Imaginary numbers

[blackhole123]

We just learned what these are today in math. I get the concept of it, but im not sure of its application.

 

Lets say you have a equation with a negative inside a square root. You turn it into an i. Im not sure how this would help you. I havn't taken calculas yet so maybe i'll learn it then?

[ajb]

When I first come across complex numbers I had no idea just how important they are.

 

On the mathematics side there is a lot of machinery developed for complex analysis, which comes in useful in may branches of mathematics and physics.

 

On the physics side complex numbers are fundamental in quantum mechanics. Generically, you dont see just Plancks constant it is always multiplied by i.

 

Complex numbers are also very useful when describing waves and oscillations. They are used to describe phases.

 

 

One important thing to realise with complex numbers is that they are algebraically closed. That is any polynomial with complex coefficients has a complex root. This is not true of the real numbers, i.e. you can have a polynomial with real coefficents whose roots are not real numbers. In fact they will be complex numbers.

 

For a simple exmple consider

 

[math]z^{2} +1 =0 [/math].

 

You see that all the coefficents are real numbers. Question, what are it's roots? You qucikly realise that [math] z = \pm i[/math] are the roots, which are imaginary numbers.

 

If you do something similar with a polynomial with complex coefficient, you will see that the roots are also complex. That is you will never "leave" the field of complex numbers by considering roots of polynomials.

[woelen]

In engineering and electronics, the use of complex numbers also is very widespread.

 

In electronics, you can model linear capacitors and resistors as frequency dependent linear resistors:

 

resistor with resistor R: [math]Z_R = R[/math]

capacitor with capacitance C: [math]Z_C = 1/i \omega C[/math]

inductor with inductance L: [math]Z_L = i \omega L[/math]

 

Here Z is the generalized impedance (complex resistance) and [math]\omega[/math] is the frequency of the electrical signal across the component (2*pi*f).

 

You can do calculations with these complex imedances, as if you were computing currents and voltages in ordinary resistor networks. These computations then yield output voltages or currents, which e.g. can be written as

 

[math]V_{out} = H(\omega) V_{in}[/math]

 

Here, H(ω) is some freqency dependent complex number. The absolute value of H equals the gain (or attenuation) of the signal at freqency ω, while the argument of H(ω) equals the phase shift between output and input at frequency ω.

 

Another important application of complex numbers in engineering is in the determination of the poles (and zeroes) of a linear system. The location of poles in the complex plane gives a lot of information about a system's stability and usefulness. Poles in the right half plane (with positive real part) indicate that the system is unstable.

[Klaynos]

Electronics is very interesting when considered in this way because it is a very nice application of wave mechanics, which leads onto us being able to model many many wave systems with relatively simple electronics systems... Which we could not do without complex numbers...

[alan2here]

You forgot the most important use of Imaginary numbers

http://alan2here.deviantart.com/gallery/digitalart/fractals/

[[Tycho?]]

What do imaginary numbers have to do with fractals?

[alan2here]

It's the Mandlebrot, Julia, Nova, Barnsley, Newton etc... type fractals, they are verry much related to numbers which have real and imaginary parts. This is verry evident even when just using fractal art software like Ultrafractal not just when delving into the maths.

[swansont]

The Euler formula uses of imaginary numbers, that ajb mentioned in passing (waves and oscillations).

[D H]

Alan, fractals are pretty mathematical art. This hardly qualifies as the most "important use of imaginary numbers".

[[Tycho?]]

It's the Mandlebrot, Julia, Nova, Barnsley, Newton etc... type fractals, they are verry much related to numbers which have real and imaginary parts. This is verry evident even when just using fractal art software like Ultrafractal not just when delving into the maths.

 

Ok, I still fail to see how thats the most important use of imaginary numbers. It looks more like the least important use.

[ajb]

The Euler formula uses of imaginary numbers, that ajb mentioned in passing (waves and oscillations).

 

Yes of course, I would say that this is one of the most useful things you can do with complex numbers. If you can't remember multiangle formule for the trigonometric functions etc, you can work them out explicitly from the properties of exponentials.

[blackhole123]

Ok I get that imaginary numbers are important but I'm still not sure how they are used. Like when my math teacher explained it she said that you cant take the square root of a negative number so i was made up to represent this. I just fail to see how this makes the equation any more solvable. You still can't have the square root of a negative and just because you put an i down doesn't change this.

[[Tycho?]]

Ok I get that imaginary numbers are important but I'm still not sure how they are used. Like when my math teacher explained it she said that you cant take the square root of a negative number so i was made up to represent this. I just fail to see how this makes the equation any more solvable. You still can't have the square root of a negative and just because you put an i down doesn't change this.

 

Look it up on wikipedia. Imaginary numbers are immensly useful in physics. They can hugely simplify almost anything to do with waves (which is a huge chunk of physics right there), they are pretty much required for circuit stuff in electrical engineering. I dont have any personal examples beyond this, but I'm sure there are a ton. Hugely useful.

[uncool]

;316624'']Ok, I still fail to see how thats the most important use of imaginary numbers. It looks more like the least important use.

 

Looks to me like someone's trying to get traffic to their site.

=Uncool-

[John Cuthber]

Fractals, like those generated using imaginary numbers, are used for data compression which is reasonably important in this www world. I don't know if the particular fractals they use are calculated using imaginary numbers.

 

 

"I just fail to see how this makes the equation any more solvable. You still can't have the square root of a negative and just because you put an i down doesn't change this."

That's a fair point but I think there are 2 answers. Firstly the mathematicians are happy because now they can write down an answer. If they had to keep on making up "strange" numbers in order to answer lots of equations I think they would have given up, but just one weird number i, seems to do the job.

The other reason is that ignoring the fact that i doesn't exist and finishing the calculation seems to work in quite a lot of physics. If it makes the equations work, perhaps it must be (in some sense) real.

[woelen]

The use of complex numbers is a tool. The tool helps us to solve many physical problems, but it does not make the numbers more real.

 

The example I gave for electronic circuits is a nice demonstration of this. None of the voltages and currents in the circuit really is complex or imaginary, they all are real.

 

But, the use of euler's formula (exp(ix) = cos(x) + i sin(x)), combined with the use of properties of linearity can be used to solve many problems in a very elegant and compact way.

 

You will also notice that in non-linear physics the use of complex numbers is much more limited, unless the non-linear system is regarded as an infinite dimensional linear system. But the latter approach in many cases is just as cumbersome as the direct non-linear approach.

[sourkraut]

complex number are not only a tool for science and engineering they describe our world better than the real number system can. But complex numbers do not work alone. They need the real number system. if you agree that 3 = 3/1 then you agree that 3 units is the same if it is written with one real number or two real numbers. Then you can start to wrap your brain around the fact the complex numbers exist as pairs ie. (a,b) of real numbers. Complex numbers just give us the second dimension to our every day world of numbers.

I hope that gave you enough info to search for a better insight on complex numbers. there is A LOT of great explanations out there. It is helpful sometimes to get a launching pad to set off on your search for the mind altering break through in the world of math. enjoy your trip.

[the tree]

blackhole

You still can't have the square root of a negative and just because you put an i down doesn't change this.

Why not? Not, why doesn't it change this, but why can't you have a square root of a negative number? The reason, holds no more weight than why you can't have some-natural-number minus some-bigger-natural-number.

 

The use of complex numbers is a tool. The tool helps us to solve many physical problems, but it does not make the numbers more real.

A very important point to remember is that numbers aren't real in the first place, real and imaginary are very misleading terms.

 

 

The example I gave for electronic circuits is a nice demonstration of this. None of the voltages and currents in the circuit really is complex or imaginary, they all are real.

This article covers that application as well as a lot of the rest of the very basics of complex numbers:

http://scienceblogs.com/goodmath/2006/08/i.php

 

woelen

But, the use of euler's formula (exp(ix) = cos(x) + i sin(x)), combined with the use of properties of linearity can be used to solve many problems in a very elegant and compact way.

Hmm? All that's been given to me is [math]e^{kix}=\cos(kx)[/math] (unless k is negative).

Also, I haven't been given an explanation as to why, I don't suppose you could explain it?

[D H]

Hmm? All that's been given to me is [math]e^{kix}=\cos(kx)[/math] (unless k is negative).

Also, I haven't been given an explanation as to why, I don't suppose you could explain it?

 

Woelen was correct. Euler's formula is easy to prove using the Taylor expansions of exp(x), sin(x), and cos(x). To start, here are the expansions (if you want the derivations, just ask):

 

[math]\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}[/math]

 

[math]\sin(x) =

\sum_{n=0}^\infty (-1)^n \frac {x^{(2n)}}{(2n)!}[/math]

 

[math]\cos(x) =

\sum_{n=0}^\infty (-1)^n \frac {x^{(2n+1)}}{(2n+1)!}[/math]

 

Expanding exp(ix),

 

[math]\exp(ix) = \sum_{n=0}^\infty \frac {i^nx^n}{n!}[/math]

 

Split the series into even and odd parts.

 

[math]\exp(ix) =

\sum_{n=0}^\infty \frac {i^{(2n)}x^{(2n)}}{(2n)!} +

\sum_{n=0}^\infty \frac {i^{(2n+1)}x^{(2n+1)}}{(2n+1)!}[/math]

 

Simplify using [math]i^{(2n)} = (-1)^n[/math], [math]i^{(2n+1)} = (-1)^n i[/math]:

 

[math]\exp(ix) =

\sum_{n=0}^\infty (-1)^n\frac {x^{(2n)}}{(2n)!} +

i \sum_{n=0}^\infty (-1)^n\frac {x^{(2n+1)}}{(2n+1)!}[/math]

 

or

 

[math]\exp(ix) = \cos x + i\sin x[/math]

[vori]

I have tried to explain this and following topics with some of the documents at

http://www.goretaivor.com/Science/Mathematics/index.php

 

I am really frustrated that if you want to learn something you have to go through all forums you can find collecting crumbs instead of having one place where topics would be properly explained.

[iNow]

I am really frustrated that if you want to learn something you have to go through all forums you can find collecting crumbs instead of having one place where topics would be properly explained.

 

I advise this area then:

 

http://www.scienceforums.net/forum/forumdisplay.php?f=149

 

 

It is being added to all of the time.

[Bignose]

I have tried to explain this and following topics with some of the documents at

http://www.goretaivor.com/Science/Mathematics/index.php

 

I am really frustrated that if you want to learn something you have to go through all forums you can find collecting crumbs instead of having one place where topics would be properly explained.

 

have you tried MathWorld? Wikipedia or any other encyclopedia?

 

But, seriously, I have my doubts that there could ever be a "one place" where you could have everything. I mean, just on this one topic of imaginary numbers (complex analysis), there are several multi-hundred page books listed as still being in print on Amazon.com. Add to that all the journal articles and out of print texts on complex analysis, and we're talking hundreds of thousands of pages. And, all that on just one small part of mathematics. Collecting it all together is probably an impossible task.

[alan2here]

Can someone tell me simply what happens when you square a complex (2D) number.

 

For example my number is n

 

n = (2, 3)

 

o = n ^ 2

 

o = ?

[insane_alien]

if you have number x+iy you do (x+iy)(x+yi) you can use FOIL for this form like an ordinary equation.

[ajb]

Can someone tell me simply what happens when you square a complex (2D) number.

 

For example my number is n

 

n = (2, 3)

 

o = n ^ 2

 

o = ?

 

You mean the square root of a vector?

 

You don't have a natural product on vectors. Thus, I don't know what your squaring operation means.