Why mathematicians accepted imaginary numbers: a maths EPQ idea
A title to start from
Why did mathematicians come to accept imaginary numbers, and what can they do that real numbers cannot?
Why it works as an EPQ
The cubic formula forced square roots of negative numbers into calculations with real answers, which gives a clear historical and mathematical argument.
Scope and difficulty
Solid. Solid. Further Maths students will meet complex numbers; Maths-only students should allow time to learn the basics first.
The maths
Builds on these A Level topics: Algebra and functions · Trigonometry.
You would learn:
- Complex numbers and the Argand diagram
- Cardano's formula for cubics
- De Moivre's theorem
One possible plan
- Solve a cubic with three real roots by Cardano's method and show where √−1 appears.
- Show how complex numbers turn rotations into multiplication.
- Use de Moivre's theorem to derive a trigonometric identity.
- Judge whether 'imaginary' numbers are less real than negative numbers or irrationals.
Pitfalls
- Making it a history essay with no worked mathematics.
- Errors with √(ab) = √a√b for negative numbers.
Where to start reading
- An Imaginary Tale (Paul Nahin)
- Search for: casus irreducibilis Bombelli cubic
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All pure maths ideas · all 93 ideas
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