May 27, 2009

For many in the real world, e gets no love. e, the base of the natural logarithm, actually has much more power than its other, better-known irrational cousin, pi or π. Most who have taken middle school math can tell you that pi is the ratio between the circumference and diameter of a circle,

No one can doubt the coolness of this relationship, and how its result is a crazy irrational number that never changes with the size of the circle.

The other main irrational number–e–however, does far more than explain a simple geometrical relation.


As x increases from zero (exclusive) to infinity, the function increases from negative infinity to positive infinity. (thanks Wikipedia)

Those who have taken calculus may remember that the graph, , has a derivative of…wait for it…. No other function has this property. In essence, this means exponential functions have the distinct ability to incorporate their results into the next iteration of initial conditions (Thomasina  uses this idea in Tom Stoppard’s excellent play Arcadia).

Since the exponential function involves continual change, it’s perfect for modelling things like population growth, continually compounded interest, and a large array of physical processes from heat transfer to fluid flow, really anything that involves a feedback loop. Differential equations is an area of math that entirely relies upon the exponential function, and differential equations is vitally important for pretty much every field of engineering.

Huge swaths of our modern world rely upon a very simple (albeit never-ending) number. No other number has that kind of power, especially not puny pi or the golden ratio.

For those of you who wasted your time memorizing digits of pi, here are the first two million of e.


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