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I have read an article who claims that complex numbers does not exist and any physics based on complex numbers are bogus science!

I am surprised that in 21st century there are still such idiots dare to write like that. Anyway, his key argument is the following:

Consider the system of two equations $$\begin{cases}(x-1)^2+(y-1)^2=1 \\ x+y=0\end{cases}$$ the first of which is a circle centered at (1,1) of radius 1 and the second of which is a straight line passing (0,0) and (1,-1). The system has complex roots, but the geometric objects have no intersection. Therefore, the complex roots are bogus.

Question: What is wrong with his argument?

Answer: Only when we consider $x$ and $y$ as real numbers, the two equations correspond to the two geometric objects.

There are three ways to understand complex numbers.

The simplest way is that $a+\sqrt{-1}b$ is the point $(a,b)$ on a plane. In this case $(x-1)^2+(y-1)^2=0$ and $x+y=0$ each describes a relation between two points and therefore both are equations for a two-dimensional figure in a four-dimensional space. We cannot visualize four-dimensional space!

The second way is to consider $a+\sqrt{-1}b$ as the polynomial $ax+b$ in the space of polynomials in such a way that we identify the polynomials $x^2$ and $-1$. The two equations are therefore relations among two polynomials.

The third way is to consider $a+\sqrt{-1}b$ as the matrix $\begin{pmatrix}a & b \\ -b & a\end{pmatrix}$. The sum and product of two complex numbers are the sum and prodcut of two matrices.

Why does the author write such article? Because Stephen Hawking talks about complex numbers in physics and he also argues that we do not need God in science!