Is There an Alternative Addition for Complex Numbers or Quaternions?

[Gerenuk]

For complex numbers or quaternions, is there another two variable function, that behaves like addition? I mean a function f with
$$af(b,c)=f(ab,ac)$$
$$f(a,b)=f(b,a)$$
$$f(a,f(b,c))=f(f(a,b),c)$$

And which conditions have to be imposed to make this function uniquely $f(a,b)=a+b$?

 

[trambolin]

You mean like 2x2 matrices?

 

[Gerenuk]

Not sure what you ask.
a, b and c are complex numbers (first case) or quaternions (second case).
So in fact these variables are either *special* 2x2 matrices or 4x4 matrices. But I don't see how this matters.

 

[trambolin]

If you represent them matrices, addition and multiplication is also defined by matrix algebra.

 

[Gerenuk]

I'm not sure what you mean. I need an answer for the function f. So $f(a,b)=?$
apart from normal addition.

 

[trambolin]

Let $f:\mathbb{C}\times\mathbb{C}\to \mathbb{R}^{2\times 2},\ \ z=a+ib, w=c +id$,

$f(z,w) = \begin{pmatrix}a &-b\\ b &a \end{pmatrix}+ \begin{pmatrix}c &-d\\ d &c \end{pmatrix} = \begin{pmatrix}a+c &-(b+d)\\ (b+d) &a+c \end{pmatrix} \Longrightarrow z+w = (a+c) +i(b+d)$

Similarly, for multiplication

$g(z,w) = \begin{pmatrix}a &-b\\ b &a \end{pmatrix} \begin{pmatrix}c &-d\\ d &c \end{pmatrix} = \begin{pmatrix}ac-bd &-ad-bc\\ ad+bc &ac-bd \end{pmatrix} \Longrightarrow z\cdot w = (ac-bd) +i(ad+bc)$

I think I made my point and I am running out of arguments. Can you also be specific about what you are asking for?

 

[Gerenuk]

I really don't understand what you read in my question. Feel free to let me know where I missed out a point.

I try to repeat it more precisely: Find _all_ functions
$f:\mathbb{C}\times\mathbb{C}\to \mathbb{C}$
which for all complex numbers a, b, c satisfy
$$af(b,c)=f(ab,ac)$$
$$f(a,b)=f(b,a)$$
$$f(a,f(b,c))=f(f(a,b),c)$$
This function should not be trivial addition.
$$f(a,b)\neq a+b$$

 

[trambolin]

For complex numbers or quaternions, is there another two variable function, that behaves like addition?

I guess another means you look for something else than plain addition. And in the last post you ask all the functions. Are you trying to ask something about the underlying algebra? The set you are looking for is the set of all linear, commutative and associative functions.

Or you want to find an isomorphism for complex numbers other than 2x2 matrices?

 

[Gerenuk]

The set you are looking for is the set of all linear, commutative and associative functions.

I think that's what I am looking for. Can you give one example?

And second question: Which additional contraints do I have to add, so the there will be no solution to f apart from the trivial addition example.

 

[Ben Niehoff]

You don't want linear functions, but rather homogeneous functions of degree 1. Specifically, symmetric, associative, first-order homogeneous functions. Here's one example:

$$f(a,b) = \sqrt{a^2 + b^2}$$

and here's another

$$f(a,b) = \frac{ab}{a + b}$$

These work for a, b real; haven't tried for complex, but I would assume that either they work or they can easily be modified to work. The first is the cartesian distance function. The second is an operation known in electrical engineering as "parallel", often written

$$a \; || \; b$$

which is used to find the equivalent resistance of several resistors in parallel.

You can probably come up with more examples. As for what property would restrict f(a,b) to be simply addition, I'm not sure.

 

[Gerenuk]

Thanks! I thought about the examples more. The question now is if there are any solutions which are not related by
$$h(f(a,b))=h(a)+h(b)$$
for a special function h(x). In particular
$$h(x)=x^n$$
seems to work, but this structure is isomorphic to the normal multiplication and addition.