Uses of complex multiplication?

[Prem1998]

The only thing which makes complex numbers different from 2-dimensional vectors or any other two-component mathematical object is their multiplication, right?
Complex multiplication has uses in rotations but we can easily achieve that using polar co-ordinates. And, their other applications in co-ordinate geometry can be done by replacing them with vectors.
And, their remaining use is in quantum mechanics where wave amplitudes are complex. But, do we ever need to multiply these wave amplitudes? I mean, if we don't, then this use of theirs can also be achieved by two numbers separated by commas and enclosed in brackets without ever needing to define square root of negative numbers.

 

[FactChecker]

The only thing which makes complex numbers different from 2-dimensional vectors or any other two-component mathematical object is their multiplication, right?

You should say that they have the right kind of multiplication. After all, vectors have dot products and cross products, which are not nearly as good.

Complex multiplication has uses in rotations

This is the main point. Relating the fundamental algebraic operation of multiplication to the fundamental geometric operation of rotation gives profound results. It means you can always divide because every rotation has a rotation in the opposite direction. It means you always have a square root because every rotation has a half rotation. (Likewise for all other roots.) It means that periodic and circular behavior can be represented by the fundamental operation of repeated multiplication. The consequences are profound.

but we can easily achieve that using polar co-ordinates.

Complex numbers have polar coordinates.

And, their other applications in co-ordinate geometry can be done by replacing them with vectors.

Not nearly as well.

And, their remaining use is in quantum mechanics where wave amplitudes are complex.

They have many more applications to all periodic and circular behavior. And because Fourier series, spectral analysis, and Laplace transformations are so useful, the complex analysis that they are based on is very powerful.

But, do we ever need to multiply these wave amplitudes? I mean, if we don't, then this use of theirs can also be achieved by two numbers separated by commas and enclosed in brackets without ever needing to define square root of negative numbers.

You are saying that replacing complex numbers would require several constructs and operations, all handled differently, not closed under the operations. That is a very poor exchange.

PS. Because of its advantages, the approach of complex analysis has been extended to higher dimensions in the fields of "Geometric Analysis" and "Geometric Calculus". It consolidates a lot of physics into a more concise and methodical representation. For instance, Maxwell's equations are just one equation in Geometric Algebra. I would hesitate to recommend studying Geometric Algebra because it is not main-stream and there is a learning curve. But it is good to be aware of it.

 

[Prem1998]

You should say that they have the right kind of multiplication. After all, vectors have dot products and cross products, which are not nearly as good.This is the main point. Relating the fundamental algebraic operation of multiplication to the fundamental geometric operation of rotation gives profound results. It means you can always divide because every rotation has a rotation in the opposite direction. It means you always have a square root because every rotation has a half rotation. (Likewise for all other roots.) It means that periodic and circular behavior can be represented by the fundamental operation of repeated multiplication. The consequences are profound.Complex numbers have polar coordinates.Not nearly as well.They have many more applications to all periodic and circular behavior. And because Fourier series, spectral analysis, and Laplace transformations are so useful, the complex analysis that they are based on is very powerful.
You are saying that replacing complex numbers would require several constructs and operations, all handled differently, not closed under the operations. That is a very poor exchange.

My point is: If we exclude the application of complex numbers in rotations ( which also can be achieved by co-ordinate geometry and trig), then can't all of their other applications be achieved by defining complex numbers just as an ordered pair of numbers without relating them to square roots of negative numbers?

 

[FactChecker]

My point is: If we exclude the application of complex numbers in rotations ( which also can be achieved by co-ordinate geometry and trig), then can't all of their other applications be achieved by defining complex numbers just as an ordered pair of numbers without relating them to square roots of negative numbers?

That would be much more difficult, to the point that some things would be nearly impossible (Unless you are just talking about changing the notation. But that would be silly.) For instance, the Taylor series at x₀=0 (the Maclaurin series) of the function f(x) = 1/(x² + 1) stops converging for |x| > 1 because -i and +i are zeros of the denominator. How would you explain that using co-ordinate geometry and trig?

 

[PeroK]

My point is: If we exclude the application of complex numbers in rotations ( which also can be achieved by co-ordinate geometry and trig), then can't all of their other applications be achieved by defining complex numbers just as an ordered pair of numbers without relating them to square roots of negative numbers?

The set of complex numbers is isomorphic to the set of ordered pairs of real numbers with multiplication defined by:

$(a,b)(c,d) = (ac-bd, ad+bc)$

If your question is why don't we do that and forget about $i$ then the answer is partly convention, but mostly convenience.

In particular, taking $\mathbb{C}$ as the algebraically closed extension of the real numbers was the original purpose of complex numbers. Why would you deny their importance and properties as numbers?

It's so convenient to know that matrices and linear operators have eigenvalues, expressible in the notation of numbers, rather in the notation of ordered pairs of real numbers.

If your question is based on a mistrust or dislike of complex numbers, then you should lose that mistrust and embrace them as one of the most useful tools we have at our disposal.

 

[Prem1998]

That would be much more difficult, to the point that some things would be nearly impossible (Unless you are just talking about changing the notation. But that would be silly.) For instance, the Taylor series at x₀=0 (the Maclaurin series) of the function f(x) = 1/(x² + 1) stops converging for |x| > 1 because -i and +i are zeros of the denominator. How would you explain that using co-ordinate geometry and trig?

I was just looking for a problem which can only be solved if you take the square root of negative numbers. If the thing that you siad about that Maclaurin series can only be proved by complex numbers, then, yes, this is an application where complex numbers are not just for convenience.
But talking about their more practical aplications in quantum mechanics, Fourier series and slectral analysis, do we ever get to multiply complex numbers while using complex numbers in these fields? Do we ever get to use the property i^2=-1 in doing some practical work of complex numbers(except rotations)? Or are these applications just about handling numbers with two components? Because complex numbers are more about i^2=-1 than they are about being numbers having two components.

 

[PeroK]

I was just looking for a problem which can only be solved if you take the square root of negative numbers. If the thing that you siad about that Maclaurin series can only be proved by complex numbers, then, yes, this is an application where complex numbers are not just for convenience.
But talking about their more practical aplications in quantum mechanics, Fourier series and slectral analysis, do we ever get to multiply complex numbers while using complex numbers in these fields? Do we ever get to use the property i^2=-1 in doing some practical work of complex numbers(except rotations)? Or are these applications just about handling numbers with two components? Because complex numbers are more about i^2=-1 than they are about being numbers having two components.

Have you ever done any complex maths? I do $i^2 = -1$ several times, every day!

 

[pwsnafu]

I was just looking for a problem which can only be solved if you take the square root of negative numbers.

Of course. The problem that started it all! The cubic equation requires you to take square root of negative numbers to get the answer out. For example $x^3 - 15x = 4$ has $x=4$ as the only real solution. But the Cardano formula gives $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$. Things like this forced the mathematicians at the time to consider square roots of negatives seriously.

For a slightly more modern example, explain why the homogeneous solutions of a second order linear differential equation transitions from pairs of exponentials when the discriminant is positive, to exponential multiplied by trigonometrics when the discriminant is negative.

 

[Prem1998]

Have you ever done any complex maths? I do $i^2 = -1$ several times, every day!

I was not talking about using i^2=-1 while solving maths. I can, right now, invent three dimensional numbers and call the third component j and make the all math relationships and do the math all day.
I was talking about whether the fact that i^=-1 is used in the practical applications of complex numbers or not (except rotations). So, does quantum mechanics use complex numbers because it needs the sqaure roots of negative numbers or does it use it because it needs numbers with two components? Did you ever use i^2=-1 while doing quantum mechanics maths? The same things for Fourier transform and spectral analysis.
So, basically, complex numbers are used in many areas of science, but the thing which they're all about, i.e. i^2=-1, is that thing used in those areas of science?

 

[PeroK]

I was not talking about using i^2=-1 while solving maths. I can, right now, invent three dimensional numbers and call the third component j and make the all math relationships and do the math all day.
I was talking about whether the fact that i^=-1 is used in the practical applications of complex numbers or not (except rotations). So, does quantum mechanics use complex numbers because it needs the sqaure roots of negative numbers or does it use it because it needs numbers with two components? Did you ever use i^2=-1 while doing quantum mechanics maths? The same things for Fourier transform and spectral analysis.
So, basically, complex numbers are used in many areas of science, but the thing which they're all about, i.e. i^2=-1, is that thing used in those areas of science?

Of course $i^2 = -1$ is used all over quantum mechanics and the linear algebra that supports it.

The point you're missing is that in the ordered pairs, you have:

$(0,1)^2 = (0, 1)(0,1) = (-1, 0)$

You don't get rid of the square root of a negative number, you simply encapsulate it in a different form.

 

[FactChecker]

I was not talking about using i^2=-1 while solving maths. I can, right now, invent three dimensional numbers and call the third component j and make the all math relationships and do the math all day.
I was talking about whether the fact that i^=-1 is used in the practical applications of complex numbers or not (except rotations). So, does quantum mechanics use complex numbers because it needs the sqaure roots of negative numbers or does it use it because it needs numbers with two components? Did you ever use i^2=-1 while doing quantum mechanics maths? The same things for Fourier transform and spectral analysis.
So, basically, complex numbers are used in many areas of science, but the thing which they're all about, i.e. i^2=-1, is that thing used in those areas of science?

I don't understand why you are concerned about that one equation, i² = -1. It is a small part of the subject. The real answer is that Laplace transforms, Fourier transforms, spectral analysis, and many other subjects are intimately tied to the complex number system. How can you have the complex plane and its algebra without including i² = 1?
And yes, there are other ways to do things, but they are not nearly as convenient or intuitive, so what is the point?

 

[Prem1998]

Of course $i^2 = -1$ is used all over quantum mechanics and the linear algebra that supports it.

The point you're missing is that in the ordered pairs, you have:

$(0,1)^2 = (0, 1)(0,1) = (-1, 0)$

You don't get rid of the square root of a negative number, you simply encapsulate it in a different form.

So, you get quantum mechanics problems where you have to multiply the complex wave amplitudes and actually use the fact that i^2 =-1 rather than using 'i' just as a tool to get two-component numbers required to describle wave amplitudes, right? Then, it's fine. Then, complex numbers are really a basic requirement rather than a tool for convenience.
By the way, by using the ordered-pair numbers, I wasn't talking about just changing the notation and still use square roots of negative numbers in some other way. I was talking about defining complex numbers just as an ordered pair without linking them to square root of -1 or any other multiplication rule, given that their multiplication was not of any use in science (if we exclude rotations, which i found as the only use of i^2=-1) according to me until sometime ago.

 

[FactChecker]

But talking about their more practical aplications in quantum mechanics, Fourier series and slectral analysis, do we ever get to multiply complex numbers while using complex numbers in these fields?

In studying feedback systems, differential equations, and control laws, it is necessary to know when they are stable. That almost always requires factoring a denominator completely. I don't know how often the roots of the denominator include ±i, but there are almost always complex number roots. And that is all I have to say on this subject.

 

[lavinia]

Take a look at this. It reviews the application to complex manifold theory to physics.

http://projecteuclid.org/download/pdf_1/euclid.bams/1183544081

 

[jedishrfu]

Historically, Charles Steinmetz championed the use of complex numbers in Electrical Engineering for impedance in AC circuits. Prior to his approach engineers would setup and solve differential equations to get the answer they needed. However using the Steinmetz scheme it was reduced to simpler algebraic manipulation via the phasor (phase vector):

https://en.wikipedia.org/wiki/Phasor

electrical impedance:

https://en.wikipedia.org/wiki/Electrical_impedance

and more on Steinmetz:

https://en.wikipedia.org/wiki/Charles_Proteus_Steinmetz

 

[mfb]

And, their remaining use is in quantum mechanics where wave amplitudes are complex. But, do we ever need to multiply these wave amplitudes?

Yes. Quantum field theory has higher powers of the complex values of the fields.

Real and imaginary part separated by a comma would not represent the physics, where the global phase is arbitrary. Amplitude and phase separated by a comma would work, but where is the point? It would make working with the expressions much more messy.

 

[Svein]

The only thing which makes complex numbers different from 2-dimensional vectors or any other two-component mathematical object is their multiplication, right?
Complex multiplication has uses in rotations but we can easily achieve that using polar co-ordinates. And, their other applications in co-ordinate geometry can be done by replacing them with vectors.

Of course there are other alternatives to complex numbers. An example that springs to mind is quaternions (https://en.wikipedia.org/wiki/Quaternion).

 

[lavinia]

And, their remaining use is in quantum mechanics where wave amplitudes are complex. But, do we ever need to multiply these wave amplitudes?

Wave amplitudes are multiplied in Quantum Mechanics - for instance when one computes the Hermitian inner product of two states.

Another example is the state of a not entangled two particle system. It can be expressed as a tensor product of the individual particle states. The complex coefficients of this state with respect to a basis are products of the coefficients of the individual states.

Also a quantum mechanical state space may be thought of as a vector space over the complex numbers. That is: it is an abstract vector space whose scalars are complex numbers. From this point of view the complex numbers are not vectors but rather are coefficients in field of numbers.

 

[jedishrfu]

Of course there are other alternatives to complex numbers. An example that springs to mind is quaternions (https://en.wikipedia.org/wiki/Quaternion).

As another historical note, William Rowan Hamilton, the Irish mathematician and the discoverer of quaternions wanted to refashion physics with quaternions as the base.

However other folks notably Heaviside and Gibbs disagreed citing its "unnecessary complexity" and refashioned quaternions into vector analysis borrowing a lot of ideas including the i,j,k notation we often use today.

More recently, quaternions were making a comeback in computer graphics and simulations programming.

https://en.m.wikipedia.org/wiki/Quaternion

And then there's octonians...

 

[FactChecker]

More recently, quaternions were making a comeback in computer graphics and simulations programming.

In many flight simulators, quaternions have always been used to avoid the gimble lock of Euler angles. But the quaternions are the three-dimensional equivalent of the complex numbers in two dimensions. They are not an alternative to complex numbers in two dimensions as @Svein implied.

 

[olivermsun]

My point is: If we exclude the application of complex numbers in rotations ( which also can be achieved by co-ordinate geometry and trig), then can't all of their other applications be achieved by defining complex numbers just as an ordered pair of numbers without relating them to square roots of negative numbers?

By the way, by using the ordered-pair numbers, I wasn't talking about just changing the notation and still use square roots of negative numbers in some other way. I was talking about defining complex numbers just as an ordered pair without linking them to square root of -1 or any other multiplication rule, given that their multiplication was not of any use in science (if we exclude rotations, which i found as the only use of i^2=-1) according to me until sometime ago.

But that's a huge use. The fact that multiplications induce changes in amplitude and phase that agree with i^2=-1 is pretty fundamental. In fact, you kind of get the trig identities for free once you have that.

 

[Prem1998]

Thanks for all the replies. If 'i' is not being used only as a component separater, then that settles the matter for me.