 ## Wednesday, December 28, 2011

### 95: Happy e Day

Before I start, I'd like to thank listener "Transfinitematrices",
who wrote another nice review of this podcast on iTunes.  Now, on to
the podcast.
You may have heard of 'pi day', the celebration held at geeky locales
every March 14th in honor of the constant pi, equal to approximately 3.14,
which describes the ratio between the circumference & diameter of a
circle.   But why should pi have all the fun?  Other mathematical constants
deserve their own holidays too.  Probably the best-known transcendental
constant after pi is 'e', equal to approximately 2.718.  And as you would
guess, a select group of ubergeeks on the net have designated February 7 as
e Day.  The traditional e Day celebration consists of E-clairs and
brown-Es.  But, just like we have to give thenks on Thanksgiving before
digging into the yummy food, we should review the mathematical significance
of e before starting on our tasty desserts.
What is the definition of e?  To understand the most straightforward
definition, let's first review the basic operation of calculus, the
derivative.  The derivative measures the slope of a curve, or how steeply
it is rising, at any point.  Suppose you ask the question:  is there any
curve whose derivative is precisely equal to the height of the curve at
each point?  As it turns out, the function that meets this criteria is f(x)
= e to the x power, or constant multiples of that.  The inverse function,
f(y) = natural log of y, written as ln y, also has a useful property: its
derivative at any point y is 1/y.  Due to these properties, a lot of
calculus calculations involving exponentials and logarithms can be
simplified by basing them around e.
Another consequence of these properties is that e tends to show up in
various calculations involving continuous probability.  When the
mathematician Jacob Bernoulli first discovered e in the late 1600s, he was
working on the 'compound interest problem'.  Let's assume you put \$1 in the
bank at 100% interest.  If you just receive the 100% at the end of the
year, you will get \$1 more, for a total of \$2.  But if you receive 50% of
that interest at each of two points during the year, you end up with \$1 *
1.5^2, or \$2.25.  You might argue that the 'fairest' system would be one
that would pay interest in as many small installments as possible, to yield
the maximum amount, or "continuous compounding".  Because the n intervals
each get 1/nth of the interest, the total doesn't approach infinity as one
adds more and more intervals-- it gradually converges.  And the amount it
converges on, equal to the limit of (1+1/n) to the nth power as n
approaches infinity, is precisely equal to e dollars.  In general, an
account with an interest rate of R will multiply its contents by e to the
Rth power each year under continuous compounding.
e also plays a role in a surprising equation known as Euler's Identity,
discovered by Swiss mathematiciam Leonard Euler in the 1700s:  e to the (pi
* i) + 1 = 0.  This equation brings together five of the most famous
mathematical constants:  1 and 0, our familiar pi, the imaginary number i
defined as the square root of -1, and of course e.  Several polls of
professional mathematicians have labeled this equation as the "greatest
equation ever", for the way it brings together such seemingly unrelated
constants into one neat formula.  Some have gone rather overboard, as in
this quote from Stanford professor Keith Devlin:  "Like a Shakespearean
sonnet that captures the very essence of love, or a painting that brings
out the beauty of the human form that is far more than just skin deep,
Euler's equation reaches down into the very depths of existence."
There are lots of other surprising facts about e, which you can find in
the references in the show notes.  But I think you've now learned enough to
justify that eclair, so go ahead and dig in.
And this has been your math mutation for today.

• e at wikipedia
• Euler's Identity at wikipedia
• Blog on e day