What is the use/need of complex numbers in real life if they are imaginary?
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Here’s an answer that I can’t believe I haven’t seen yet, namely why complex numbers were invented in the first place. Complex numbers are the correct setting in which to solve polynomial equations.
Here’s an incredibly common problem. Say you have a function f(x) depending on a single variable x. Maybe f outputs the height of a plane at time x, or f outputs the profit of selling x units of a product, or whatever. Very often something we want to do is know what value of x (if any) achieves a specified value c for f(x). When does the plane get to a certain height, how many units do we need to sell to make a certain profit, etc.
That is, we want to solve f(x) = c for x. Or, if we let g(x) = f(x) – c, we’re trying to solve g(x) = 0. Here’s the trick. If we’re lucky, g(x) is a [polynomial](https://en.wikipedia.org/wiki/Polynomial). And even if it isn’t, we can in many circumstances meaningfully approximate g(x) by a polynomial, with different choices for a polynomial approximation available depending on how accurate we want the approximation to be (see [Taylor polynomials](https://en.wikipedia.org/wiki/Taylor%27s_theorem) for more on this).
So now let’s assume g(x) is a polynomial. Then we’re trying to solve a polynomial equation. Over the real numbers, this is not always possible. For instance x^2 + 1 = 0 has no solution over the real numbers (and solving this is what complex polynomials were invented to do). Here’s the punchline. **Any polynomial equation over the complex numbers has a solution**. You can also get for free that any polynomial equation has a “full set” of solutions over the complex numbers (This result is called the [fundamental theorem of algebra](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra)). This gives you at least two fruitful options. One is to then relate the solutions over the complex numbers back to the real numbers to try and better understand or potentially solve your original problem. Or you can see if your original problem is better framed over the complex numbers, and you get these new solutions for free.
This also works even better if you know a little calculus. There we might be trying to optimize our function f(x), when do we make the most profit, when does the plane get to its highest point, etc. This turns out to be tightly connected to solving the equation f'(x) = 0. Then if f(x) is a polynomial, it turns out f'(x) is also a polynomial, and we’re back to solving polynomial equations again.
The preceding discussion can also be generalized to polynomials in more than one variable, and doing so lands you at my area of study, algebraic geometry!
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