# 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 $\pi$?

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What is a "version of pi"? – Qiaochu Yuan Jan 13 '13 at 1:12
The fascination with the digits and approximations/"representations" of $\pi$ never ceases to amaze me! – amWhy Jan 13 '13 at 1:15
Changed it to representations! – Bossman759 Jan 13 '13 at 1:21
What is a "representation of pi"? – Qiaochu Yuan Jan 13 '13 at 1:26
And which are "most representations" ? – Yves Daoust 16 hours ago

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.

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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

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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.

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The first sentence of this answer is very misleading. It makes it look as if "any decimal representation of [this number] cannot be exact" is the definition of "irrational". This false meme is strangely persistent in view of the fact that it is never taught in classrooms or textbooks. To say that a number is irrational means it's not an integer over another integer. – Michael Hardy May 21 '13 at 12:49

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?

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Yes, I guess am kind of uncertain with those digits considering that there are many different approximations of pi. – Bossman759 Jan 13 '13 at 1:15
But all of them yield the same sequence of digits. They wouldn't be valid if they didn't. – Michael Hardy Jan 13 '13 at 1:22
Coincidentally, the accuracy of computations of $\pi$ is currently a hot topic on StackOverflow: How to determine if my calculation of $\pi$ is accurate. – user53153 Jan 13 '13 at 6:12

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

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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.)

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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.

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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.

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