# Confidence in the first $k$ decimal places of a product after multiplying $N$ irrational numbers together

If I multiply $N$ irrational numbers together to generate a product $P$, where the irrational number are specified to a working precision of $m$ decimal digits, how many decimal digits, $k$, should I have confidence in for $P$?

For example, how large of an $N$ can I get away with standard double floating point precision to have a confidence of $k$ decimal digits in the product (about m ~ 15 stable decimal places)?

Roughly speaking, if you have an error of a factor of $1 \pm 10^{-15}$ in each multiply, the standard deviation is a factor of $1 \pm \frac 1{\sqrt 3}10^{-15}$ (assuming a uniform distribution). You need about 3 of these to lose the first digit, 300 to lose the second.

It isn't worth being too careful in this calculation, because if your number is stored in decimal, the fractional error changes by a factor 10 between a number that starts 10000 and 99999. In binary it is only a factor of 2, but the principle is the same.