Mathematics behind a calculator Mathematics behind a calculator
I have tried googling around for a bit, but I lack the technical terms and mathematical knowledge to find results that gives me mathematical intuition about how a calculator work mathematically. 
If you have time, please give me a short or long description on how a calculator works. If it is a long and thorough answer, I will start a bounty on 50 rep to award this answer with. 
Otherwise, it will suffice with terms so that I may research further on what is "in" a calculator. 
I sincerely hope that I have not misunderstood the concept behind a calculator, please leave me a comment if so.
PS I want to know how a calculator calculates more advanced expressions, not just division but, for example, $e$, $\sum$, $\int$, trigonometric functions and et cetera.
 A: *

*Many of these calculations are done using series. For example, the
value of $e$ (if not saved in a hard-drive) can be calculated as: $$
   e = \sum_{n=0}^{\infty} \frac{1}{n!} $$ Altough there are many other
examples, for sine, cosine, and et cetera, that are calculated using
series. Note that a calculator can't sum infinite terms, so they will
sum a finite (however large) number of parcels to give you an decent
approximation of the actual value.

*For integrals and derivatives, there are a lot of algorithms who can do this, and to better understand them you can start studying a topic called numerical analysis, which is a "way" of evaluating integrals, derivatives and other operators, using computer algorithms.
At the end of the day, all of those reduce to the simplest operations such as addition or multiplication. Lets think, for example, in a simple, intuitive algorithm to evaluate the derivative of $f(x) = x^2$ at $x=2$:


*

*Define an small $h$, such as $h = 0.000001$.

*Use of the derivative formula. Calculate $(2+0.000001)^2 - 2^2$

*Divide the result by $0.000001$.

*Return the value of the fraction.




You can do this calculation and you'll verify that approximates the value of the derivative.
However, there are much more complex, efficient and exact algorithms out there to evaluate these calculations. Numerical Analysis is an very interesting topic.
A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
 $\ds{\large\bullet}$ Constants like $\ds{\expo{}}$, $\ds{\ln\pars{2}}$, $\ds{\pi}$, etc$\ldots$ can be kept, with some finite number of decimals,  in a 'hard disk'.
$\ds{\large\bullet}$ Many functions are evaluated in 'small intervals'. For example, $\ds{\root{x}}$ just need to be evaluated in $\ds{\pars{0,1}}$ because
$\ds{\root{x} = 1/\root{1/x}}$. Similarly, $\ds{\ln\pars{x} = -\ln\pars{1/x}}$.
Newton-Rapson Method is helpful for evaluations in $\ds{\pars{0,1}}$.
A: The computational unit underlying computer mathematics is the Arithmetic Logic Unit (ALU), which is a combinatorial circuit that performs the basic functions of addition, subtraction, multiplication, division, negation (inversion), absolute value and a few other functions on binary numbers.  Person-centuries have gone into optimizing these circuits for speed, accuracy, and functionality.
As @FelixMartin points out, it is very simple to store constants such as $e$, $\pi$, etc.
Beyond such a high-level "explanation," one would need to know precisely what you're seeking.
