Dear Tekle Gemechu

The equation x+2 = 3 has a solution in Z, x = 1.

The equation 3x = 1 has no solutions in Z. We extend the domain into Q, the set of rational numbers. we obtain x = 1/3 ∈ Q.

Also, x2 = 3 has no solutions in Q. We extend the domain into R, the set of real numbers, we obtain x = ±√3 ∈ R.

Similarly, the equation x2 = - 2 has no solutions in R. We extend the domain into C, the complex field of numbers set, we obtain the solutions x = ±√-2 = ± i√2 ∈ C.

One needs to study the properties of this new field of complex numbers.

It is available in any introductory course about complex numbers.

Each number is an ordered pair z = (a,b).

a is called the real part and b is called the imaginary part.

For simplicity, we write z= a+ ib where i = (0,1) and sometimes represented

by i = (0,1) = √-1.

you can see that z = (0, √2) = i√2 satisfies the equation x2 = - 2.

In fact, using the multiplication of complex pairs ( not the usual multiplication in R), (a,b)(c, d) = (ac - bd, ad + bc), we have (0, √2)2 = (0, √2).(0, √2)

= ( 0x0 - √2x√2, 0x√2 - √2x0) = (-2, 0) = -2 is real without imaginary part.

For simplicity, we use the language of ( i√2)2 = 2(i2) = -2.

In brief, you can imagine a complex number (a, b) = a + ib as a point in the plane

with real part a and imaginary part b.

Regards

Real roots are special complex roots, those with imaginary part 0. The mention of "complex" only indicates that one doesn't have to leave C to find the zeros.

https://math.stackexchange.com/questions/552132/is-every-real-root-also-a-complex-root-complex-zeros-and-fundamental-theorem-o

I think the question is about the meaning of complex zeros in real, or even integer, polynomials that represent/model real-world systems; it is not about the existence of complex zeros in such polynomials.

In physics, it is known that if a polynomial energy equation has complex zeros, this may signify the presence of energy sources or sinks in the examined system; in other words the system either loses or gains – or both – energy due to some interaction with its environment.

Further, there may be the case that the system energy loss and gain are balanced. In quantum mechanics, such systems are described by the so-called PT-symmetric Hamiltonians that are invariant under the combined action of parity (space reflection) and time-reversal.

Dear all

Issam Kaddoura

Manal Kadi Kanaan

Spiros Konstantogianniss

thanks!