How are the digits of an irrational number computed?

Question

I have a question about irrational or just long sequences of rational numbers.

What method/algorithm is used to determine what digit will come next in the sequence, I mean how do they know for sure? It is a random sequence after all right?

Just tell me how they find the next number in "pi". I know they do it with computers and math, but what is the theory behind knowing what number comes next.

Answer 1

Only for a tiny fraction of irrational numbers there is an algorithm that produces the digits in their decimal expansion, because there is only a countable number of algorithms!

Nevertheless, every real number has a decimal expansion, which is unique except for terminating decimals like $1=0.999\cdots$.

There are several methods for $\pi$.

Answer 2

If you consider the example of $\pi$, there are various formulas which converge at $\pi$. I don't remember one about $\pi$ but we can define $e$ as $$\lim_{x\ \rightarrow \infty} \huge( \normalsize 1 + \frac{1}{x}\huge )\normalsize ^x$$ As you begin to take higher and higher values of $x$, you will notice that the answer begins to get nearer and nearer to $e$.

Answer 3

One simple formula for pi is $$\frac\pi4=1-\frac13+\frac15-\frac17+\frac19\cdots$$ This formula comes from calculus. It also uses the fact that, in radians, $\tan(\pi/4)=1$.
The formulas that people actually use approach $\pi$ much more quickly than this one.