Thread: Meaningful representation of "e"?

Not sure if this is worthy of its own thread, but I couldn't find any recent related threads, so a here's a quick question...

Is it valid to algebraically derive something like "e=???" from identities in Euler's formula?

For example, we know from Euler's formula. So could I derive, say, ?

How in the who knows what did you get that result?

Edit: Oh I think I see now, but no that is not how you deal with exponents at all. You may want to brush up on your exponent algebra.

Edit again: I'm sorry I suppose that is valid. Just strange to see it that way I guess.

Yes it's valid to but why on earth would you want to?

Originally Posted by TheObserver

How in the "who knows what" did you get that result?

Edit: Oh I think I see now, but no that is not how you deal with exponents at all. You may want to brush up on your exponent algebra.

It took me a while to understand your first sentence. A bit too many interrogatives clumped up together Anyway, I'm not sure if this is right or not...

is the original identity, well... the one obtained from the actual famous one.

, by simply rooting both sides.

, by properties of the imaginary unit.

I don't see what's wrong here, if you could point it out.

Originally Posted by nano

Yes it's valid to but why on earth would you want to?

It's a mix of interest and boredom, I'll admit. It could be a meaningful representation of Euler's number and I never see it anywhere. Call it number mysticism if you'd like but I thought it was kind of a beautiful identity. Also, I think I'm edging on something big (or not) but I'll have to see now.

Well it is no more beautiful or fundamental than Euler's identity since it's merely a reformulation. And the reason you never see it expressed as such is because it's a pretty useless form. There are countless expressions for e which are not only useful but shed a certain level of insight into the nature of the constant.

That's not me playing down your curiosity however.

Originally Posted by nano

Well it is no more beautiful or fundamental than Euler's identity since it's merely a reformulation. And the reason you never see it expressed as such is because it's a pretty useless form. There are countless expressions for e which are not only useful but shed a certain level of insight into the nature of the constant.That's not me playing down your curiosity however.

Fair enough. Checked it on Wolfram Alpha; at least it's valid.What doesn't check out, however, is e=1^(-i/(2pi)).

Indeed. A simpler form of what you've done there is as follows:



The fallacy is that you've applied the following rule for a negative x



The problem being that this relation only generally holds for non-negative x (and you can see why).

Originally Posted by nano

Indeed. A simpler form of what you've done there is as follows:



The fallacy is that you've applied the following rule for a negative x



The problem being that this relation only generally holds for non-negative x (and you can see why).

*sigh* Sorry if this is insanely stupid of me, really, but I'm confused as to where that applies in my equation? I don't see where what I did relates to that rule. Again, sorry. I'm not that great with math.

Originally Posted by halorealm

*sigh* Sorry if this is insanely stupid of me, really, but I'm confused as to where that applies in my equation? I don't see where what I did relates to that rule. Again, sorry. I'm not that great with math.

You said that the following relation did not check out:

Now consider how you got to that equation, step-by-step. You went from here:

which is correct, and replaced (-1) with

presumably through the logic that I listed above. That is where you went wrong.

Ah, I see now. That's not what I did though.

I went directly from Euler's formula for the identity .

Again, just rooting both sides.

The left side simplifies.

And rearranging the power on the right side by properties of imaginary numbers.

I suspect the problem concerns an exponent law that doesn't apply to complex numbers... maybe?

Or maybe something really obvious I missed.

Yes, it's effectively the same thing: your exponentiation is invalid. You should give this a read:
Exponentiation - Wikipedia, the free encyclopedia

That's quite a long Wiki article. I wonder what you're thinking... "Ah. That should keep him away fo a while" Thanks for the help; it's much appreciated.

Not at all. I love discussing elementary algebra; it's really putting my PhD to good use

Anyway, you don't need to read the entire article. Have a glance at the stuff on real powers and that involving complex numbers.
It's very easy to go wrong with complex numbers when you apply what seem like intuitive results from real (or, typically, natural) numbers. Here's another nice one:

Originally Posted by nano

Here's another nice one:

Yes. I thought about that some time ago. That's obviously not true. This question might be above me, but doesn't that still essentially mean the natural logarithm is a multi-valued function?

Originally Posted by halorealm

Originally Posted by nano

Here's another nice one:

Yes. I thought about that some time ago. That's obviously not true. This question might be above me, but doesn't that still essentially mean the natural logarithm is a multi-valued function?

Yes. You can always add 2kπi, for any integer k, when taking a natural log.

Okay. And since 0 is included in that set, can be solely described by for (or for you Tauists here).

That's correct. This is reflected in the Riemann surface for the logarithm:
http://upload.wikimedia.org/wikipedi...urface_log.jpg