How accurate are most representations of pi? 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 $\pi$?
 A: 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.
A: As you have been said alredy, no decimal representation will be exact, but we can know its error. If we extend those representations to any representation, we will have a perfect acuracy, but it won't be a numeric value. For example, you can represent it, by words, just as you said, by the ration of diameter/circunference, and that's exact.
About more mathematicals representations, there are a lot of them, in the form of infinite series, like the one made by Ramanujan, that converges to the actual value very fast:
$$\frac{1}{\pi}=\frac{2\sqrt2}{9801}\sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}}$$
Another exact one, by Leibniz:
$$\pi=4\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$
Several values of the zeta function, specially with integers, have $\pi$ as the result (with a factor), like $\zeta(2)$ or $\zeta(4)$, which can be calculated by Fourier series.
And lots more, more info here: http://en.wikipedia.org/wiki/Pi
A: 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:
$$ \pi - 3.14 \le 10^{-2} $$

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.
A: 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
A: According to my memory, that last "9" is actually "8979323...",
so it is pretty good.
A: This is a rounded representation with all exact digits, so you can infer
$$3.141592653585<\pi<3.141592653595.$$
(Equality is not possible as $\pi$ has an infinite decimal expansion.)
A: In third century BCE, Archimedes' had calculated value of $\pi$; In 16$^{th}$ century Ludolph van Ceulen used the same method as Archimedes had extended the accuracy of $\pi$ for 35 decimal places; later in 2011 the accuracy were improved to 5 trillion decimal places. However this is still an approximation, since $\pi$ is irrational number, i.e. has infinite number of decimals. Although, this approximation is very good relatively to contemporary computational capabilities. 
A: An interesting approximation of $\pi$ is $\sqrt{2} +\sqrt{3}=3.146...$. Although it's only accurate to the hundreth's place, it fascinated me when I first learned it because it used square roots to approximate a transcendtal number. 
A: The perfect sphere of Si 28 weighs 1000 grams. It has a specific gravity of 2.3296. So, it's volume is 429.2582 cubic centimeters. It has a diameter of 9.375 centimeters. This  gives a value of 3.12576282169 for the Pi. This is very close ti Babylonian Value of Pi
https://www.academia.edu/34987009/Value_of_Pi_Babylonians_were_the_most_accurate_-_So_says_the_Si-28_Sphere
