I used that text as a little test for the AIs at DDG. Here’s from Claude 3 Haiku.

Here is the translation to English:

Complex Numbers and Polynomials

Something that hardly anyone knows is that complex numbers, in addition to their application in Complex AC Calculations (known to HTL students), also have a second, important use.

In mathematics, there is often the problem of finding a number x for a given function f(x) such that f(x) = 0 holds.

This is often the case for polynomials, as many important mathematical functions can be well approximated by polynomials.

The Fundamental Theorem of Algebra states that if we allow x to be a complex number, then there is always a solution to the problem such that f(x) = 0.

Restriction: but the polynomial must not consist of only a constant term, i.e., f(x) = 5 would be inadmissible. The x must appear at least once in the polynomial. Otherwise, there are really no restrictions :-)

PS: In the example given in the image (f(x) = x^2 + 2 = 0), a solution would be approximately x = √2 i ≈ 1.4 i.

Then I asked: “Please make it understandable to a general audience and phrase it like a pop-science author would.”

Here is the translation in a more accessible, pop-science style:

The Hidden Power of Complex Numbers

Most people think complex numbers are just some obscure math concept, useful only for electrical engineers. But these mysterious numbers actually have a hidden superpower - they can solve all sorts of important math problems.

You see, in math, we often want to find a number x that makes a function f(x) equal to zero. This comes up a lot when working with polynomials - those nice, smooth equations made up of powers of x.

The problem is, sometimes there just doesn’t seem to be any real number x that does the trick. That’s where complex numbers swoop in to save the day!

The Fundamental Theorem of Algebra proves that as long as we allow x to be a complex number (a number with both a real and imaginary part), then there will always be a solution that makes f(x) equal to zero. No exceptions!

So complex numbers unlock the ability to solve all kinds of polynomial equations that would be impossible with just regular old real numbers. It’s like they have a secret mathematical superpower.

Of course, there is one small catch - the polynomial can’t be a simple constant, like f(x) = 5. It has to actually contain the variable x somewhere. But other than that, complex numbers can tackle any polynomial problem you throw at them.

So the next time you see an imaginary number like √(-1), remember - it’s not just some weird math trick. It’s a powerful tool for solving important equations. Complex numbers may seem strange, but they’re secretly the heroes of the math world!