fredag 22 november 2013

What is $\pi$ and $e$?

There are two numbers with a special stature in mathematics:
where $\exp(t)$ is the exponential function usually defined by 
  • $\exp(t)=\lim_{n\rightarrow\infty}(1+\frac{t}{n})^n.$     (*)
The geometric definition of $\pi$ does not give direct information about its numerical value and the definition of $e$ may appear to be ad hoc without connection to either geometry or physics.

In BodyandSoul as Constructive Calculus both $\pi$ and $e$ are constructively defined through solutions of basic initial value problems solved by time stepping. More precisely, $\pi$ is defined as the smallest positive root of the equation $\sin(t)=0$, where $u(t)=\sin(t)$ and $v(t)=cos(t)$ is the solution to the initial value problem modeling a harmonic oscillator: 
  • $\frac{du}{dt} -  v =0$ and $\frac{dv}{dt} + u=0$ for $t > 0$, $u(0)=0$ and $v(0)=1$.  (**) 
Further $u(t)=\exp(t)$ is the solution to the basic initial value problem (expressing "exponential growth" with the growth rate $\frac{du}{dt}$ equal to $u$ itself):
  • $\frac{du}{dt}=u$ for $t > 0$ and $u(0)=1$.   (***)
In particular, 
  • $(1+\frac{t}{n})^n$ 
is the result of solving (***) by time stepping with time step $\frac{t}{n}$ with
  • $u(t+\frac{t}{n})=u(t)+\frac{t}{n}u(t) = (1+\frac{t}{n})u(t)$, 
which converges to $\exp(t)$ as the time step $\frac{t}{n}$ goes to zero with $n$ tending to infinity.

Introducing and defining the numbers $\pi$ and $e$ this way, gives both an understanding why they are so fundamental (by expressing basic properties of solutions to basic mathematical equations connecting to basic physics) and also shows how the (decimal expansions of the) numbers can effectively be computed.

PS1 Notice that once $\sin(t)$ and $\cos(t)$ have been defined as solutions of (**), it follows that $(\cos(t),\sin(t))$ can be geometrically interpreted as the coordinates of a point moving along a unit circle with unit speed and thus $t$ a measure of angle as arc length. The geometric interpretation of $\pi$ thus follows from numerical algebra and not the other way around as in Standard Calculus.

PS2 In Standard Calculus $\sin(t)$ and $\cos(t)$ are defined geometrically as quotients of the lengths of the sides of right-angled triangles again without access to the numerical values except for a few special values of the angle $t$.

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