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pi

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Idea

The real number π\pi \in \mathbb{R} is the length of one semi-circle of the unit circle. Hence 2π2 \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.

Definition via real analysis

The real number π\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 :sin_\mathbb{R} : \mathbb{R}\to\mathbb{R}, which itself is the unique solution to the differential equation/initial value problem

sin =sin sin_\mathbb{R}'' = -sin_\mathbb{R}
sin (0)=0 sin_\mathbb{R}(0) = 0
sin (0)=1. sin_\mathbb{R}'(0) = 1 .

Alternatively, π\pi may be defined as 12τ\frac{1}{2} \tau , where τ\tau is the period of the same function sin.

An alternative description of sinsin is as the unique function \mathbb{R}\to\mathbb{R} continuous at 00 and otherwise satisfying

1x 2sin(x)x1 1 - x^2 \leq \frac{sin(x)}{x} \leq 1

and

sin3x=3sinx4sin 3x sin 3 x = 3 sin x - 4 sin^3 x

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.)

Tangential Remarks

Either of the definitions of sinsin above mentions a thing, and claims some properties for it, in order to indirectly describe some other thing (in this case, π\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 π\pi to make sense — particularly, that the roots of sinsin are isolated, and that there is a positive root. These might make a decent exercise in a first course in classical analysis.

Definition via complex analysis

A related but perhaps more conceptual description is via complex analysis. Recall that the standard exponential function is defined by the formula

exp(z)= n0z nn!=1+z+z 22+z 36+\exp(z) = \sum_{n \geq 0} \frac{z^n}{n!} = 1 + z + \frac{z^2}{2} + \frac{z^3}{6} + \ldots

for all complex numbers zz (as an absolutely convergent series), and satisfies the equations

  • exp(z+w)=exp(z)exp(w)\exp(z + w) = \exp(z) \cdot \exp(w),

  • exp(z¯)=exp(z)¯\exp(\widebar{z}) = \widebar{\exp(z)}.

It follows that for ziz \in i \mathbb{R}, where z+z¯=0z + \widebar{z} = 0, we have |exp(it)|=1{|\exp(i t)|} = 1.

Hence texp(it)t \mapsto \exp(i t) defines a Lie group homomorphism ϕ\phi from the additive group \mathbb{R} to the multiplicative group S 1S^1 of unit complex numbers. It may be checked that this is a local diffeomorphism: a local inverse to zexp(z)1z \mapsto \exp(z) - 1 in a neighborhood of 00 is given by

zlog(1+z)=zz 22+z 33z \mapsto \log (1 + z) = z - \frac{z^2}{2} + \frac{z^3}{3} - \ldots

Thus ϕ:S 1\phi: \mathbb{R} \to S^1 maps a small neighborhood UU of 00 onto a small neighborhood VV of 11 in S 1S^1. As S 1S^1 is connected, we have S 1= n1V nS^1 = \bigcup_{n \geq 1} V^n where V nV^n consists of nn-fold products of elements in VV, and in fact a finite union n=1 mV n\bigcup_{n=1}^m V^n suffices since S 1S^1 is compact. It follows easily that the homomorphism ϕ:S 1\phi: \mathbb{R} \to S^1 is surjective.

In particular, there exists some element tt \in \mathbb{R} such that ϕ(t)=1\phi(t) = -1. We may define π\pi to be the least positive tt such that ϕ(t)=1\phi(t) = -1 (cf. Euler’s formula e πi+1=0e^{\pi i} + 1 = 0). Indeed, it is slightly nicer just to observe that ker(ϕ)\ker(\phi) is nontrivial (since S 1S^1 has torsion elements whereas \mathbb{R} does not) and must be a discrete subgroup, generated by a unique positive τ\tau, and one defines π=τ/2\pi = \tau/2.

These considerations suggest why mathematicians sometimes consider τ=2π\tau = 2\pi to be “more fundamental” than π\pi, or even that 2πi2\pi i is the truly fundamental quantity, being (up to sign) the unique generator of the kernel of exp: ×\exp: \mathbb{C} \to \mathbb{C}^\times. This is also reflected in the familiar contour integral formula

|z|=1dzz=2πi\int_{{|z|} = 1} \frac{d z}{z} = 2\pi i

that on some level pervades complex analysis.

“Archimedean” definition

A sequence that approximates π\pi that was available to Archimedes (using his “method of exhaustion”) involves calculating perimeters of inscribed 2 n2^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=0x_1 = 0, x n+1=2+x nx_{n+1} = \sqrt{2 + x_n}. A typical term has nested square roots, like x 4=2+2+2x_4 = \sqrt{2 + \sqrt{2 + \sqrt{2}}}. This sequence rapidly approaches 22 in the limit; put

y n=2 n2x n.y_n = 2^n \sqrt{2 - x_n}.

The statement is that y ny_n is half the perimeter of the inscribed 2 n+12^{n+1}-gon for n1n \geq 1, so that y ny_n is an increasing convergent sequence which tends to π\pi in the limit. Incidentally, simple high-school algebra shows

y n=2 k=2 n2x ky_n = 2 \prod_{k=2}^n \frac{2}{x_k}

and thus we may deduce Vieta’s formula (see below).

Archimedes used a similar idea, in effect viewing 2π2\pi as squeezed between the perimeters of inscribed and circumscribed polygons with 32 n3 \cdot 2^n sides each (starting with a hexagon). By considering n=5n = 5, he was eventually led to his famous estimate 3+1071<π<3+173 + \frac{10}{71} \lt \pi \lt 3 + \frac1{7}.

Other formulas

  • Leibniz formula: π4=113+15\frac{\pi}{4} = 1 - \frac1{3} + \frac1{5} - \ldots. Based on the power series for the arctangent

    arctan(x)=xx 33+x 55,\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots,

    this series for π/4\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:

    π4=4arctan(15)arctan(1239)\frac{\pi}{4} = 4\arctan(\frac1{5}) - \arctan(\frac1{239})
  • Integral formulas (based ultimately on the Riemann integral, the Henstock integral, the Lebesgue integral, or the like):

    • Also based on the arctangent:

      π=4 x=0 1dxx 2+1. \pi = 4 \int_{x=0}^1 \frac{\mathrm{d}x}{x^2 + 1} .
    • Or the arcsine?:

      π=2 x=0 1dx1x 2 \pi = 2 \int_{x=0}^1 \frac{\mathrm{d}x}{\sqrt{1 - x^2}}

      (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:

    • Vieta’s formula:
π=2×22×22+2×22+2+2×22+2+2+2×\pi= 2\times\frac{2}{\sqrt{2}}\times \frac{2}{\sqrt{2+\sqrt{2}}}\times \frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\times\cdots

This was perhaps the first infinite product in the history of mathematics.

  • Euler’s product formula:

    sin(x)=x n1(1x 2π 2n 2)\sin(x) = x \prod_{n \geq 1} (1 - \frac{x^2}{\pi^2 n^2})

    One point is that putting x=π2x = \frac{\pi}{2}, one can derive another famous formula, namely

  • Wallis’s formula:

    π2=lim n2123434565672n2n12n2n+1\frac{\pi}{2} = \lim_{n \to \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \ldots \cdot \frac{2 n}{2 n - 1} \cdot \frac{2 n}{2 n + 1}

    Alternatively, this may be derived from Stirling's approximation? for n!n!, or rather starting from an asymptotic formula which follows from Stirling’s approximation:

    (2nn)2 2nπn.\binom{2 n}{n} \sim \frac{2^{2 n}}{\sqrt{\pi n}}.
  • Euler series: let ζ(s)\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:

    cot(x)=1x2 n1xπ 2n 2x 2.\cot(x) = \frac1{x} - 2\sum_{n \geq 1} \frac{x}{\pi^2 n^2 - x^2}.

    Expanding each of the summands in a geometric series and rearranging terms, one may derive

    xcot(x)=12 m1ζ(2m)x 2mπ 2m.x \cot(x) = 1 - 2\sum_{m \geq 1} \zeta(2 m)\frac{x^{2m}}{\pi^{2 m}}.

    On the other hand, one has

    xcot(x)=ix+2ixe 2ix1=1+ n2B n(2ix) nn!x\cot(x) = i x + \frac{2 i x}{e^{2 i x}-1} = 1 + \sum_{n \geq 2} B_n \frac{(2 i x)^n}{n!}

    where the B nB_n are the Bernoulli numbers. By comparing coefficients, one arrives at

    2π 2mζ(2m)=(1) m2 2m(2m)!B 2m-\frac{2}{\pi^{2 m}} \zeta(2 m) = (-1)^m \frac{2^{2 m}}{(2 m)!}B_{2 m}

    or, after rearranging,

    π 2k=(1) k1(2k)!2 2k1B 2kζ(2k)\pi^{2 k} = (-1)^{k-1} \frac{(2 k)!}{2^{2 k - 1}B_{2 k}}\zeta(2 k)

    which has as special cases the formulas π 2=6ζ(2)\pi^2 = 6\zeta(2), π 4=90ζ(4)\pi^4 = 90\zeta(4), π 6=945ζ(6)\pi^6 = 945\zeta(6).

  • Continued fractions: the regular continued fraction expansion of π\pi is highly irregular, but several generalized continued fraction expansions listed in Wikipedia include

    π=41+1 22+3 22+5 22+7 22+9 22+\pi=\textstyle \frac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}

Irrationality and transcendence

Famously, π\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:

tan(z)=z1z 23z 25z 27.\tan(z) = \textstyle \frac{z}{1 - \textstyle \frac{z^2}{3 - \textstyle \frac{z^2}{5 - \textstyle \frac{z^2}{7 - \ldots}}}}.

The basic inference was that tan(z)\tan(z) must therefore be irrational for rational values of zz, and since tan(π/4)=1\tan(\pi/4) = 1, it follows that π/4\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 qtan(q)\sqrt{q}\; \tan(\sqrt{q}) (derived from the above), he shows this expression is irrational for rational qq, and therefore not even π 2\pi^2 can be rational. He goes on to speculate that π\pi is in fact transcendental.

The transcendence of π\pi was finally established by Lindemann in 1882, by adapting methods of 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:

Theorem

If α 1,,α n\alpha_1, \ldots, \alpha_n are algebraic numbers linearly independent over \mathbb{Q}, then exp(α 1),,exp(α n)\exp(\alpha_1), \ldots, \exp(\alpha_n) are algebraically independent over \mathbb{Q} (which is to say that the field (exp(α 1),,exp(α n))\mathbb{Q}(\exp(\alpha_1), \ldots, \exp(\alpha_n)) has transcendence degree nn over \mathbb{Q}).

A proof is given in Wikipedia. It follows that if π\pi and therefore iπi\pi were algebraic, then exp(iπ)=1\exp(i\pi) = -1 would be transcendental, which is absurd.

References

A whole chapter on π\pi written for the mathematically literate is given in

  • Ebbinghaus et al., Numbers, GTM (Readings in Mathematics) 123, Springer (1991).

The material on the Euler series for π 2n\pi^{2 n} in terms of the zeta function was extracted from

  • Kenneth Ireland and Michael Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84, Springer-Verlag (1982).

The proof of irrationality of π\pi given by Legendre is said to be given in the following treatise:

  • A.-M. Legendre, Eléments de géométrie (11th edition), Note IV. F. Didot, Paris (1817). Translation by Charles Davies: (web)

Last revised on October 5, 2018 at 07:35:54. See the history of this page for a list of all contributions to it.