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Euler's formula for e^x

We want to prove the identity \begin{equation}\label{eq:euler_e} e^x = \lim_{n\rightarrow\infty} \left( 1+\frac{x}{n}\right)^n.

Practice page

This page will help you master the calculations required to perform the day-of-the-week trick. Click on the buttons to generate random numbers, months, years or dates, and to check your results. (If the page seems faulty, check that have JavaScript enabled in your web browser.)

How does it work? A detailed example

Let's see how this works using the birth date of Nelson Mandela, 18 July 1918, as an example.

Doubling the cube using origami — proof

Figure 1

First we show that the first three steps amount to dividing the height of the square into three equal parts.

Trisecting the angle using origami — proof

In the image on the right the angle between the blue line and the bottom edge is the angle to be trisected, and we must show that $\alpha =\beta =\gamma$.

The following applet, kindly provided by Chris Jennings, allows you to play with the paramters governing the diffusion rates and see a pattern evolve. The applet requires that your browser support at least the Java 6 runtime. Click here to install the latest version of Java. Unfortunately, this applet does not currently work on all Intel-based Macs.