27 Oct '07 13:46

If i^2= (-1), what is i^i? i to the ith power.

- Joined
- 28 Dec '04
- Moves
- 52874

slatington, pa, usa- Joined
- 02 Aug '04
- Moves
- 8702

Halfway- Joined
- 28 Dec '04
- Moves
- 52874

slatington, pa, usa- Joined
- 25 Oct '02
- Moves
- 20443

Out of my mind27 Oct '07 15:241 edit

He used another way of writing i:*Originally posted by sonhouse***What is n in that equation?**

And how does PI enter into this? interesting.

Also, what does exp mean? exponent? if so, what exponent?

i = e ^ (PI * i / 2)

Since 1 = e ^ (2*PI*i) you can add extra factors of e ^ (2*PI*i), hence the n.

So i^i = e ^(-PI/2) mod e ^ (2*PI)- Joined
- 23 Aug '04
- Moves
- 24791

tinyurl.com/y2c6j2t3- Joined
- 07 Sep '05
- Moves
- 35068

27 Oct '07 17:261 edit

It's not part of the defiition of i, but it's an extremely useful general result:*Originally posted by AThousandYoung***i is part of the definition of i? That doesn't make sense.**

cos(*theta*) +*i*sin(*theta*) = e^(*i theta*)

He's using the case where*theta*= PI/2. Then the left-hand side is equal to*i*.- Joined
- 28 Dec '04
- Moves
- 52874

slatington, pa, usa27 Oct '07 17:291 edit

I'm getting PI i'd thinking about this:*Originally posted by mtthw***It's not part of the defiition of i, but it's an extremely useful general result:**q

cos(*theta*) +*i*sin(*theta*) = e^(*i theta*)

He's using the case where*theta*= PI/2. Then the left-hand side is equal to*i*.

So theta is just any angle and he made that angle PI/2.

You said 'extremely useful'. can you show some samples of the usefulness of this? Also, are there other identities like this that can be useful?

Other angles of theta, or equivalences?

what about definitions in radians? Are there any generalizations to be made there?- Joined
- 02 Aug '04
- Moves
- 8702

Halfway- Joined
- 02 Aug '04
- Moves
- 8702

Halfway- Joined
- 07 Sep '05
- Moves
- 35068

27 Oct '07 17:451 edit

Or almost any serious applied mathematics. It crops up in wave theory, quantum mechanics, fluid dynamics, instability calculations...*Originally posted by Palynka***Yep, someone that needs to deal with dynamic optimization should be familiar with it. It's called de Moivre's theorem.**- Joined
- 28 Dec '04
- Moves
- 52874

slatington, pa, usa27 Oct '07 17:58

Nice. Thanks for that. I Goog'd De Moivre and found this neat wiki:*Originally posted by mtthw***Or almost any serious applied mathematics. It crops up in wave theory, quantum mechanics, fluid dynamics, instability calculations...**

http://en.wikipedia.org/wiki/De_Moivre's_formula

This piece goes into it in some detail.

BTW, my present read is 'The Riemann Hypothesis"

by Karl Sabbagh. Anyone read this? Great read if you are into maths.

I think my next maths book will be 'the joy of i'. I think there is a book with something like that title.- Joined
- 11 Nov '05
- Moves
- 43938

27 Oct '07 18:031 editCalculating with i ( as sqrt(-1) ) is only an extension of the real number system. What you can do with R you can do with C.

...and more:

With the complex number system (C) you can calculate square root of any number (including complex numbers), not only positive numbers. You can calulate arcsin with any number (including complex numbers), not only numbers between -1 and 1. You can take logaritm of any number (including complex numbers), not only positive numbers. And so on.

With i = sqrt(-1) you can calculate i^i, sqrt (i), log (i) with 'no problem'. But you cannot calculate i/0 but who can?

But one pecularity of the complex number is that no complex number is larger or smaller than any other number. Another one is that negative infinity is the same as positive infinity.

Complex numbers are fun!- Joined
- 02 Aug '04
- Moves
- 8702

Halfway27 Oct '07 18:061 edit

I love the fact that -e^(pi.i)=1. (also shown by de Moivre's formula)*Originally posted by sonhouse***Nice. Thanks for that. I Goog'd De Moivre and found this neat wiki:**

http://en.wikipedia.org/wiki/De_Moivre's_formula

This piece goes into it in some detail.

BTW, my present read is 'The Riemann Hypothesis"

by Karl Sabbagh. Anyone read this? Great read if you are into maths.

I think my next maths book will be 'the joy of i'. I think there is a book with something like that title.

It's one of those things that show you how mathematics can be so beautiful and understand why some cultures saw glimpses of the divine in them. Three of the most important mathematical constants combined into such a simple result.- Joined
- 28 Dec '04
- Moves
- 52874

slatington, pa, usa27 Oct '07 18:09But one peculiarity of the complex number is that no complex number is larger or smaller than any other number. Another one is that negative infinity is the same as positive infinity.

I follow everything but that one. It sounds like you are saying all complex numbers are equal?? And that (+inf) + (-inf)=0? and (+inf) * (-inf)= (-inf)^2?- Joined
- 09 Jun '07
- Moves
- 46058

at home27 Oct '07 18:51

I didnt know that!*Originally posted by FabianFnas***Calculating with i ( as sqrt(-1) ) is only an extension of the real number system. What you can do with R you can do with C.**

...and more:

With the complex number system (C) you can calculate square root of any number (including complex numbers), not only positive numbers. You can calulate arcsin with any number (including complex numbers), not only ...[text shortened]... er one is that negative infinity is the same as positive infinity.

Complex numbers are fun!

But isnt it a matter of definition?

What if I define the 'size' of a complex number C where C = R +iI as

sqrt(I^2 + R^2) ... does that have a use?

ðŸ™„