesting stuff.   On the way to church, I talked with the kids a little about this article, and posed the following question (based loosely on the article):

Voyager 1, being about 12.5 billion miles away, is the most distant man made object from earth.   If it were in orbit around the earth at this distance, how many places of pi would you have to use to calculate the circumference of that orbit to within 1 inch?

I meant for the question to be rhetorical, to illicit guesses and discussion, but to my amazement, the kids jumped on it.   My wife, the math PhD, gave them a very quick primer on significant digits with some examples:

diameter (inches)	estimated pi	estimated circumference (inches)	actual circumference (inches)	error (inches)
100	3	300	314.1592...	14.1592,,,
100	3.1	310	314.1592...	4.1592...
100	3.14	314	314.1592...	0.1592...

They quickly realized they needed as many decimal places as there were digits in the diameter.   The rest of their solution went something like the following:

• We need the diameter in inches!  
• What is 12.5 billion miles in inches?
• No, *25* not 12.5 - we need the diameter, not radius! dork.
• I don't know, but I think there around 5,000 feet in a mile.
• That means there are about 60,000 inches in a mile.
• How did you get that?  5*12 is 60, then times a 1,000.  duh!
• So what's 25 billion times 60,000?
• Well 6 times 25 is 150...
• So how many zeroes do we add?
• A billion has 9 zeroes.   60,000 has 4.  We add 13!
• So it's around 150 with 13 zeroes.  That's 16 digits.
• Is the answer 16?

They got the correct answer.   I was amazed and full of pride.  They had worked together, used approximation techniques, and came up with the solution.

Anyway, turns out, NASA uses 3.141592653589793, or pi to 15 places for high precision orbital calculations.  I would hazard a guess this was chosen around the fact that a 64-bit double precision floating point number in IEEE 754 format (the most common format used in computing) is only accurate to 15.95 decimal digits and not because they needed to calculate the circumference of the orbit of Voyager 1.

Mathematicians Jörg Arndt and Christoph Haenel have suggested that for most "cosmological" scale calculations that 39 digits are all that's needed because that’s what's needed to calculate the circumference of the observable universe to within one atom’s diameter.

As of October 2014, the record for calculating the most digits of pi is 13,300,000,000,000.   That's a pretty big pie.