Understanding the complex numbers entity

[DrDolittle]

complex numbers

My conception of complex numbers entity goes like this.

For going from city a to city b(both are real worlds).To travel from city a to city b we can go through city c (let us say,it is say a complex world).By going through a complex world the path travelled is minimal(analogous to complexity in solving the equation).Hence I view complex numbers entity as just a mathematical tool to exist like laplace domain.Am i right with my conception?

Regards
drdolittle

 

[Mina Ayman]

complex numbers

I think your concept is right to a great extent. I think also that complex analysis was developed to generalize some problems from 1D space to 2D space.
You can download a book about complex analysis at
https://www.math.gatech.edu/~cain/winter99/complex.html

 

[chirkut_iis]

Re: complex numbers

I feel that rather than being just an analytical tool, complex quantities are absolutely indispensable in many areas of physics and hence real world. Eg. the wave function in abstract hilbert spaces of quantum mechanics needs to be complex. Of course this is a slightly ambiguous statement, but specific examples could also be given.

regds,
C

 

[tantoun2004]

Re: complex numbers

Before quantum mechanics was developped , we used complex numbers especially Euler formula as a tool , convert every trigonometric into complex then take real part to see the physical thing.
But after Quantum mechanics we knew that the probability amplitude of the events MUST be complex, Schrodinger equation gives us complex function psi so we changed our view to consider complex functions in physics not only their real part.
Regards,

 

[ms.ms]

Re: complex numbers

I think complex number are a natural extension of the set of real number, conceptually not very different from the extension to negative numbers or fractional numbers of the set of positive integers.
A negative delta time means, for example, than an event occurred before another, nothing else.
A complex number, for example, indicates an oscillation (e^ix) instead of an endless grow.
First of all complex numbers are necessary to mathematicians to solve something like sqrt(-1), even if they proved very useful to many other fields.

 

[cmunikat]

Re: complex numbers

Dear all,

After reading the posts I can say that I feel the same as ms.ms, and may add that some abstractions are so much inside us, so common, that we simply do not realize until we read the words ms.ms said (negative numbers and so on).

I agree also with the comparisons with transformations.

In a philosophical way, I'd say this is the meaning. However they are very much useful tools for us.

Cmunikat

 

[the_jackal]

Re: complex numbers

Dr. Dolittle, first i did not quite understand your analogy or what is it that your conception is? For instance "I view complex numbers entity as just a mathematical tool to exist like laplace domain", is vague to me. We know that Laplace Transforms are a generalization of CTFT which in turn provides us with a representation of signals as linear combinations of complex exponentials of the form exp(st) where s=jw.

 

[truebs]

Re: complex numbers

Certainly , your perception is close but not excatly right !

Complex number are more than just mathematical tool, it allows us to express various quantities (Physical ) in a form that can be dealt for analysis. For example , Phasors ,Tensors used to eloborate Antenna Theory ,or Electromagnetics. There are various situations where just real numbers can't satisfy the needs for anlaysis and it becomes mandatory for us to include complex number.

One thing i'd like to explain is that ,Lapalace is really just a tool ,without it analysis would be tough but not impossible. We can choose some other transfor or anlayse directly.

But..Complex numbers if not used ,will make a number of theories un - analysable.

 

[DrDolittle]

Re: complex numbers

Laplce transformation is not just a mathematical tool. It is world in itself where the parameters have a special connotation. Complex Numbers entity may not have such a concrete picture, but I would never stop telling Laplace domain with sucha a shortcoming.

Regards
drdolittle