I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root.
I don't see why the reason is "since $y^3 − py − q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$)."
Secondly, I don't see how this fact led to the need to introduce complex numbers. It seems like the book is hinting at this. But maybe I am wrong. I don't see why it was necessary to introduce complex numbers due to cubic equations if they were shown to have at least one real solution. Then the real solution could be enough, right?
Thank you for your explanation!
14.2 Quadratic Equations
The usual way to introduce complex numbers in a mathematics course is to point out that they are needed to solve certain quadratic equations, such as the equation $x^2 + 1 = 0$. However, this did not happen when quadratic equations first appeared, since at that time there was no need for all quadratic equations to have solutions. Many quadratic equations are implicit in Greek geometry, as one would expect when circles, parabolas, and the like are being investigated, but one does not demand that every geometric problem have a solution. If one asks whether a particular circle and line intersect, say, then the answer can be yes or no. If yes, the quadratic equation for the intersection has a solution; if no, it has no solution. An "imaginary solution" is uncalled for in this context.
14.3 Cubic Equations
The del Ferro-Tartaglia-Cardano solution of the cubic equation
$$
y^3 = py + q
$$
is
$$
y = \root{3}\of{ \frac{q}{2} + \sqrt{ \left(\frac{q}{2}\right)^2 – \left(\frac{p}{3}\right)^3 } } +
\root{3}\of{ \frac{q}{2} – \sqrt{ \left(\frac{q}{2}\right)^2 – \left(\frac{p}{3}\right)^3 } }
$$
as we saw in Section 6.5. The formula involves complex numbers when $(q/2)^2 – (p/3)^3 < 0$. However, it is not possible to dismiss this as a case with no solution, because a cubic always has at least one real root (since $y^3 – py – q$ is positive for sufficiently large positive $y$ and negative for sufficiently large negative $y$). Thus the Cardano formula raises the problem of reconciling a real value, found by inspection, say, with an expression of the form
$$
\root{3}\of{ a + b\sqrt{ -1 } } +
\root{3}\of{ a – b\sqrt{ -1 } }
$$
The quote is from the Mathematics and its history book pp 276-277.
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Even though the cubic has real coefficients the formula for the roots requires complex numbers.
That seemingly contradictory fact is probably what delayed discovering the formula. The book on the history of mathematics you quote from should discuss that.
Read about the casus irreducibilis on wikipedia.