How many Decimal Places are Needed For Accuracy to a Given Number of Significant Figures?
For example: $1234/3.141 = 392.8$, but $1234/3.14159$ changes the result to $392.7$, so how do I am using enough digits of $\pi$ for my result to be accurate to a given number of significant figures?
There is no guaranteed answer, but if you carry one more you will usually be OK and if you carry two more you will almost always be OK. The reason there is no guaranteed answer is that the less precise calculation can be very close to a breakpoint. Maybe the result is $1.233500005$, which we round up to $1.234$. If the result decreases very slightly we will cross the breakpoint to $1.2334999995$ which will round down to $1.233$.
"Significant places" is often shorthand for an approximate error bound. If you keep four places, the fractional error is between $\pm 0.00005$ when the number is $9999$ and $\pm 0.0005$ when the number is $1000.$ That is a difference of a factor $10$. If you care about the accuracy you need to maintain the error bound explicitly and compute the range that the correct answer could be in. Often we don't want to go to that much work, so just use the significant place rules and call it good enough.
