The irreducible case and imaginary numbers

Observation — The "irreducible case" and imaginary numbers

When $D<0$ the formula asks us to take the square root of a negative number: the mathematicians of the 1500s were disgusted by it. But the beauty is that, even in this case, the three real roots exist (because $\Delta_3>0$): the formula expresses them by passing through imaginary quantities which, on adding up, nevertheless give real numbers. It is the first historical example in which “to reach the real one must pass through the imaginary”. Cardano himself reckoned with $\sqrt{-15}$ in the Ars Magna, declaring them “as subtle as they are useless”. Bombelli (1572) made a consistent computation of them: we shall return to this in the chapter Complex numbers (Boyer, ch. 13).

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Topics: Second-degree equations
Concepts: Discriminant of the cubic · Third-degree equation · Imaginary numbers
People: Rafael Bombelli · Gerolamo Cardano