What Does i^i Really Mean in Complex Numbers?

[trevdna]

i^i means WHAT?!

My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

i^i = e^(-pi/2)

Which is cool, I suppose. But the more I think about it, the more I wonder:

what the heck does it mean?

For that matter, what does anything raised to an imaginary power mean?
For instance:
e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

 

[g_edgar]

For e^(ix) one approach is to note that for real constants a, y=e^(ax) is the solution of y'(x) = ay(x), y(0)=1. So for complex number i, to evaluate e^(ix) we want complex-valued function y such that y'(x) = iy(x), y(0)=1. The solution is y(x) = cos(x) + i sin(x).

 

[mathman]

e^(ix) = cos(x)+i*sin(x) may not be intuitive, but you can prove it by expanding both sides into power series and see they are the same.

 

[chaoseverlasting]

My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

i^i = e^(-pi/2)

Which is cool, I suppose. But the more I think about it, the more I wonder:

what the heck does it mean?

For that matter, what does anything raised to an imaginary power mean?
For instance:
e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

Why are real numbers any more 'real' that complex numbers? They are just convenient representations of things that we observe around us.

The function $$e^x$$ gives us an easy/convenient way to represent other quantities. I assume you're taking some form of coaching for competetive examinations from your post. At this level, there arent a lot of examples that I can think of. It's a very useful function when dealing with geometrical rotation of a quantity or geometrical figures in general (eg. a circle of unit radius or other regular polygons).

You can also define any possible mathematical function in terms of $$e^x$$ with something called Fourier Analysis, which is one of the corner stones of signal analysis and digital electronics.

In Electrical engineering, I doubt you can go through a single topic without encountering the exponential function in some form. Its also lends itself to very very convenient representations of AC waveforms and provides a graphical analogy to the same (Phaser Diagrams).

There are countless examples of the use of the exponential function in 'real' life. Hope that helps.

 

[Jamma]

My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

i^i = e^(-pi/2)

Which is cool, I suppose. But the more I think about it, the more I wonder:

what the heck does it mean?

For that matter, what does anything raised to an imaginary power mean?
For instance:
e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

It doesn't mean that, $$i^{i}$$ means an infinite number of different values.

Think about square roots, or things raised to the power of 1/2. Then this means a set of two different numbers (except for 0). For irrational numbers, you can get an infinite number of (complex) values.

For complex numbers, we can define a general power $$z^{v}$$ as:

$$z^{v}=e^{v.log(z)}$$

and log(z) can have an infinite number of values. This definition of exponentiation is basically just a rule which is consistent with all the older laws for use with real numbers.

So in your case:

$$i^{i}=e^{i.log(i)}=e^{i((-1/2+2n).\pi .i)}$$

for n any natural number. So $$e^{-\pi /2}$$ is an answer, but there are many other values also.

 

[sEsposito]

For that matter, what does anything raised to an imaginary power mean?
For instance:
e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

I like to think of imaginary numbers to be a convenient way to "extend the number-line" so to speak... If you look back in history to when we first conceived the "real" numbers, to the time when we formalized them into the set $$R$$ and then to the point where imaginary numbers came into play -- none of it is intuitive. It's only intuitive to us because we're comfortable with it.

There's a nice documentary on this subject, I think it was put out by BBC... It's on YouTube somewhere. You'll find it if you search "Mathematics Documentary".

 

[Tac-Tics]

For that matter, what does anything raised to an imaginary power mean?

What does it mean for a number to be raised to a fractional power? if 2^2 = 2 x 2, what the heck does 2^(1/2) mean?

Generalizations don't always make sense. The power of generalizing in math is that you throw away some of your assumptions (exponentiation has a literal interpretation in terms of multiplication) while maintaining handy invariants (x^(a+b) = x^a * x^b) in order to describe a larger (and often more abstract) class of mathematical objects (real numbers, as opposed to the integers).

Exponentiation is interesting, too, in that it loses one interpretation (iterated multiplication) and gains a new one (converting rectangular form to polar form).