discussion: are there any places that complex numbers are necessary?

[C3PO]

so you can just use vectors algebra, why to put complex numbers, that they called as they are? complex can be existed only on plane, when 3D vectors can be also in all the space, & even 4D & more if you need to use the data like that...

When calculating asymmetric 3-phase AC loads, using complex numbers reduced 3 pages of trigonometry calculations to 1/2 a page of complex numbers calculations. (I have that old exam paper somewhere around the house)

 

[Gumok]

But I think I understand DGMP's question now... I would reformulate it - you ask: "Is there any part of physics that can be done only with complex numbers, without any possible replacement (even if it would take to make calculations harder)?"

Well... I'm not really sure. There are many fields of physics using them for comfort (that means that you can avoid them somehow) but there are also many that were derived only through complex algebra and by complex algebra we need to use them on.

 

[csanders]

The difference between these two types of motion is very profound and occurs because, in the second case, the solutions of the quadratic equation are complex and involve the square root of -1. We will look at these in more detail presently.

http://plus.maths.org/content/101-uses-quadratic-equation-part-ii

It's been 15 years since I did the math, but some where in there comes Euler’s Identity: e^(i*pi) + 1 = 0

EDIT: I should have read further, they kind of explain it later in the article.

 

[DGMP]

When calculating asymmetric 3-phase AC loads, using complex numbers reduced 3 pages of trigonometry calculations to 1/2 a page of complex numbers calculations. (I have that old exam paper somewhere around the house)

thats what I meant. you can just use trigonometry to avoid the complex. even if the calculations are becomes impossible..

 

[Urwumpe]

thats what I meant. you can just use trigonometry to avoid the complex. even if the calculations are becomes impossible..

Yes, and you can also use numeric iterative solutions of differential equations to avoid trigonometry.

 

[DGMP]

Yes, and you can also use numeric iterative solutions of differential equations to avoid trigonometry.

you now turn toward computing...

 

[fireballs619]

But I think I understand DGMP's question now... I would reformulate it - you ask: "Is there any part of physics that can be done only with complex numbers, without any possible replacement (even if it would take to make calculations harder)?"

Well... I'm not really sure. There are many fields of physics using them for comfort (that means that you can avoid them somehow) but there are also many that were derived only through complex algebra and by complex algebra we need to use them on.

Hm, if this is the question I think the answer is that no, no such part of physics exists. Here is my reasoning, which may very well be wrong.

Of course i has no physical meaning, since the square root of -1 has no physical meaning. It is then prudent to ask what we mean by "multiplication". We mean a system of scaling and rotating numbers (or vectors, rather) in a plane. We can say that multiplying by a positive number rotates 360 degrees and scales by some factor. Thus, the product going to face in the same direction as it did before the operation.

A negative number would mean rotation of 180 degrees in the plane and some scaling. Thus, the product is in the opposite direction (positive by negative=negative), while two rotations of 180 degrees would bring it back to the original direction (negative by negative = positive).

Now we can think of some rotation that, when applied twice will bring you to -1 in the plane. That is, some number when multiplied by itself will bring you to negative one, or the square root of negative one. This rotation would have to be 90 degrees, because we want two of these rotations to sum to 180 degrees.

Thinking of it like this, complex numbers represent the rotation, magnitude, and other properties of angles. I think one could avoid using complex numbers when you think of it like this. It of course makes calculations more cumbersome and harder, which is why we use i to begin with.

Just my thoughts, I haven't really ever studied anything like this.

 

[Max Pain]

Hm, if this is the question I think the answer is that no, no such part of physics exists.

I just want to point out what has already been said earlier:

In quantum Physics, the wave function is by its very nature complex. It consists of a real and imaginary part.

AFAIK in classical physics (that is mechanics, electrodynamics and relativity) complex numbers are just handy but not strictly necessary. All the physical quantities, that can be measured, are real. Quantum mechanics, however, would not work without complex numbers (or using a different concept which is mathematical equivalent to complex numbers).

You can see it by looking at the schroedinger equation, which has the imaginary unit i built in. That means mathematically that you can't just separate the real and imaginary part of the solution.

 

[fireballs619]

I just want to point out what has already been said earlier:



AFAIK in classical physics (that is mechanics, electrodynamics and relativity) complex numbers are just handy but not strictly necessary. All the physical quantities, that can be measured, are real. Quantum mechanics, however, would not work without complex numbers (or using a different concept which is mathematical equivalent to complex numbers).

You can see it by looking at the schroedinger equation, which has the imaginary unit i built in. That means mathematically that you can't just separate the real and imaginary part of the solution.

But can't you just think of the complex number and any operation done to it as the scaling and rotation of a vector?

 

[Urwumpe]

But can't you just think of the complex number and any operation done to it as the scaling and rotation of a vector?

Yes. And you can drive a car by thinking about servoactuators, electric currents and mechanics.

This is much more complicated, but possible.

 

[fireballs619]

Yes. And you can drive a car by thinking about servoactuators, electric currents and mechanics.

This is much more complicated, but possible.

Well of course, that's why we all love using complex numbers :lol:

I'm not arguing that we SHOULD be thinking of it so, only that it could be thought of like that.

 

[Urwumpe]

Well of course, that's why we all love using complex numbers :lol:

I'm not arguing that we SHOULD be thinking of it so, only that it could be thought of like that.

Well, if you are programmer responsible for making a car engine work, you might want to use complex numbers for many calculations... but not for all

 

[Max Pain]

But can't you just think of the complex number and any operation done to it as the scaling and rotation of a vector?

Yeah, of course you can define and use a mathematical structure which is equivalent to complex numbers. I, for myself, don't see the point in doing so. You can always "avoid" using complex numbers by calling it by a different name.

The point I am making is:
All the laws of classical physics (equation of motion, maxwells equations) are written in real differential equations and the laws of quantum mechanics use complex differential equations.

 

[thepenguin]

Why has nobody suggested solving quadratic equations where the discriminate is negative? This was the original purpose of complex numbers.

for example, I found an equation that you must use complex to solve:
x = cosx

haven't you already disproved your own statement? Are you looking for more examples, or am I missing something here?

 

[csanders]

*taps mic* Hello. This thing on?

http://plus.maths.org/content/101-uses-quadratic-equation-part-ii

It's been 15 years since I did the math, but some where in there comes Euler’s Identity: e^(i*pi) + 1 = 0

EDIT: I should have read further, they kind of explain it later in the article.

That link is titled: "101 uses of a quadratic equation: Part II." It's a very well done article. If you haven't read it (also part I), I highly recommend it.

 

[asbjos]

Complex numbers in real life?

Do scientists use complex numbers in calculations regarding real science? E.g. to represent something which actually exists and not is imaginary?

If so, could you point me to some articles or references?

 

[tl8]

Do scientists use complex numbers in calculations regarding real science? E.g. to represent something which actually exists and not is imaginary?

If so, could you point me to some articles or references?

Science less so, but definitely Electrical Engineering. It is very helpful in describing AC voltages and their relation to each other.

 

[Einion Yrth]

QED couldn't exist without complex numbers. Whether photons and electrons are "real" of course could be a matter for debate.

 

[Urwumpe]

Do scientists use complex numbers in calculations regarding real science? E.g. to represent something which actually exists and not is imaginary?

If so, could you point me to some articles or references?

What is real science compared to unreal science? :blink:

Most of the kinematics in modern Computer games is based on complex numbers... quarternions to be correct.

The same applies to navigation, satellites, etc.

The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices. In addition, unlike Euler angles they are not susceptible to gimbal lock. For this reason, quaternions are used in computer graphics,[11] computer vision, robotics, control theory, signal processing, attitude control, physics, bioinformatics, molecular dynamics, computer simulations, and orbital mechanics. For example, it is common for the attitude-control systems of spacecraft to be commanded in terms of quaternions. Quaternions have received another boost from number theory because of their relationships with the quadratic forms.

Also, complex numbers are very often used in mathematical problems which lead to more efficient solutions of more or complex equation systems, from Sudoku to computational fluid dynamics.

 

[Hlynkacg]

Electrical engineering, specifically the behavior of AC systems, and various problems associated with calculating motion and spacial relationships feature practical applications of complex numbers.

Not sure if that's "real" enough for you but it is what it is.

---------- Post added at 08:37 ---------- Previous post was at 08:37 ----------

Oops

:ninja:'ed