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Hello everyone, Legend here, you know me ( ͡° ͜ʖ ͡°)

Well, I have been investigating quantum physics last few years and I even have some home-made experiments proving that newton's law's of physics do not work when working with extremely small objects like atoms/electrons, even molecules. So, in order to understand quantum mechanics i needed to understand imaginary numbers. Basically it's not so hard to learn how to operate with them but what i do not fully understand is why do they work?
I mean, they are not real. Squared number must be positive, if this statement is not always true for any x (y=x^2) then our math is a lie and we need new math. Imaginary numbers are simply just and indication and incentitive for us to start thinking in some other direction. So, where do they come from? An Ithalian mathematician, Rafael Bombelli invented them in 16. century. His predcessor was a gambler used math's and equations in order to be good at gambling (mostly statistics and posiblities), so he came up with a formula for calculating the roots (solutions) of qubic equations.
Unfortuonately, this formula didn't work for him, not always because he would sometimes get square roots of negative numbers while solving equations and he didn't know how to operate with them, so he always got stuck at that part. Problem was that he experimentaly knew that the equation he was working on had 3 real solutions (because it's cubic it has to have 3 roots - fundamental law of algebra, doubled solutions counted (when discriminant = 0)). So the equation definitely had 3 real solutions, but while solving it via formula he couldn't get those solutions because he fucking had negative numbers. Well, Bombelli decided to solve this via equation, so he used some algebra to write complex part with something, he didn't use "i" but something. So, he actually saw that if he would solve equation with that something, he would have - something and + something on the end which crosses each other and he is only left with real solutions, and that was fucking amazing. Later in quantum physics, even compressing stupid digital pictures like .jpeg does you need to use imaginary numbers. But why do they fucking work. They shouldn't. You can't just change maths as you want because you can't come up with a solution -_- I want an explanation on why do imaginary numbers work in REAL life.

Because they believed in themselves.

There were a few grammatical errors in there.
Honestly, you should break that down more.
I've learned walls of text are more likely to be glossed over.
Interesting research, but my answer is:
Sometimes, math can break everything you know.
You can only have conjectures.
So enjoy the 3.14 we call life.

If you can use it in theory for a real life scenario...there's no reason why it shouldn't work for said real life scenario, for real.

They work because they're not real, but there is no other solution to the problem, breaking our laws of math. They work because some lazy bastard decided to break the law(also I=square root -1)

Complex numbers are real fun to mess around with. It's basically a new number system to mess around with that works. It makes the "impossible" possible.
There are many uses for complex numbers inside mathematics and probably physics, but not many in the real world. For example, finding all 3 roots of a cubic which only intersects the x-axis once. You do this by deducting one of the complex roots, and using the complex conjugate of it to find the other.

WHY do they work though? No clue. Imaginary numbers can turn into "real" numbers easily though when the imaginary part is multiplied with another, as the square root of -1 squared is just -1. I haven't learned too much about complex numbers so far, but I have recently been studying it