[marginalia_nu]

They are very much equivalent. You can express complex numbers with a 2x2 matrix of real numbers.

  R = 
     1 0 
     0 1
 
  I = 
     0 1
    -1 0

I and R form a basis that spans something that behaves like a complex plane.

You have the expected identities

R^2 = R

IR = I

I^2 = -R

I^3 = -I

I^4 = R

Transposition is complex conjugate. You can put them into exponentials, e^Ix = cos(x) + I sin(x); everything works as you would expect.

[sdenton4]

It's really inelegant though! To multiply two complex numbers, we need to embed one in a 2x2 matrix while keeping the other as a 2d vector, and then do the matrix vector multiplication. And furthermore, it's not like every 2x2 matrix corresponds to a complex number...

[adrian_b]

That is not correct.

When complex numbers are represented in the matrix form mentioned by the other poster above, the multiplication of the complex numbers corresponds to the multiplication of the matrices, not to multiplications of vectors by matrices.

The matrices of this form, having just 2 parameters instead of the 4 parameters of a 2x2 matrix, are obviously just a subset of the general 2x2 matrices.

However this subset is closed to matrix addition and multiplication, so it is a field isomorphic to the complex numbers.

This representation of the complex numbers is useful to understand that the multiplication rule for complex numbers is not some random arbitrary rule, but it is the same as the rule for matrix multiplication, which is also not a made-up rule, but it results from function composition, when you compose the functions of a vector expressed by the multiplication of a matrix with the vector.

Actually the construction of the complex numbers goes like this, if you consider the transformations of the 2-dimensional vectors that correspond to a scaling and a rotation of the vector (which are a subset of the linear transformations expressed by multiplication with a non-singular 2x2 matrix of general form), and you compute the matrices that perform such transformations, you arrive at 2x2 matrices with 2 parameters of the form shown by the poster above.

This set of transformations happens to have the algebraic structure of a commutative field. Because the matrices are defined by only 2 parameters, you can simplify the operations by keeping just the 2 parameters and using rules for operations expressed directly with them.

Thus you obtain the standard rules for operations with complex numbers.