The real number $\pi \in \mathbb{R}$ is the length of one semi-circle of the unit circle. Hence $2 \pi$ is the circumference of the full unit circle.
This number plays a central role in trigonometry, but – especially via its relation to complex analysis via Euler's formula – it also appears in many other situations.
The real number $\pi \in \mathbb{R}$ (occasionally called Ludolph’s number —make that very occasionally, nowadays— or Archimedes’ constant) is the minimal positive root of the sine function $sin_\mathbb{R} : \mathbb{R}\to\mathbb{R}$, which itself is the unique solution to the differential equation/initial value problem
Alternatively, $\pi$ may be defined as $\frac{1}{2} \tau$, where $\tau$ is the period of the same function sin.
An alternative description of $sin$ is as the unique function $\mathbb{R}\to\mathbb{R}$ continuous at $0$ and otherwise satisfying
and
although this makes practical calculations rather difficult. (If you really care to know why this characterization works, you can look here. Or at this nForum discussion.)
Either of the definitions of $sin$ above mentions a thing, and claims some properties for it, in order to indirectly describe some other thing (in this case, $\pi$). To properly make sense of either definition, it is necessary to ensure that one description or the other actually does define some thing, that there is in fact exactly one such thing so defined, and that it has the needed properties for the definition of $\pi$ to make sense — particularly, that the roots of $sin$ are isolated, and that there is a positive root. These might make a decent exercise in a first course in classical analysis.
A related but perhaps more conceptual description is via complex analysis. Recall that the standard exponential function is defined by the formula
for all complex numbers $z$ (as an absolutely convergent series), and satisfies the equations
$\exp(z + w) = \exp(z) \cdot \exp(w)$,
$\exp(\widebar{z}) = \widebar{\exp(z)}$.
It follows that for $z \in i \mathbb{R}$, where $z + \widebar{z} = 0$, we have ${|\exp(i t)|} = 1$.
Hence $t \mapsto \exp(i t)$ defines a Lie group homomorphism $\phi$ from the additive group $\mathbb{R}$ to the multiplicative group $S^1$ of unit complex numbers. It may be checked that this is a local diffeomorphism: a local inverse to $z \mapsto \exp(z) - 1$ in a neighborhood of $0$ is given by
Thus $\phi: \mathbb{R} \to S^1$ maps a small neighborhood $U$ of $0$ onto a small neighborhood $V$ of $1$ in $S^1$. As $S^1$ is connected, we have $S^1 = \bigcup_{n \geq 1} V^n$ where $V^n$ consists of $n$-fold products of elements in $V$, and in fact a finite union $\bigcup_{n=1}^m V^n$ suffices since $S^1$ is compact. It follows easily that the homomorphism $\phi: \mathbb{R} \to S^1$ is surjective.
In particular, there exists some element $t \in \mathbb{R}$ such that $\phi(t) = -1$. We may define $\pi$ to be the least positive $t$ such that $\phi(t) = -1$ (cf. Euler’s formula $e^{\pi i} + 1 = 0$). Indeed, it is slightly nicer just to observe that $\ker(\phi)$ is nontrivial (since $S^1$ has torsion elements whereas $\mathbb{R}$ does not) and must be a discrete subgroup, generated by a unique positive $\tau$, and one defines $\pi = \tau/2$.
These considerations suggest why mathematicians sometimes consider $\tau = 2\pi$ to be “more fundamental” than $\pi$, or even that $2\pi i$ is the truly fundamental quantity, being (up to sign) the unique generator of the kernel of $\exp: \mathbb{C} \to \mathbb{C}^\times$. This is also reflected in the familiar contour integral formula
that on some level pervades complex analysis.
A sequence that approximates $\pi$ that was available to Archimedes (using his “method of exhaustion”) involves calculating perimeters of inscribed $2^n$-gons (squares, octagons, etc.) in a unit circle, using half-angle formulas. This brief description is, for that matter, available to high-school students, so we can just summarize the result.
Define a sequence by the recursion $x_1 = 0$, $x_{n+1} = \sqrt{2 + x_n}$. A typical term has nested square roots, like $x_4 = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$. This sequence rapidly approaches $2$ in the limit; put
The statement is that $y_n$ is half the perimeter of the inscribed $2^{n+1}$-gon for $n \geq 1$, so that $y_n$ is an increasing convergent sequence which tends to $\pi$ in the limit. Incidentally, simple high-school algebra shows
and thus we may deduce Vieta’s formula (see below).
Archimedes used a similar idea, in effect viewing $2\pi$ as squeezed between the perimeters of inscribed and circumscribed polygons with $3 \cdot 2^n$ sides each (starting with a hexagon). By considering $n = 5$, he was eventually led to his famous estimate $3 + \frac{10}{71} \lt \pi \lt 3 + \frac1{7}$.
Leibniz formula: $\frac{\pi}{4} = 1 - \frac1{3} + \frac1{5} - \ldots$. Based on the power series for the arctangent
this series for $\pi/4$ obviously converges slowly, but more sophisticated expressions in terms of the arctangent have better convergence rates, such as the following first observed by Machin:
Integral formulas (based ultimately on the Riemann integral, the Henstock integral, the Lebesgue integral, or the like):
Also based on the arctangent:
Or the arcsine?:
(but this is an improper integral? if one is using the Riemann integral).
Of course, you can come up with many modifications of these (including proper Riemann integrals based on the arcsine).
Product formulas:
This was perhaps the first infinite product in the history of mathematics.
Euler’s product formula:
One point is that putting $x = \frac{\pi}{2}$, one can derive another famous formula, namely
Wallis’s formula:
Alternatively, this may be derived from Stirling's approximation? for $n!$, or rather starting from an asymptotic formula which follows from Stirling’s approximation:
Euler series: let $\zeta(s)$ be the Riemann zeta function. By applying the logarithmic derivative to Euler’s product formula above, one may derive the following partial fraction expansion for the cotangent:
Expanding each of the summands in a geometric series and rearranging terms, one may derive
On the other hand, one has
where the $B_n$ are the Bernoulli numbers. By comparing coefficients, one arrives at
or, after rearranging,
which has as special cases the formulas $\pi^2 = 6\zeta(2)$, $\pi^4 = 90\zeta(4)$, $\pi^6 = 945\zeta(6)$.
Continued fractions: the regular continued fraction expansion of $\pi$ is highly irregular, but several generalized continued fraction expansions listed in Wikipedia include
Famously, $\pi$ is an irrational number, although proving this fact is no triviality. The first substantive idea of proof was given by Lambert, who discovered the following continued fraction expression for the tangent:
The basic inference was that $\tan(z)$ must therefore be irrational for rational values of $z$, and since $\tan(\pi/4) = 1$, it follows that $\pi/4$ is irrational. This was not a completely rigorous proof since Lambert had failed to prove the irrationality of certain infinite continued fractions, but Legendre provided the missing lemma in 1806. Actually Legendre proves more: applying his lemma to the continued fraction for $\sqrt{q}\; \tan(\sqrt{q})$ (derived from the above), he shows this expression is irrational for rational $q$, and therefore not even $\pi^2$ can be rational. He goes on to speculate that $\pi$ is in fact transcendental.
The transcendence of $\pi$ was finally established by Lindemann in 1882, by adapting methods of Charles Hermite who had established the transcendence of e in 1873. This definitively proved the impossibility of solving the problem of “squaring the circle” by means of ruler-and-compass constructions. A far more general theorem was then established by Weierstrass:
If $\alpha_1, \ldots, \alpha_n$ are algebraic numbers linearly independent over $\mathbb{Q}$, then $\exp(\alpha_1), \ldots, \exp(\alpha_n)$ are algebraically independent over $\mathbb{Q}$ (which is to say that the field $\mathbb{Q}(\exp(\alpha_1), \ldots, \exp(\alpha_n))$ has transcendence degree $n$ over $\mathbb{Q}$).
A proof is given in Wikipedia. It follows that if $\pi$ and therefore $i\pi$ were algebraic, then $\exp(i\pi) = -1$ would be transcendental, which is absurd.
A whole chapter on $\pi$ written for the mathematically literate is given in
The material on the Euler series for $\pi^{2 n}$ in terms of the zeta function was extracted from
The proof of irrationality of $\pi$ given by Legendre is said to be given in the following treatise: