[I see JB has some of the same questions I do. Oh, well... the post was already written before I saw that, so here it is.]
Why do we only round the value of pi in most instances to 3.14?
Who is "we"?
That's a shitload of error that is accepted that gets compounded more and more the more calculations are involved.
It's acceptable when it's a good enough approximation for the purpose at hand. 3.14 was often used in the days of slide-rule calculations since the typical slide 10" rule was good to about three digits of precision.
When you need a better approximation, or you need to control numerical precision issues, use a more precise approximation for pi. When you don't, any added precision is meaningless.
A circle is about 4 inches in diameter. What's its area?
Let's see... 4 inch diameter is 2 inch radius, and area is pi time radius squared, so it's
(2 in)
2 x 3.1415926535897932384626433832795 = 12.566370614359172953850573533118 in
2.
Um... no.
About 4 inches has one digit of precision. The diameter could be anywhere between 3.5 inches and 4.5 inches and still be right to a single digit of precision. A 3.5-inch diameter circle has an area 9.6211275016187417927918453612935 in
2. A 4.5-inch diameter circle has an area of 15.904312808798328269717132127852 using a value for pi with 32 digits of precision.
4 in
2 x 3.14 = 12.56 in
2, which is certainly precise enough considering the precision of the input data. In fact, if you round the value of pi to 3, you get an area for that circle of 12 in
2, and that's not any less accurate than 12.56, It has the advantage of being very easy to calculate without the use of a calculator, slide rule, or
We should be using ~60 decimal places of pi because that would give us the precise measurement we need that could calculate a circumference the size of the universe to within a Planck lengths diameter.
That doesn't make sense, but we could know the circumference of a sphere to within a Planck length only if we knew its diameter to better than approximately 1/3.14 of a Planck length. Guess what... we don't.
But humans are lazy and they generally cant compute when things are more than 4 or 5 numbers.
With digital calculators and computers it's fairly easy, but the precision of the result is still limited by the least-precise parameter in the calculation, so any extra digits of precision stated in the result is meaningless. So why bother?
Sloppy work. Good is the enemy of better and better is the enemy of PERFECT. Aim for perfection or accept your answers will be wrong.
Not necessarily sloppy at all. Anyone who assumes the data is perfect is deceiving himself. When multiplying a measurement good to three digits of precision by a parameter good to 60 digits of precision, the product has only three digits of precision (maybe four in some circumstances). So the extra work is wasted (and, worse, can give a false sense of precision that isn't real). Look at the example above: using the Windows calculator with its value for pi with 32-digits of precision to calculate the area of a 4" circle with 1 digit of precision gives an answer that could legitimately be between 9.6 and 16 (to two digits). 12 is a perfectly good answer to the stated precision. Anything beyond that has no meaning and should be avoided in serious work.