How accurate are most representations of pi?

Question

I understand that $\pi$ is the ratio of a circle's circumference to it's diameter and it is equal to about 3.14159265359(According to Google) but how accurate is this and most representations of $\pi$?

Answer 1

According to my memory, that last "9" is actually "8979323...", so it is pretty good.

Answer 2

Recall that $\pi$ is not rational, that is, any decimal representation of $\pi$ cannot be exact. However, we can estimate the maximum amount of error which occurs.

A version of $\pi$ which is accurate to $n$ digits after the decimal place has a maximum error of $10^{-n}$.

For example, $3.14$ is accurate to $2$ digits after the decimal place. The maximum error is therefore $\dfrac{1}{100}$, or $0.01$.

You can see why this is initutively, consider the following:

$$ \pi = 3.14??????????? \cdots$$

Where $?$ represents any decimal place. Therefore, the error is:

$$ \pi - 3.14 = 0.00??????????? \cdots $$

Obviously, no matter what value of $?$ is put in, we have:

$$ \pi - 3.14 \le 10^{-2} $$

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Let's check out your example, we have $\pi \approx 3.14159265359$. I cannot speak for the fact whether the last digit is rounded, so I will ignore that. We have: $\pi \approx 3.1415926535$. The maximum error is $10^{-10}$. This is a maximum error of $0.00000000001$. For any practical application, you have more than enough accuracy.

Answer 3

Since a slightly better approximation is $\pi\approx3.141592653589793$, the error in the approximation $3.14159265359$ is clearly very small:

$$3.14159265359-3.141592653589793=0.000000000000207=2.07\times 10^{-13}\;,$$

and since in fact $\pi>3.141592653589793$, the actual error is smaller than this. In short, it’s a very good approximation.

Answer 4

If you write $3.14159$ and those digits are correct, then the number of digits tells you how accurate it is. Since what I wrote gives five digits after the decimal point, if we assume the last digit is rounded, then the error is no bigger than $0.00001/2$, so that's how accurate it is.

But I wonder what was intended in this question. Could it be that some uncertainty in these digits was suspected?

Later edit: See http://en.wikipedia.org/wiki/Approximations_of_%CF%80 and http://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80