July 23

[admiral sab]

I'm loving mine. Of course that pic is way old, she's 6 months now and that's two months. Perhaps an update is in order...

[e of pi]

What kind of editing was done to that image, btw. It's always looked a little odd. Anywho, thanks Zeke. But, as a student of Algebra II, heading into discrete math by way of testing out of Geometry, what does the new part of mine mean? I'm not sure if I should be offended or not.

[evay]

Originally Posted by e of pi

But, as a student of Algebra II, heading into discrete math by way of testing out of Geometry, what does the new part of mine mean? I'm not sure if I should be offended or not.

the english major takes a stab at it: that "e of pi" is considered an "irrational" number, and proving said number irrational is an exercise left to the student. Sort of like how many angels can sit on the head of a pin: calculate area of pin P, calculate area of angels' arses A, divide. the actual math is left as an exercise for the student. (roughly paraphrased from Heinlein's Friday)

[Zeke]

Yeah, I'm being a little sneaky here. A classic trick of math professors is to leave proving some theorem as an exercise -- thus saving the class time it would take to show the proof. Typically this is done when the proof in question is obvious, tedious, or repetitive of one already shown. But occasionally a prof will forget how a proof is done, and leaving it as an exercise is one way to weasel out of it.

Now, the natural way to read "e of pi" is either e times pi or e over pi. And we know that e and pi are transcendental, but combinations are another story. For example, we don't even know for sure that e+pi and pi^e are irrational, let alone transcendental. (It would be very strange if they were rational, of course.) So, rather than just give him the member title "Irrational" without proof, I decided to acknowledge there was something to prove... while not, of course, proving it myself.

Which is not to say I couldn't, of course. As a matter of fact, I have discovered a truly marvelous proof, which this post is too narrow to contain.

[evay]

I know pi is used in measuring circles, and i is the square root of -1 (so the square root of -4 is 2i), but what's "e" in mathematical terms?

[Celeste]

math makes my head hurt.

[PointyHairedJedi]

Originally Posted by Celeste

math makes my head hurt.

And dentistry makes my mouth hurt.

[Scooter]

Originally Posted by evay

I know pi is used in measuring circles, and i is the square root of -1 (so the square root of -4 is 2i), but what's "e" in mathematical terms?

It's a constant roughly equal to 2.7 and has to do with natural logarithms. If I remember correctly e is the limit as n approaches infinity of (1 +1/n)^n. Right, math-heads?

Not very applicable to my chosen field of ancient history. But if anyone asks I can always say Caesar invented it. People usually believe that. (He did invent military cryptography, which has as mathematical basis, so we're not too far afield...)

[Asky]

you could always look it up on Wikipedia or something and use your newfound knowledge to pretend to be smarter than you really are...

[Zeke]

Originally Posted by evay

I know pi is used in measuring circles, and i is the square root of -1 (so the square root of -4 is 2i), but what's "e" in mathematical terms?

Scooter's correct, except about Caesar. (And he is right about Caesar where cryptography is concerned. Caesar devised the Caesar shift, rot-3 in modern parlance, which is why I had the Romulans use it.)

Anyway, e is as deeply embedded in calculus as pi is in geometry. y = C × e^x is the only function which is its own derivative. As such, it's the best base for logarithms (just as pi is the best basis for angle measurement), so logs to the base e are called natural. Common logs, i.e. to base 10, are strictly for engineers and the occasional number theorist.

e also arises in accounting. If you have $1 in the bank at 100% simple interest, your balance will be $2 in a year. If the interest is compounded semiannually, that rises to $2.25. Compound it monthly and you'll end up with $2.61. It turns out that there's a limit to this process, representing a sort of continuous compounding of interest, and that limit is e dollars.

A third way e can be found is through infinite series. e is the sum of the series 1/0! + 1/1! + 1/2! + 1/3! + ..., where the exclamation point is a factorial sign. (4! = 4 x 3 x 2 x 1, and so on. Before you ask, there is a double factorial sign, and it's the only kind of double exclamation point I can stand.) Any of these three properties of e can be, and has been, used as its definition; you can derive the other properties from the one you start with.

As you can see, it's easy to get me talking about this stuff. I did some quality math-related ranting in this LJ post, for anyone interested.

[Scooter]

Originally Posted by Zeke

Anyway, e is as deeply embedded in calculus as pi is in geometry. y = C × e^x is the only function which is its own derivative. As such, it's the best base for logarithms (just as pi is the best basis for angle measurement), so logs to the base e are called natural. Common logs, i.e. to base 10, are strictly for engineers and the occasional number theorist.

Wow, that is the most lucid explanation of a mathematical concept I've seen in ages. Cool.

I bet if we'd been born with 2.7 fingers, then base 10 would never have even been invented. Except that base 10 would be base e, since our numbering system would have developed using an e-based place value...