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Simple numerical methods for calculating the digits of Pi
How the letter 'pi' came in mathematics?
When I calculate the value of $22/7$ on a calculator, I get a number that is different from the constant $\pi$.
Question: How is the $\pi$ constant calculated? (The simple answer, not the Wikipedia calculus answer.)
$\pi$ is not equal to $22/7$. As a matter of fact, it cannot be expresses as a ratio of two whole numbers at all. It is a transcendental number, i.e. not algebraic, i.e. there exists no algebraic equation $\displaystyle\sum_{i=0}^Na_ix^i = 0$ that has $\pi$ as its root. We can only approximate the value of $\pi$. This has been proven by Ferdinand von Lindemann in the 19th century. I suggest you read more about transcendental numbers to get more familiar with the concept.
In answer to your second question, NOVA has an interactive exhibit that uses something like Archimedes method for approximating $\pi$. Archimedes method predates calculus, but uses many of its concepts.
Note that "simple" and "calculus" are not disjoint concepts.
Pi ($\pi$) is a mathematically defined, a priori constant, one definition being the ratio of the circumference of a circle to its diameter. In some places, it also has (had) a (mathematically inaccurate) legal definition. The mathematical definition is universally agreed upon. In the decimal number system, it has an infinite decimal expansion, and so cannot be represented (mathematically) exactly as a decimal number (in a finite amount of space). For most non-mathematical purposes, (e.g. architecture, agriculture), accuracy past a few decimal places is not necessary. For astronomy, aviation & aeronotics, more accuracy is usually needed -- maybe ten or twenty decimal places (please correct me if I'm wrong). The estimation of $\pi$ has a very long history (compared to the span of recorded history and of the history of mathematics). There are in turn many books on Pi. For example, The Joy of Pi & A History of Pi to name a few.
There are even more methods of calculating $\pi$, including, amazingly, some relatively recent developments. Perhaps the easiest method, if you want to avoid advanced mathematics (and calculus is too advanced) and take a few things on faith, is to use simple geometry and rely on a trigonometry function (which could be argued is circular reasoning since we will use the fact that $360$ degrees equals $2\pi$ radians). You can use, for example, the area of a regular $n$-gon (for $n$ large) with vertices on a circle of radius $1$ as an approximation for $\pi$. This area is then $$ A_n=n\cdot\sin\frac{\theta_n}{2}\cos\frac{\theta_n}{2}=\frac{n}{2}sin\theta_n \qquad\text{for}\qquad \theta_n=\frac{360\text{ deg}}{n}=\frac{2\pi}{n}\text{ (rad)} $$ (draw a triangle from the center to two adjacent vertices, bisect this triangle by a line from the center to the midpoint of the vertices, calculate lengths, the area of this whole triangle, and multiply by the number of these which is $n$). With a little calculus knowledge, we can also verify that in fact (when $\theta_n$ is in radians!), $$ \lim_{n \to \infty}A_n= \lim_{n \to \infty}\frac{sin\frac{2\pi}{n}}{\frac{2}{n}}= \lim_{x \to 0}\frac{\sin\pi x}{x}=\pi\;. $$ A more recently found formula (Bailey–Borwein–Plouffe, 1995) whose statement requires not so much math is $$ \pi=\sum_{n=0}^{\infty} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right) \left( \frac{1}{16} \right)^n $$ which converges very quickly to the answer, i.e., not very many terms are needed to get any given desired accuracy, since as can be seen, the $n$th term is (much) less than $16^{-n}$, so that the first $n$ terms easily give $n$ hexadecimal (base 16) digits, or $n\log{16}\simeq{1.2n}$ decimal places of accuracy.
The (early) history of approximations to $\pi$ can also be (roughly) traced by the its (various) continued fraction expansions, e.g. $\pi = 3 + \frac{1}{7+}\frac{1}{15+}\frac{1}{1+}\frac{1}{292+}\dots$, with the number of terms used increasing (roughly!) with historical time; so $3$ is the simplest approximation, then $22/7$, then $333/106$, etc.
As already mentioned by others, $\pi \neq 22/7$. To answer your question, how to compute digits of $\pi$, $\pi$ can be obtained as a limit of sequence of numbers which can be computed to great accuracy.
For instance, the Madhava-Leibniz formula can be used to compute the digits of $\pi$. In fact, it was one of the first ways to compute the digits of $\pi$. $$\pi = 4 \times \left(1 - \frac13 + \frac15 - \frac17 + \frac19 - \cdots \right)$$ If we truncate the above series at $n$-terms, i.e. if we denote $$x_n = 4 \times \left(1 - \frac13 + \frac15 - \frac17 + \frac19 - \cdots + (-1)^{n} \frac1{2n+1}\right)$$ error analysis will give you that the $|| x_n - \pi|| \leq \left( \frac2n \right)$. The convergence of this series is hence slow. If you want an accuracy upto $2$ digits, you need to take of the order of $200$ terms.
There are other well-known series/ products which converge to $\pi$.
