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I've been wondering, I've learned that equations may have solutions or roots in complex numbers. However,

I've not entirely understood, how the transition between real and complex solutions occurs. That an equation has solutions in either real or complex reads that the equation "extends" to real as well as complex, but I don't understand how the transition between the two occurs. That is, how is it possible that an equation has sols in both domains?

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Let's start with a really simple example.

Think about trying to solve the quadratic equation $$ x^2 - 2x + 2 = 0 . $$ You try the quadratic formula, and discover that the discriminant $$ b^2 - 4ac = 4 - 4 \times 1 \times 2 = -4 $$ is negative. That means you can't take its square root, so the equation has no roots.

That's the seventh grade algebra answer. But if later you learn that there are things called complex numbers, and that the real numbers are a subset of the complex numbers, then you can think of the coefficients of that quadratic equation as complex numbers. You check that the quadratic formula still works for the complex numbers, and then you conclude that your equation has two roots, $$ x = 1 + i \text{ and } x = 1 -i . $$

(You don't really have to go back to check the proof of the quadratic formula. You can just verify that these two values really do solve the equation.)

The idea of thinking about one number system as part of another isn't really new to you. The equation $$ x + 2 = 0 $$ has no solution in the positive integers, but if you invent the negative integers and then work in the set of all integers the solution is $x = -2$.

Similarly, the equation $$ 5x = 3 $$ has no solution in the integers, but does have the solution $x=3/5$ if you enlarge the number system to include fractions (rational numbers).

You need another extension, to the real numbers, in order to find a number that satisfies $x^2 - 2 = 0$.

Now lets get back to the complex numbers. They were invented to extend the real numbers in order to solve one particular equation: $$ x^2 + 1 = 0 . $$

Then we discovered something wonderful. Not only does the quadratic formula work in the complex numbers to solve all quadratic equations, it turns out that in the complex numbers every polynomial (with real or complex coefficients) can be completely solved: it will have as many solutions (roots) as its degree, if you count multiple roots properly. That theorem is called the Fundamental Theorem of Algebra (FTA).

(The FTA is not quite as useful as it sounds. Although it guarantees that fifth degree polynomials have five roots it does not provide a formula like the quadratic formula to find them. Abel and Galois proved there is no such formula.)

The extension of analysis from the real to the complex domain goes much deeper than the study of polynomials. The real functions $e^x$ and $\sin$ and $\cos$ and many others work for complex values too.

Finally, you might ask whether there are number systems that contain the complex numbers. The answer is "sort of". The quaternions come next.

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