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Pardon my ignorance and lack of thorough understanding, but I am missing a piece of the puzzle when it comes to complex numbers and can't seem to find an answer. I have been trying to understand complex numbers for about a year.

First some background. So Wikipedia states:

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and .... Complex numbers allow solutions to certain equations that have no solutions in real numbers.

I wanted to highlight how it says "real numbers". So it seems this assumes the foundation of having real numbers, as opposed to only relying on discrete integers or something like that. So far I haven't been able to find any explanation as to why it's strictly tied to real numbers and not something else.

But despite that, the main thing I have been unable to find so far is an example of a polynomial like the original one from the 1700's or so, which results in a cubic equation that has a negative radical. I was hoping this brief history of complex numbers would show an example polynomial, but as far as I can tell it results in a positive value.

The so-called casus irreducibilis is when the expression under the radical symbol in $w$ is negative.

Don't know why they didn't include a proper example, though I could be misreading it.

In order to try to get at why complex numbers were invented, beyond that history and beyond explanations like this, I would like to more deeply understand what they were trying to do with solving the polynomials that were real-number based. So they already assume real numbers (and I'm not sure why like I mentioned), which is a big thing to assume, and they have the other factoring and polynomial solving techniques to use which are based on other mathematical assumptions. I would like to go through their steps and understand what the meaning of solving real-number-based polynomials and running into the corner of having a negative radical. I don't see when that situation would arise and when it does, what it feels like.

Wondering if one could point me to an example or reference to help me in better learn this, or if it's straightforward even provide an example polynomial with some background and explanation.

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The standard example is contained in Bombelli's Algebra: $x^3-15x-4=0$. If you apply Cardano's formula to it you get that a root of this polynomial is $\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$.

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Just expanding on José Carlos Santos's answer a bit:

A solid reference which explains the matter very well is John Stillwell, Mathematics and Its History (second edition 2002), p.258f.: this even shows a page from Bombelli's manuscript (1572).

(A flabbier reference which explains it less well is my answer here: What's the point of obtaining the complex roots of a real polynom if I can't see them?.) :)

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