February 06, 2010

Celebrating e-day

Tomorrow 2/7 is e-day.

e or Euler's number is a number that is approximately 2.718281828, so that is why someone chose 2/7 as an e-day. But e is an irrational number, so its decimal expansion is never-ending and never-repeating.

Why is this number e so important that people have even named a day after it?

If you've studied calculus, you already know at least part of the story. But even if you haven't, I'll try to unravel at least the most basic feature of e.

Consider the exponential function ex. It is graphed below.

It has one remarkable property: when you draw a tangent to it at any particular point, the SLOPE of that tangent is always the value of the function ex at that point. See below two examples:

A tangent at 0.69 with slope 2

A tangent at -0.69 with slope 0.5

This feature is usually expressed this way: ex is its own derivative, or the derivative of ex is ex. There exists NO other function with that property!

Here's also an interesting explanation about one fundamental property of e as it relates to growth: An intuitive guide to exponential functions. This guide is meant for BEGINNERS. It's not based on calculus. Instead, it starts by looking at a basic system that doubles after an amount of time, and refines this basic system to arrive at the idea of e.

But that's just for starters. The number e has popped up in all kinds of interesting places for mathematicians over the years. For example

One famous equation ties in e, Pi, 1, 0, and the imaginary unit i — five important numbers in mathematics:

eiπ + 1 = 0

See even more representations of e (infinite series, continued fractions, infinite products, and special limits). It truly is quite a number! I don't claim to understand why it is involved in all these things - like I said, it seems to "pop up" in all kinds of places. But maybe you can see a glimpse of why it is so special.

Curiously, in some places, e-day means something different from a day dedicated to the number e. But to celebrate the e-day in honor of the number e, whatever your language, I suggest making or baking a food that either starts with "e" or has "e" as a prominent part of its name... such as chEEsE or browniEs. It's your choice!
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