How do you program irrational numbers like Pi into a computer or calculator if they don't have an absolute value?

But \pi does have a well-defined value.  It doesn't have a terminating decimal expansion, no, but then again neither does 1/3.  Decimal expansions are a way of representing numbers, not the numbers themselves.

Most use an approximation, the same way humans do calculation with Ï€.  They can only hold so many digits of Ï€ at a time-- so they'll store something like: 3.141592653589793115997963468544185161590576171875 Rather than: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089... (Notice the discrepancy-- it's close to Ï€, but not exactly correct) Some programs can actually recognize Ï€ using mathematical principles, like, say Maple or Mathematica.  For these special cases, you can store an algorithm to produce the necessary accuracy that you need (when outputing specific numeric values), or would simply write an exact formula using the Ï€ symbol.  But that's a case where computers really DO use the exact value of Ï€-- they just don't actually bother to calculate any numerical results with it. For instance, if you asked Maple to calculate the area under the curve y = sqrt(16 - x^2), it would tell you: 8Ï€ Versus a standard calculator, which might tell you an approximation: 25.13274123

There are lots ways to calculate pi, and not all of them are strictly geometric. Many of them are series, for example. Here's a very inefficient, but simple series that converges to pi: \pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \cdots If you add up a couple hundred terms, you'll get 3.14. Of course, nearly all computer programs only need a few digits of pi, so instead of calculating pi, they just store it, like Joshua Engel says. Many other important functions are routinely calculated as series. For example, the natural exponential is e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{3!} x^3 + \cdots It turns out that since most math libraries used in common programming languages represent real numbers as limited-precision floats, they don't have to bother using a method that can calculate an arbitrary number of digits. Just getting, say, 15 digits is good enough. For this purpose, it's more efficient to just pre-calculate some interpolating polynomials and store the coefficients in the same way pi is stored. So long as the function calls spit out floats that are correct to within their precision, everything's dandy. Other irrational numbers are calculated in other ways. Square roots, for example, can be calculated pretty efficiently using the "Babylonian method", which turns out to be equivalent to using the Newton-Raphson method. Arbitrary real powers can be calculated by reducing them with the formula below to logarithms and exponentials, which are typically implemented using the approach above. x^y = e^{y \ln x} Sines and cosines can be calculated with interpolating polynomials too. In some particularly simple processor architectures which don't support multiplication, though, there are alternative algorithms to calculate these: http://en.wikipedia.org/wiki/CORDIC.

There are algorithms to calculate pi to a certain number of digits. There are also whole trilogies written just to prove the algorithm and to calculate the approximate error. Let me share one with you. it is well known that [math] \arctan(1) = \frac{\pi}{4} [/math] It is also well known that [math] \arctan(x) = \sum_{k=0}^{+\infty}(-1)^k\frac{x^{2k+1}}{2k+1}[/math] So [math]\pi = 4\sum_{k=0}^{+\infty}\frac{(-1)^k}{2k+1} [/math] Now we only need to check how fast it converges. We can use the fact that it is alternate converging: Pi is in between two successive terms, and to know the average local difference, we need to know the difference between the two successive terms and divide by two. This difference equals: [math]\frac{4}{2k+1}[/math] Now for the calculations, we assume that we have used all 1.5 billion computers in the world to calculate 3 billion terms per second since the beginning of time. This means we know the sum of the first 2.12*10^36 terms. applying the previous equation yields: [math]8.8\cdot 10^{-37}[/math] or: we now know 36 digits of pi. ugh. Luckily, there are better algorithms. Let's derive one. We know a special formula involving tangents: [math] \tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} [/math] or, rewritten: [math]\arctan(a) + \arctan(b) = \arctan\left(\frac{a+b}{1-ab}\right)[/math] for example: [math]\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right)= \arctan\left(\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}}\right) = \arctan(1) = \frac{\pi}{4}[/math] Interesting... Suddenly, we have a formula for Pi that converges faster: [math] \pi = 4\sum_{k = 0}^{+\infty} \frac{(-1)^k}{(2k+1)\cdot 2^{2k+1}} + \frac{(-1)^k}{(2k+1)\cdot 3^{2k+1}} [/math] Now the difference evaluates to: [math]\frac{1}{(2k+1)\cdot 2^{2k+1}} \approx \frac{1}{4^{k+1}}[/math] which yields, when k = 2.12*10^36, around 10^36.3 decimals. Which is more than we'll ever need.

People have algorithms for calculating pi to the required number of digits.  Unlike some mathematical series, these converge quite fast.  There is the Album of Algorithmitic Techniques, which is a set of web pages that contain documented text and REXX code for doing things like square roots, and other things that REXX misses. On the other hand, even most rational numbers can not be handled by REXX exactly: you need to do 2/3 rather than 0.66666666667

Referencing a previous answer, the coincidence of infinite series with Pi is fascinating - and the explanation for it is fairly hard to grasp. (I think it might be related to "squaring the circle": I might have got hold of it for 30 seconds one time:). It was proven differently by Euler and by Ramanujan, to name two of many . Euler solved the "Basel Problem" (in 1735) using the simplest series of all: I'm just referring to addition of fractional exponents (reciprocals of square numbers) in parentheses - not the fancy stuff to the left of that. This series produces a continuously convergent value that is pi^2/6. It is extremely simple to program this (could be demonstrated in Excel, for example). I wrote a calculator that can do many of these natural number series "at the drop of a hat": http://naturalnumbers.org/#ZT I can certainly give you the source code.