Do we really need complex numbers?

[PeterBushMan]

There is an axis for Imaginary numbers, and there is an axis for real numbers.

So we can just use the X and Y axes, what is the difference? They can do the same thing.

 

 

[joigus]

Sum of complex numbers can be implemented by 2-vectors. But product of complex numbers cannot. Product of complex numbers is equivalent to inner product (real part) and vector product (imaginary part). So their algebraic properties package more than ordinary 2-vectors. You can, of course, define special operations for 2-vectors that replicate all the properties of complex numbers. You can also do it with matrices, but in the end you would be doing the same thing.

[studiot]

2 hours ago, PeterBushMan said:

There is an axis for Imaginary numbers, and there is an axis for real numbers.

So we can just use the X and Y axes, what is the difference? They can do the same thing.

 

 

You have posted this in Applied Mathematics, and it is true that most if not all applications can be handled in other ways.

Perhaps you do not know the role played by complex numbers in Pure Mathematics.

Perhaps the simplest answer is that number systems were developed in sequence starting with the simplest counting numbers,

Going through systems including

The full set of integers.

The ratioanl numbers (fractions)

The real numbers

The imaginary numbers

The complex numbers (do you know the difference between imaginary numbers and complex numbers ?  What does complex mean ?

 

Now each of these systems was introduced when it bcame apparent that there were equations in the simpler system, that had no solutions in the simpler system.

 

For instance the equation 2x = -30 has no solution in the counting numbers or the positive integers, the equation 2x = 1 has no solution in the integers, the equation 2x²-20x+1=0 has no real solutions  - both solutions are complex.

 

So complex numbers are necessary to provide solutions to all quadratic equatuions.

Some equations of higher order do not even have a solurion formula withoug involving complex numbers.

 

It is just fortunate happenstance that having developed complex numbers they find many useful applications elsewhere.

 

Of course the process has not stopped there, there are even more complicated 'number' than complex ones.

 

 

[uncool]

Strictly speaking, no, you do not "need" the complex numbers for anything.

You can write everything you do in term of pairs of complex numbers, which just so happen to have the properties of the complex numbers.

You could go even further; real numbers are equivalence classes of sequences of rational numbers (under an equivalence relation of, approximately, "limits to the same number").

And further still; rational numbers are equivalence classes of pairs of integers (under an equivalence relation of "same ratio").

And yet further; integers are equivalence classes of pairs of rational numbers (under an equivalence relation of "same difference").

But it's far more convenient to not constantly be going through this entire hierarchy to make even simple statements. It's much, much easier to simply define a set of numbers once, and then deal with all of the ramifications by repeatedly using that definition. And the complex numbers turn out to be useful enough to do so - much more than you might expect, in fact.

Edited October 22, 2022 by uncool

[Sensei]

Quote

Do we really need [xyz]?

Use the right kind of tool for the right kind of job..

[studiot]

5 hours ago, studiot said:

2x²-20x+1=0 has no real solutions  - both solutions are complex.

My apologies that equation should have been 20x²-2x+1=0.

[Markus Hanke]

10 hours ago, PeterBushMan said:

So we can just use the X and Y axes, what is the difference? They can do the same thing.

This is true in classical physics, but as it turns out it is not true in quantum mechanics. You can construct a class of experiments where real-valued QM (replace complex numbers by pairs of real ones) makes predictions that are different from complex-valued QM, thereby opening up a way to test this experimentally.

Turns out, complex Hilbert spaces are an essential feature of any QM formalism that describes the world accurately (within that domain):

https://arxiv.org/abs/2101.10873

[PeterBushMan]

21 hours ago, studiot said:

You have posted this in Applied Mathematics, and it is true that most if not all applications can be handled in other ways.

Perhaps you do not know the role played by complex numbers in Pure Mathematics.

Perhaps the simplest answer is that number systems were developed in sequence starting with the simplest counting numbers,

Going through systems including

The full set of integers.

The ratioanl numbers (fractions)

The real numbers

The imaginary numbers

The complex numbers (do you know the difference between imaginary numbers and complex numbers ?  What does complex mean ?

 

Now each of these systems was introduced when it bcame apparent that there were equations in the simpler system, that had no solutions in the simpler system.

 

For instance the equation 2x = -30 has no solution in the counting numbers or the positive integers, the equation 2x = 1 has no solution in the integers, the equation 2x²-20x+1=0 has no real solutions  - both solutions are complex.

 

So complex numbers are necessary to provide solutions to all quadratic equatuions.

Some equations of higher order do not even have a solurion formula withoug involving complex numbers.

 

It is just fortunate happenstance that having developed complex numbers they find many useful applications elsewhere.

 

Of course the process has not stopped there, there are even more complicated 'number' than complex ones.

 

 

The complex numbers (do you know the difference between imaginary numbers and complex numbers ?  What does complex mean ?

---- imaginary numbers  are in the form Yi.

---- complex numbers  have two parts X+Yi, X is a real number.

 

I only can understand complex number in that way - First we invented imaginary number which is not exist. then we try to find some uses of it.

 

 

[studiot]

1 hour ago, PeterBushMan said:

The complex numbers (do you know the difference between imaginary numbers and complex numbers ?  What does complex mean ?

---- imaginary numbers  are in the form Yi.

---- complex numbers  have two parts X+Yi, X is a real number.

 

I only can understand complex number in that way - First we invented imaginary number which is not exist. then we try to find some uses of it.

 

 

 

That's exactly right, well done.  +1

'Complex' means made of more than one part and as you say 'The Complex Numbers' are made of two parts.

 

What did you think of the other information poeple provided ?

[joigus]

15 hours ago, Markus Hanke said:

This is true in classical physics, but as it turns out it is not true in quantum mechanics. You can construct a class of experiments where real-valued QM (replace complex numbers by pairs of real ones) makes predictions that are different from complex-valued QM, thereby opening up a way to test this experimentally.

Turns out, complex Hilbert spaces are an essential feature of any QM formalism that describes the world accurately (within that domain):

https://arxiv.org/abs/2101.10873

Interesting. +1

People have been thinking about alternatives to complex formulations of QM for ages. I think Schrödinger himself initially thought that the presence of complex numbers in his equation might point to some flaw in his argument.

There's always the trivial possibility of replacing the number 1 by the 2x2-dim identity matrix, the number i by a 2x2-dim antisymmetric matrix with determinant 1, and say something with philosophical undertones like "oh, the number i doesn't exist; it's just a 2x2 matrix!"

[joigus]

On 10/22/2022 at 11:52 AM, joigus said:

Sum of complex numbers can be implemented by 2-vectors.

Sorry, I meant addition. A false friend tricked me.

A little bit more on this fascinating --at least to me-- topic:

Suppose that, for some reason, you are repulsed by numbers which are the square root of a negative real number. You can always obtain a numeric structure that's totally equivalent to complex numbers by means of the following trick:

Complex numbers "secretly" are 2x2 real matrices. Now, 2x2 real matrices can always be uniquely expanded into a symmetric part and an antisymmetric part.

Here's how you do it. Introduce the special matrices that are going to be respective stand-ins for 1 and i:

\[ E=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \]

\[ I=\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right) \]

and define a complex number \( z \) with real part \( x \) and imaginary part \( y \) --and its conjugate-- as "secretly,"

\[ z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \]

\[ z^{*}=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \]

Then, the absolute value (squared) of \( z \) is,

\[ z^{*}z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right)\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right)=\left(\begin{array}{cc} x^{2}+y^{2} & 0\\ 0 & x^{2}+y^{2} \end{array}\right)=\left(x^{2}+y^{2}\right)E \]

The product of \( z \) and \( z' \) --another complex number-- is

\[ zz'=\left(xx'-yy'\right)E+\left(xy'+x'y\right)I \]

etc.

Now, whether complex numbers are "secretly" 2x2 real matrices, or conversely 2x2 real matrices "secretly" are complex numbers is, of course, totally immaterial from a purely mathematical POV.

Errata

It should be:

\[ z=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \]

\[ z^{*}=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \]

So I guess my answer is: Yes, we do need complex numbers. We can dress them as 2x2 real matrices if we want, but we need them is some disguise or another.

[studiot]

Actually there is more to what Markus said than meets the eye.

 

The twist in the tail of using complex numbers in QM lies in non cummutativity.

The 'full' set of complex numbers entail numbers of the form a + bi, where a and b are real numbers.

Both the real numbers and the complex numbers formed this way form an algebraic Field and enjoy all the field properties, including the 10 Field axioms.

One of the  most important consequences is the unique factorisation theorem which guarantees unique solutions to all alegbraic equations in one of these Fields.

 

Now Gauss discovered and did a lot of work on something simpler, we now call 'Gaussian Integers'.

These connect ordinary integers with Gaussian integers in the same way as the reals and the complex numbers.
So if p and q are integers then p + qi  are gaussian integers.

Now neither the ordinary integers nor the gaussian integers form an algebraic Field. They only satisfy 6 of the field axioms.
But they do form a more general algebraic structure called a Ring.

Some rings such as the integers still satisfy unique factorisation so for instance the only factorisation of -28  is -(1x2x2x7).

But in the ring p+q√5i,  the number 6+0i  factorises as follows:-  (2+0i)x(3+0i)  and also (1+√5i)(1+√5i)

Such rings of complex numbers fail to satisfy unique factorisation   -  essentially a quantum behaviour.

 

[NTuft]

 

On 10/22/2022 at 1:03 AM, PeterBushMan said:

There is an axis for Imaginary numbers, and there is an axis for real numbers.

I presume you're referring to the Argand plane?

Quote

Argand diagram refers to a geometric plot of complex numbers as points z = x + iy using the x-axis as the real axis and y-axis as the imaginary axis. [...]
https://handwiki.org/wiki/:Complex plane#Argand_diagram

Quote

 

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called real axis, is formed by the real numbers, and the y-axis, called imaginary axis, is formed by the imaginary numbers.

The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

The complex plane is sometimes known as the Argand plane or Gauss plane.

https://handwiki.org/wiki/:Complex plane

 

On 10/22/2022 at 1:03 AM, PeterBushMan said:

So we can just use the X and Y axes, what is the difference? They can do the same thing.

The Argand plane diagrams are graphing complex numbers out from the origin as vectors. We can graph vectors on our Cartesian x-y graph, too. However, the geometric interpretation of the real number line is already a construct of numbers as vectors to my mind: a number selected on the line has a magnitude, its value, and a direction, positive or negative away from 0.

For the complex plane, look at this difference:

Quote

Stereographic projections

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane. [...]

Under this stereographic projection the north pole itself is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane. We speak of a single "point at infinity" when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane.
https://handwiki.org/wiki/Complex_plane#Stereographic_projections

To my mind, this is related to the idea that with complex numbers there is no concept of larger or smaller that can be compared to how we think of real numbers as being closer or further from 0. 

 

7 hours ago, PeterBushMan said:

I only can understand complex number in that way - First we invented imaginary number which is not exist. then we try to find some uses of it.

Imaginary is just a name, given by Descartes himself I believe. Gauss did not like it -- and thought they were on equal footing, to paraphrase his writing.

+1 uncool on number sets

+1 joigus on matrices

Edited October 23, 2022 by NTuft
attributions. x-post w/studiot.

[uncool]

1 hour ago, studiot said:

The twist in the tail of using complex numbers in QM lies in non cummutativity

Err, what?

Complex numbers are entirely commutative, and any way in which noncommutativity appears in QM can appear in a "real" explanation just as much as for a "complex" one.

1 hour ago, studiot said:

Such rings of complex numbers fail to satisfy unique factorisation   -  essentially a quantum behaviour.

Um.

Two things.

First, there are plenty of subrings of the real numbers that fail to satisfy unique factorization. Consider Z[√5], and factor 4. That can be fixed in a somewhat natural way by adding (1+√5)/2; however, the "natural" fix doesn't work for Z[√7] (and this failure - among others - is a reason for the modern study of algebraic number theory, especially ideal class groups). Failure of unique factorization is not inherently related to complexity.

Second, I see no reason why a failure of unique factorization is "essentially a quantum behavior".

Edited October 23, 2022 by uncool

[joigus]

2 hours ago, joigus said:

Complex numbers "secretly" are 2x2 real matrices. Now, 2x2 real matrices can always be uniquely expanded into a symmetric part and an antisymmetric part.

Disclaimer: It wouldn't be all of 2x2 matrices. It would only be those of the form,

\[ z=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \]

Those, and not other 2x2 real matrices, are complex numbers.

33 minutes ago, studiot said:

But in the ring p+q√5i,  the number 6+0i  factorises as follows:-  (2+0i)x(3+0i)  and also (1+√5i)(1+√5i)

 

These rings have fascinated me for decades. You could get an approximation as close as desired to the real numbers by means of rational combinations of 1 and the square root of any integer number that's not a perfect square. The p and q would have to be rational, instead of integer.

Would that be a way in which the continuum can be approximated by a discrete mapping that frees physics from singularities and other similar "diseases"?

8 minutes ago, NTuft said:

Imaginary is just a name, given by Descartes himself I believe. Gauss did not like it

Absolutely spot on. Just a name. Nothing "spooky" in imaginary.

8 minutes ago, uncool said:

Complex numbers are entirely commutative, and any way in which noncommutativity appears in QM can appear in a "real" explanation just as much as for a "complex" one.

Agree. Non-commutativity: totally peculiarly quantum, but not due to complex nature.

[studiot]

59 minutes ago, uncool said:

Err, what?

Complex numbers are entirely commutative, and any way in which noncommutativity appears in QM can appear in a "real" explanation just as much as for a "complex" one.

Entirely ?

I don't think so.

But then I don't think you mean quite that literally.

Nor are real numbers entirely commutative either.  For example multiplication and square rooting are not commutative  2.√3 is not equal to √2.3

They are both commutative for either of  the two operations of addition and multiplication as befits their status as Fields, which I believe I mentioned.

But QM commutation is about the commutation of two different operators.

So can you detail your real operators in QM that do not commute ?
The dreaded 'i' seems to appear lots in this treatment.

https://quantummechanics.ucsd.edu/ph130a/130_notes/node109.html

 

1 hour ago, uncool said:

Second, I see no reason why a failure of unique factorization is "essentially a quantum behavior".

Maybe the factorisation was stretching things a bit far  as composition is involved, but I can't see what you mean by this

1 hour ago, uncool said:

First, there are plenty of subrings of the real numbers that fail to satisfy unique factorization. Consider Z[√5], and factor 4. That can be fixed in a somewhat natural way by adding (1+√5)/2; however, the "natural" fix doesn't work for Z[√7] (and this failure - among others - is a reason for the modern study of algebraic number theory, especially ideal class groups). Failure of unique factorization is not inherently related to complexity.

How is (1+√5)/2 an element of an integer ring, when division is not closed with the integers ?

[Markus Hanke]

4 hours ago, joigus said:

Sorry, I meant addition.

I think a far more interesting operation appears when one uses complex numbers as exponents - suddenly we are now dealing with rotations and scalings, which is a much richer structure than real exponents can yield. This being linear transformations, it’s not surprising that there is a close connections to certain types of matrices.

Either way, the results I linked to seem to show unambiguously that - for whatever reason - complex numbers are indispensable for QM.

[uncool]

2 hours ago, studiot said:

So can you detail your real operators in QM that do not commute ?
The dreaded 'i' seems to appear lots in this treatment.

I'd need you to first define "real operators".

As I said earlier, anything that can be defined in terms of complex numbers can instead be defined in terms of pairs of real numbers, etc.

2 hours ago, studiot said:

Maybe the factorisation was stretching things a bit far  as composition is involved, but I can't see what you mean by this

I meant that I have no idea what connection you are trying to draw between unique factorization and quantum physics.

2 hours ago, studiot said:

How is (1+√5)/2 an element of an integer ring, when division is not closed with the integers ?

Why not? Why should √5 be considered "integer", while (1+√5)/2 can't be? How are you defining "integer" outside of the actual integers?

It turns out that (1+√5)/2 actually is quite integer-like, in that it can solve monic polynomials; such numbers are actually called "algebraic integers", and act somewhat like integers do for rational numbers.

Edited October 23, 2022 by uncool

[studiot]

21 minutes ago, uncool said:

I'd need you to first define "real operators".

As I said earlier, anything that can be defined in terms of complex numbers can instead be defined in terms of pairs of real numbers, etc

And I did ask for an example of this 'anything' , whilst at the same time pointing to a well respected treatment of QM, involving plenty of imaginary and/or complex numbers.

 

But this is taking the discussion away from the key statement

2 hours ago, studiot said:

But QM commutation is about the commutation of two different operators.

 

 

2 hours ago, studiot said:

How is (1+√5)/2 an element of an integer ring, when division is not closed with the integers ?

Yes I am aware of extension algebras and arithmetic and I know tha there are some pretty exotic creations floating around there, but I am not very adept at them.
I was always given to understand that for every gain in one direction you tend to loose something in another when you add to algebraic complexity.

[joigus]

4 hours ago, NTuft said:

Imaginary is just a name, given by Descartes himself I believe. Gauss did not like it

I'm with Gauss on this one. +1

Remove all nonsense propping-up-to-mystical nuances from anything intended to clarify our understanding. Vocabulary is a good place to start.

 

1 hour ago, Markus Hanke said:

I think a far more interesting operation appears when one uses complex numbers as exponents - suddenly we are now dealing with rotations and scalings, which is a much richer structure than real exponents can yield. This being linear transformations, it’s not surprising that there is a close connections to certain types of matrices.

Either way, the results I linked to seem to show unambiguously that - for whatever reason - complex numbers are indispensable for QM.

I share your taste. I'd rather multiply complex numbers than add them. Adding complex numbers is messy; multiplying them is nice.

I also agree that complex numbers are an indispensible part of QM.

Unfortunately QM forces us, not only to add them, but normalise the result, after having added them, which is even more awkward than just adding them.

I think the problem of what mathematical representation makes our description least awkward will always be an open question. The parametrisation of the sphere is, in this respect, a cautionary note that I never forget.

[uncool]

9 hours ago, studiot said:

And I did ask for an example of this 'anything' , whilst at the same time pointing to a well respected treatment of QM, involving plenty of imaginary and/or complex numbers.

Sure.

Instead of saying "the product of 1+2i and 2+5i is -8 + 9i", you could instead say "the product of (1, 2) and (2, 5) under the operation (a, b)*(c, d) = (ac - bd, ad - bc) is (-8, 9)".

Yes, it's a trivial renaming of the complex numbers; that's part of my point.

9 hours ago, studiot said:

But this is taking the discussion away from the key statement

But QM commutation is about the commutation of two different operators.

OK, here are two real operators (as I understand it; I never received an answer as to what you meant) that fail to commute:

 

0 1
0 0

0 0
1 0

Possibly more relevant to QM, here are some real operators that act like the spin operators, if I haven't made a stupid mistake:

0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0

0 0 0 1
0 0 -1 0
0 -1 0 0
1 0 0 0

1 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 -1

where multiplication by i is replaced by multiplication by the matrix

0 -1 0 0
1 0 0 0
0 0 0 -1
0 0 1 0

All of the components of each operator are now real.

9 hours ago, studiot said:

 

12 hours ago, studiot said:

How is (1+√5)/2 an element of an integer ring, when division is not closed with the integers ?

Yes I am aware of extension algebras and arithmetic and I know tha there are some pretty exotic creations floating around there, but I am not very adept at them.
I was always given to understand that for every gain in one direction you tend to loose something in another when you add to algebraic complexity.

You seem to have quoted yourself; I'm not sure I see how this responds to my post.

[John Cuthber]

"Do we really need complex numbers?"
It depends on whether or not you want to be able to solve quadratics.
If you want solutions, then you have to put up with complex numbers.

[joigus]

Another interesting application of complex numbers inspired by physics: Materials have a refractive index. The reflected wave is given in terms of the real part, the imaginary part conveniently embodies the property of absorbance. So a metal has a considerable imaginary part in its complex refractive index.

This is a nice example of what @Markus Hanke said about complex numbers as exponents.

I see nothing "imaginary" in the ability of a material to absorb light.

Why wouldn't one want to use such a handy book-keeping device?

[MigL]

I would think that you need a character to represent the root of -1.
The character i doesn't seem like a bad choice, but j would hace done just as well.
And calling such numbers complex is aso practical, although 'strange' or 'unreal' would have also been fine.

But you do need to represent the root of -1 somehow ...

[NTuft]

Well... this is a goofy website:

https://www.gnqr.co.uk/quantum-time-home-phase-four

that clearly needs vetting. However, what I was looking for was this, which is brought out as an image on the lower part of the page after navigation:

Quote

 

JCF Gauss; 2nd letter to the Royal Society, 1831. Paragraph 24, his last statement in a long boring paper.

[24] "We have believed that we were doing the friends of mathematics a favour by this account of the principal parts of a new theory of so-called imaginary quantities. If one formerly contemplated this subject from a false point of view and therefore found a mystery darkness, this is in large part attributable to clumsy terminology. Had one not called +1, -1 and the  square-root of -1, positive, negative and imaginary counting units, but instead, say direct, inverse and lateral counting units, then there could scarcely have been talk of such darkness". (translated from the Latin by William Ewald)

[Page editor's remarks, possible additions; see page:] Try to follow that little lot in Latin without my red ink. We (as in "the Royal We") are the mathematicians here; you others are merely "the friends". Clearly, Gauss was just "taking the Mickey" and being as unhelpful as his great sense of mathematical-fun and conscience could permit him. Had the great man ever deigned to leave his observatory to speak at the Royal Society, then there would have been flags, trumpets and a packed house, but this was read out to a handful of half-asleep math's fellows who were too tired to retire to the bar. (Blink, and you missed it.) [/Ed.]

 

I can't find secondary verification of the letter from Gauss, but here's another page that quotes the same section, and has a good write-up on the topic:

The Reality of Imaginary Numbers by Brett Berry

 

From the Quantum Relativity site, here is their display of the Argand plane from that same page, highlighting the facet of multiplications as rotations (and other stuff..):

 

On 10/22/2022 at 1:03 AM, PeterBushMan said:

So we can just use the X and Y axes, what is the difference? They can do the same thing.

I think an example from the Medium article is illustrative. Given the equation y=x²+1 graphed on a Cartesian plot with two perpendicular integer axes, this quadratic equation does not cross the x-axis which would normally indicate where the solutions (x=0) are found. I think it's because the X-Y Cartesian plot of this form although seeming to graph a quadratic equation is somehow still linear. When the Argand plane uses the whole Real axis as the horizontal number line it then extends the imaginary numbers, up as "direct lateral" and down as "inverse lateral"  from the direct and inverse numbers horizontally to create the necessary non-linear dimension to find the solutions to the quadratic equations. 

I'm probably wrong and in need of proof-reading so have at it. I also encourage enlistment to go spelunking about the gnqr website for discussion but now am off topic.