Monday, March 14, 2011

Happy Pi Day. But what about e ?

Click here for the latest dish on Mathematics' second most awesome number.

BTW, is there an e day? Feb. 17 (17 is like 71, spelled backwards, heh)? Why isn't that day celebrated more? If you hurt an Electrical Engineer's feelings, do they not not cry? If you cut them, do they not bleed?

e kicks ass, much moreso than pi. For example, what is 4 raised to power of i?

Pi can't answer that.

e can.

CORRECTION? Pat Ballew pointed out that Pi can most certainly answer that, so let me amend that Pi is not required, rather there are many ways to answer a single problem. Another AMEND: Feb 7 is in fact e day, as Pat points out at his blog. Sorry, just like an Engineer, I was thinking "two decimals places" or maybe the 14th made me think that double digits were required. I don't know how my brain works sometimes, honestly. I tend to think outside the box (with one foot inside) and must remind myself that sometimes the box is just fine.

Anyway, here's how to calc 4^i using only e.

Take e^z = exp z

a^0 = 1 and a^1 = a

Since exp 0 = 1 and exp 1 = e

This translates as e^0 = 1 and e^1 = e

Now, a^b X a^c = a^(b + c) for any a, b, and c

The exponential function also satisfies this: e^x X e^y = e^(x+y)

At this point, yes, we can derive Euler's formula, but that is not the problem. (or .... challenge .... either Dale Carnegie or Tom Hopkins teach'n'preach that "challenge" is the more positive word and better for one's blood pressure)

a = e^(ln a)

Raising to the power b we have a^b = e^(b ln a)

So 4^i = e^(i ln 4) = 0.18 + i0.98

SOURCE: Mathematics 1001 by Dr. Richard Elwes pages 236-237

1 comment:

Steven Colyer said...

I addendumed. Twice. A third addendum is available upon request.