Complex numbers. Also known as imaginary numbers. The imaginary number i is the solution to √-1. And it is really used in quantum mechanics and I think general relativity as well.

A complex number is just two real numbers stitched together. It’s used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.

In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn’t, then no one would use complex numbers, because computers can’t process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.

I don’t think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn’t well-defined, but (3+4i) * (5+6i) is.

I am troubling imagining in your head how you think you made an actual argument here. It seems you’re claiming that a vector of real numbers (a,b) where have the same symbol between them such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di) or complex numbers are not reducible to real numbers.

You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be

void cpx_mul(cpx_t* r, cpx_t* a, cpx_t* b) { //FOIL float first = a->real * b->real; //real float outer = a->real * b->imaginary; //imaginary float inner = a->imaginary * b->real; //imaginary float last = -(a->imaginary * b->imaginary); //real r->real = first + last; r->imaginary = outer + inner; }

and now it is mathematically well-defined and gives identical results.

Right, but you need to specify that additional definition. Imaginary numbers are useful because they come pre-loaded with all those additional definitions about how to handle operations that use them.

Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no “hostile confusion” or “flaw,” there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren’t just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can’t give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all “pre-given.” Makes zero sense. You have not pointed out a “flaw,” you just ramble and declare victory, throwing personal attacks calling me “confused” like this is some sort of competition or something when you have not even made a single coherent point.