Algebra and abstract structures · Maths EE idea · Solid
Solving the cubic: Cardano's method and its paradox
A research question to start from
How does Cardano's formula solve the cubic, and why does it need complex numbers even when all three roots are real?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A historical and mathematical puzzle (the irreducible case) that forces a real argument about complex numbers.
Mathematics you would need
- Depressed cubics and substitution
- Complex numbers and cube roots
- The discriminant
- Trigonometric solution of cubics
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive Cardano's formula for the depressed cubic.
- Show the irreducible case leads to cube roots of complex numbers.
- Solve the same cubics by the trigonometric method and compare.
- Evaluate the two methods.
Scope and difficulty
Solid. Solid; history can frame the essay but must stay short.
Pitfalls
- Retelling the Tartaglia story at length.
- Formula manipulation without explanation.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Cardano formula casus irreducibilis; trigonometric solution cubic. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).
Make it your EE
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