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.

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