How can you raise something to the power of i?

[questionposter]

How can you raise something to the power of an imaginary number and get a real number? Every time I try to do it I get an error. Does it only work on a complex plane in some way?

 

I mean I guess it get that it just happens, but, how does that actually work out?

Edited December 21, 2011 by questionposter

[ajb]

Let [math]a[/math] be a positive real number, and [math]z [/math] be an arbitrary complex number. Then we define

 

[math]a^{z} = e^{z log(a)}[/math].

 

Then you can use

 

[math]e^{x+iy} = e^{x}\left(cos(y) + i \: sin(y) \right)[/math].

 

If that is what you are asking?

 

(corrected tiny typo)

Edited December 22, 2011 by ajb

[questionposter]

Let [math]a[/math] be a positive real number, and [math]z [/math] be an arbitrary complex number. The we define

 

[math]a^{z} = e^{z log(a)}[/math].

 

Then you can use

 

[math]e^{x+iy} = e^{x}\left(cos(y) + i \: sin(y) \right)[/math].

 

If that is what you are asking?

 

I mean I get how the logic of logarithms work, I guess in this instance it generates complex numbers that are easily visible, but I mean just in general. What if I just say

 

"[math]2^i[/math]"? What does that mean? What does i actually mean when I say " sinx+5=0 at (w/e)i"?

Edited December 22, 2011 by questionposter

[timo]

I don't think I have ever seen anything but "e" being raised to some power of "i" (but in case you wanted to do that for some exotic reason, ajb already provided a possible definition). The standard definition of the exponential function (in physics) is [math] e^x = \exp (x) := \sum_{n=0}^\infty \frac{x^n}{n!} [/math]. As you hopefully see, this definition does allow for complex-valued arguments "x".

 

Sidenote: It is also common to define terms [math]e^M[/math] with matrices "M" this way.

[ajb]

"[math]2^i[/math]"? What does that mean?

 

Well following the rules we get

 

[math]2^{i} = e^{ 0 + i \: log(2)} = cos(log(2)) + i \: sin(log(2))[/math].

 

Other easy examples you can think up yourself.

 

I will warn you that taking a complex power of a complex number is harder. You have to worry about what branch cut of the complex log to use. The standard thing to do is use the principle value, and this will reduce to the definition for positive real numbers raised to a complex power.

 

Sidenote: It is also common to define terms [math]e^M[/math] with matrices "M" this way.

 

 

As a sidenote to the above sidenote, be aware of the Baker–Campbell–Hausdorff formula and related formula.

Edited December 22, 2011 by ajb

[DrRocket]

I don't think I have ever seen anything but "e" being raised to some power of "i" (but in case you wanted to do that for some exotic reason, ajb already provided a possible definition). The standard definition of the exponential function (in physics) is [math] e^x = \exp (x) := \sum_{n=0}^\infty \frac{x^n}{n!} [/math]. As you hopefully see, this definition does allow for complex-valued arguments "x".

 

Sidenote: It is also common to define terms [math]e^M[/math] with matrices "M" this way.

 

That definition is what is used in mathematics as well (good news for physics). It works quite well in a Banach algebra.

 

How can you raise something to the power of an imaginary number and get a real number? Every time I try to do it I get an error. Does it only work on a complex plane in some way?

 

I mean I guess it get that it just happens, but, how does that actually work out?

 

[math] e^{i \pi}\ = \ -1 [/math]

Edited December 22, 2011 by DrRocket

[the tree]

I think part of the problem is a lack of an intuitive meaning for x^y when y isn't an integer. Multiplying a number by itself i*pi times just doesn't make any sense. I think the trick is kind of to view it as just another function.

Edited December 22, 2011 by the tree

[ajb]

I think the trick is kind of to view it as just another function.

 

Right, you have to think more formally and follow the mathematical rules.

[Shadow]

You might be interested in reading Where Mathematics Comes From by Lakoff and Núñez; it deals with exactly what you've asked, namely "What does [math]e^{i\pi}[/math] mean". Note that I haven't finished the book yet and reactions to it are mixed at best. But it still might be an interesting read.

[questionposter]

Ok, I get that raising to the power of i would generate some expression based on the logical properties of logarythms and trigonometry, but I don't mean that, I want to know what your doing when you raise something to the power of an imaginary value. I don't mean "what happens when you raise e to the power of i pi or I guess other things on a polar graph?", I mean it in the same way as "why does 2^2=4?"

Edited December 23, 2011 by questionposter

[ajb]

... I mean it in the same way as "why does 2^2=4?"

 

As the tree states, you cannot simply think of a number (complex or real) multiplied by itself a complex number of times. Using the log is the only sensible way that I am aware of to define raising something to a complex power. In fact, this is probably the best way to define taking non-integer powers of real numbers also.

[DrRocket]

I think part of the problem is a lack of an intuitive meaning for x^y when y isn't an integer. Multiplying a number by itself i*pi times just doesn't make any sense. I think the trick is kind of to view it as just another function.

 

It is a function, but not "just another function". It is a particularly important function. A great deal of analysis depends on that function.

[questionposter]

As the tree states, you cannot simply think of a number (complex or real) multiplied by itself a complex number of times. Using the log is the only sensible way that I am aware of to define raising something to a complex power. In fact, this is probably the best way to define taking non-integer powers of real numbers also.

 

Well, you can raise something to the power of the square root of two, why not negative 1?

[questionposter]

As the tree states, you cannot simply think of a number (complex or real) multiplied by itself a complex number of times. Using the log is the only sensible way that I am aware of to define raising something to a complex power. In fact, this is probably the best way to define taking non-integer powers of real numbers also.

 

Well, you can raise something to the power of the square root of two, why not of negative 1?

Edited December 24, 2011 by questionposter

[ajb]

Well, you can raise something to the power of the square root of two, why not of negative 1?

 

You can raise a complex number by another complex number, just the intuitive definition as a number multiplied by itself a complex number of times is now lacking.

[DrRocket]

As the tree states, you cannot simply think of a number (complex or real) multiplied by itself a complex number of times. Using the log is the only sensible way that I am aware of to define raising something to a complex power. In fact, this is probably the best way to define taking non-integer powers of real numbers also.

 

 

You can raise a complex number by another complex number, just the intuitive definition as a number multiplied by itself a complex number of times is now lacking.

 

Right.

 

The important thing is that one can define exponents, using the exponential function, in such a way that it agrees with the elementary definition for integer powers. Thus the more sophisticated idea is merely an extension of what one learns in grade school, but it is a very important extension with far-reaching ramifications.

 

The fact that the exponential function, as a function of a complex variable, is [math] 2 \pi i[/math] periodic only adds to the depth of the concepts that are involved.

[questionposter]

Is "i" literally just some imaginary figment to make up for the flaws in our math? Or is it just a number who's value we cannot comprehend?

[DrRocket]

Is "i" literally just some imaginary figment to make up for the flaws in our math? Or is it just a number who's value we cannot comprehend?

 

neither

 

Go read Foundations of Analysis by Landau.

Edited January 25, 2012 by DrRocket

[ajb]

Is "i" literally just some imaginary figment to make up for the flaws in our math?

 

It may seem like that. The first time I was taught about complex numbers "i" was presented as a book keeping device to take care of the square root of -1.

 

I tend to think more abstractly that this. Appending "i" to the real numbers gives us the complex numbers which have much better properties that just the reals. Importantly, the complex numbers are algebraically closed. That is polynomials in reals do not always have roots that are reals. The complex numbers are different. Polynomials in complex numbers always have complex roots.

 

One way to think of "i" is really notational and little more. A complex number is an order pair of real numbers, say (a,b). The notation we use is a + ib.

 

I don't think I have ever thought much deeper about it than that.

[D H]

It is a function, but not "just another function". It is a particularly important function.

I'll go one step further: It is (per Rudin) the most important function in all of mathematics.

 

 

Well, you can raise something to the power of the square root of two, why not of negative 1?

With the exponential function it's easy (for positive real values of x): [math]x^{\sqrt 2}=\exp(\sqrt 2 \ln x)[/math]. This in fact is how a positive real rasied to a non-rational power is defined. It's called analytic continuation.

 

You're going to have a very hard go at it if you want to describe [math]x^{\surd 2}[/math] without resorting to the exponential function. Without the exponential function, you are going to have to use a limit of a rational number that approaches the square root of two, Dedekind cuts, or Cauchy sequences. The only people who have to do that are math majors who have to show that the analytic continuation via the exponential and these alternative approaches are equivalent.

 

 

Is "i" literally just some imaginary figment to make up for the flaws in our math? Or is it just a number who's value we cannot comprehend?

No, it's just a misfortunate name. Mathematicians are, at there core, conservative and stodgy humanities majors who are loath to accept newfangled ideas.

 

Just take a look at the names of some the kinds of numbers that mathematicians have invented.

• 
   
  
• Irrational numbers. Irrational = not logical. The irrational numbers are numbers that don't make a lick of sense. The ancient Greeks threw Hippasus of Metapontum overboard for discovering the irrationals. That nomenclature sticks around to this day in the name "irrationals" and in the name for this symbol, [math]\surd[/math]. That is the "surd" symbol: Short for absurd.
   
  
• Negative numbers. Negative = not. The negative numbers are not numbers. Mathematicians did not kill the messenger this time around, but acceptance of the negative numbers as real took quite some time.
   
  
• Imaginary numbers. Once again, we have an unfortunate label that reflects on the reluctance of some mathematicians to accept radically new ideas. Giving the imaginary numbers a disparaging name helped ease that acceptance along.

 

The term "imaginary" is just a label. Don't take it is meaning that complex numbers are any less "real" than the real numbers themselves.

Edited January 25, 2012 by D H

[imatfaal]

No, it's just a misfortunate name. Mathematicians are, at there core, conservative and stodgy humanities majors who are loath to accept newfangled ideas.

 

Just take a look at the names of some the kinds of numbers that mathematicians have invented.

• Irrational numbers. Irrational = not logical. The irrational numbers are numbers that don't make a lick of sense. The ancient Greeks threw Hippasus of Metapontum overboard for discovering the irrationals. That nomenclature sticks around to this day in the name "irrationals" and in the name for this symbol, [math]\surd[/math]. That is the "surd" symbol: Short for absurd.
• Negative numbers. Negative = not. The negative numbers are not numbers. Mathematicians did not kill the messenger this time around, but acceptance of the negative numbers as real took quite some time.
• Imaginary numbers. Once again, we have an unfortunate label that reflects on the reluctance of some mathematicians to accept radically new ideas. Giving the imaginary numbers a disparaging name helped ease that acceptance along.

 

The term "imaginary" is just a label. Don't take it is meaning that complex numbers are any less "real" than the real numbers themselves.

 

And don't even get started on the transcendental numbers

[questionposter]

I'll go one step further: It is (per Rudin) the most important function in all of mathematics.

 

 

 

With the exponential function it's easy (for positive real values of x): [math]x^{\sqrt 2}=\exp(\sqrt 2 \ln x)[/math]. This in fact is how a positive real rasied to a non-rational power is defined. It's called analytic continuation.

 

You're going to have a very hard go at it if you want to describe [math]x^{\surd 2}[/math] without resorting to the exponential function. Without the exponential function, you are going to have to use a limit of a rational number that approaches the square root of two, Dedekind cuts, or Cauchy sequences. The only people who have to do that are math majors who have to show that the analytic continuation via the exponential and these alternative approaches are equivalent.

 

 

 

No, it's just a misfortunate name. Mathematicians are, at there core, conservative and stodgy humanities majors who are loath to accept newfangled ideas.

 

Just take a look at the names of some the kinds of numbers that mathematicians have invented.

• Irrational numbers. Irrational = not logical. The irrational numbers are numbers that don't make a lick of sense. The ancient Greeks threw Hippasus of Metapontum overboard for discovering the irrationals. That nomenclature sticks around to this day in the name "irrationals" and in the name for this symbol, [math]\surd[/math]. That is the "surd" symbol: Short for absurd.
• Negative numbers. Negative = not. The negative numbers are not numbers. Mathematicians did not kill the messenger this time around, but acceptance of the negative numbers as real took quite some time.
• Imaginary numbers. Once again, we have an unfortunate label that reflects on the reluctance of some mathematicians to accept radically new ideas. Giving the imaginary numbers a disparaging name helped ease that acceptance along.

 

The term "imaginary" is just a label. Don't take it is meaning that complex numbers are any less "real" than the real numbers themselves.

 

It makes sense that imaginary numbers aren't actually just "imaginary", but at the same time, why can I not point to where x^2+5=0? Is there perhaps some other plane of existence where it does but not in 3-d space? Or...

[ajb]

It makes sense that imaginary numbers aren't actually just "imaginary", but at the same time, why can I not point to where x^2+5=0? Is there perhaps some other plane of existence where it does but not in 3-d space? Or...

 

If you try insist on x being a real number then there is no such x. However, if you allow x to be complex then you can find such an x.

 

The "plane of existence" is the complex plane

[imatfaal]

It makes sense that imaginary numbers aren't actually just "imaginary", but at the same time, why can I not point to where x^2+5=0? Is there perhaps some other plane of existence where it does but not in 3-d space? Or...

 

On what would you point to where x^2 - 5 = 0? The intercept on a graph drawn with cartesian coordinates? Makes sense; but is it really concrete and so much less imaginary to use one real number to define a number than to use two? A position on a cartesian coordinate plane is an abstraction (and a fairly modern one Descartes) it is just less of an abstraction than that required to deal with complex numbers.

[questionposter]

On what would you point to where x^2 - 5 = 0? The intercept on a graph drawn with cartesian coordinates? Makes sense; but is it really concrete and so much less imaginary to use one real number to define a number than to use two? A position on a cartesian coordinate plane is an abstraction (and a fairly modern one Descartes) it is just less of an abstraction than that required to deal with complex numbers.

 

If you try insist on x being a real number then there is no such x. However, if you allow x to be complex then you can find such an x.

 

The "plane of existence" is the complex plane

 

Well if "imaginary" in math doesn't actually mean "made up", then "real" shouldn't mean "it exists" either right?

I guess, is there some way to see where"i" exists in reality?

Edited January 25, 2012 by questionposter