In the following sections we're going to code up two algorithms.

The first, the square root, converges quickly, but like most quickly converging algorithms, can't be generalised to any other calculation: it only works for square roots. The computing landscape is sparsely populated with these quickly converging algorithms and the discovery of such algorithms are isolated events requiring the efforts or inspiration of some genius. Understanding these algorithms doesn't give you any help discovering new algorithms, but some people turn out to be better producing new algorithms than others, and these people understand all the known algorithms. Computer programmers spend a much effort looking for fast algorithms and the discovery and the discoverers are celebrated in the same way as the discovery of new continent is celebrated by the general populace.

One of the better known algorithm people is Donald Knuth (http://en.wikipedia.org/wiki/Donald_Knuth), who is famous for offering cash rewards ($2.56, a hexadecimal dollar according to Knuth) for finding mistakes in his code. Finding a mistake by Knuth is so rare that Knuth's checks (cheques) are one of computerdom's most prized trophies. People prefer to prominently display Knuth's check (cheque) on their wall for the gasps and admiration of vistors, than to deposit the money in their bank account.

Knuth is one of the people who have pushed the concept of mathematically provably correct code. While it's obvious to everyone that a plane manufacturer must show that their plane will fly, the equivalent proof of usability in computing is difficult to demonstrate. Computer code has sufficient traps, bugs and unpredictable responses to out of bounds input, that in the absence of tools to show that code does exactly what it's supposed to do and nothing else, computer programmers rarely attempt to prove that their code is correct. (One of the diagnostic features of computer programs is that that don't work well.) It seems that proving code correct is not generally possible. Knuth once said "Beware of bugs in the above code; I have only proved it correct, not tried it." Computer programmers have given up on provably correct code and currently have adopted the less rigourous, but hopefully achievable goal of of Programming by Contract (http://en.wikipedia.org/wiki/Design_by_contract). Programming by contract is used in Eiffel and Ada (and less so in C++). In languages which don't have Programming by Contract features, programmers are encouraged to put in equivalent statements (even if only in comments) in their code. The built in Programming by Contract features of Ada make it the language of choice for applications where safety is paramount (e.g. Air Traffic Control).

The discovery of the Fast Fourier Transform (http://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) and the Monte Carlo Method (http://en.wikipedia.org/wiki/Monte_Carlo_method), both of which came out of people who worked on the Manhattan project, have revolutionised signal processing and statistics respectively, creating whole industries which would not have been possible otherwise.

The second algorithm, which calculates the value of π, uses an easy to understand and generalisable method: numerical integration. Numerical integration is a brute force method that converges so slowly that it is only usable for a small number of significant figures. Often this is enough for computing purposes (you have to accept the long time, whether you like it or not). If you're doing a calculation for which no-one has discovered a quickly converging algorithm, numerical integration will often be your only choice. While in principle you could do a few square roots by hand, numerical integration, being a brute force method, requires computers (or supercomputers) to be usable at all.