I think I have some good understanding of complex numbers. They are represented as a+ib where a,b are reals, and i² = -1. More precisely, multiplying a complex number by i causes a rotation by 90° in complex space, just like multiplying by a negative number causes a rotation by 180°.
Now there don't seem to be many use cases for complex numbers. Their defendants often mention electrical engineering, Fourier transform, physical signals, etc…
My question is, what makes complex numbers uniquely useful? For those specific applications, why not use vectors with 2 reals and do the rotations with cos and sin instead of i?
Since my question does not seem to be understood very well, let me provide more details.
For example, one thing that is original in complex numbers is multiplication. To multiply 2 numbers (a + ib) and (c + id) you actually end up with ac + ibc + iad + i²bd = (ac – bd) + i(ad + bc). However you could easily define this operation on R² as (a,b) x (c,d) = (ac – bd, ad + bc) and therefore (a,b)² = (a² – b², 2ab). From there, it is easy to redefine complex exponentiation, Euler's formula, etc… in term of basic 2D algebra using special, complex operations, just as special as dot or cross product.
If I go on, with all of complex operations, all of the results of complex numbers, what prevents me from getting rid of i and understanding everything in term of linear algebra?
Best Answer
After your edit, I can better understand your question.
Yes, if you define the multiplication as $$(a,b) \times (c,d) = (ac - bd, ad + bc)$$ $\big($and the addition as $$(a,b)+ (c,d) = (a+c, b+d)\big)$$
then you have just set up the algebraic structure of the complex field.
Since $$(0,1)\times(0,1)=(-1,0)$$
you can introduce the notation $$i=(0,1)$$
and you can simplify everything. From your point of view, you are right. But it took a few hundred years to see everything so clearly. You are standing on giants' shoulders.