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Every JEE aspirant learns that i² = −1 and then spends weeks on complex numbers. Few ask why anyone accepted such strange numbers in the first place. The answer is not that mathematicians liked abstraction. It is that a working formula for cubic equations kept producing square roots of negative numbers on the way to perfectly real answers. The person who first wrote down how to calculate with them was an engineer named Rafael Bombelli. This post tells his story, works the famous cubic, and covers the mistakes students still make.
Who was Rafael Bombelli?
The MacTutor biography from the University of St Andrews says he was born in January 1526 in Bologna, the eldest of six children of a wool merchant. He had no university education. He trained under an engineer-architect and made his name draining marshes, especially in the Val di Chiana, a project that gave him a high reputation as a hydraulic engineer. He also consulted on attempts to drain the Pontine Marshes near Rome.
While waiting for marsh work to restart around 1557, he began an algebra book. He felt the subject's bitter disputes came from the lack of a clear, self-contained text that people without advanced training could read.
What did he learn from Diophantus?
On a visit to Rome, a professor named Antonio Maria Pazzi showed him a manuscript of Diophantus's Arithmetica. The two began translating it and finished five books. MacTutor reports that 143 of the 272 problems in Book III of Bombelli's Algebra come from Diophantus, and that Bombelli openly credited him. You can try some of those problems in our post on Diophantus and the Arithmetica.
The first three books of L'Algebra were published in 1572, and Bombelli died soon after.
Why did square roots of negatives appear at all?
The Cardano-Tartaglia formula solves cubics of the form x³ = px + q. As MacTutor's history of quadratic, cubic and quartic equations explains, when Cardano applied it to x³ = 15x + 4 he got an expression involving √−121, even though x = 4 is obviously a solution. In modern form, the formula gives
x = ∛(q/2 + √(q²/4 − p³/27)) + ∛(q/2 − √(q²/4 − p³/27)).
With p = 15 and q = 4: q²/4 = 4 and p³/27 = 3375/27 = 125. So the square root is √(4 − 125) = √−121, and
x = ∛(2 + √−121) + ∛(2 − √−121).
Cardano had called this kind of calculation useless. MacTutor notes that Bombelli studied this "irreducible case" in detail.
How do Bombelli's rules rescue the formula? A worked example
Bombelli wrote √−n as "più di meno" (plus of minus) and −√−n as "meno di meno" (minus of minus). In L'Algebra, which you can read in a scanned Bologna printing on archive.org, he lists rules such as "più di meno via più di meno fa meno": plus of minus times plus of minus makes minus. In modern terms, i × i = −1.
Now write √−121 = 11i. We need the cube root of 2 + 11i. Try 2 + i:
(2 + i)³ = 8 + 3(4)(i) + 3(2)(i²) + i³ = 8 + 12i − 6 − i = 2 + 11i.
Similarly (2 − i)³ = 2 − 11i. So
x = (2 + i) + (2 − i) = 4.
The imaginary parts cancel and the formula gives the real root. Check: 4³ = 64 and 15 × 4 + 4 = 64. Dividing x³ − 15x − 4 by (x − 4) leaves x² + 4x + 1, whose roots are −2 ± √3. All three roots are real, yet the formula had to pass through complex numbers to reach them.
How did Bombelli feel about his own idea?
He admitted his doubts in the book. In the scanned text he says the idea will seem to many more sophistic than real, that he held that opinion himself, until he found a demonstration with lines. MacTutor gives a fuller English translation and notes that his Algebra even contains a geometric argument for why minus times minus makes plus.
What mistakes do students make with complex numbers?
Multiplying square roots of negatives directly. √−4 × √−9 is not √36 = 6. Write 2i × 3i = 6i² = −6 first.
Forgetting i³ = −i. In the expansion of (2 + i)³ this sign is easy to lose. Write i², i³ and i⁴ as −1, −i and 1 before simplifying.
Stopping at one root. After finding x = 4, factor and solve the quadratic for the other two.
For JEE practice, our guide to complex numbers for JEE covers the modern methods.
What can a student take from Bombelli?
He was a practical engineer who trusted a calculation once he could check it, and who wrote for readers who were not experts. Both habits help: test strange results by substituting back, and explain your working so someone else could follow it. When a step in a complex number problem stops making sense, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Try it: expand (1 + 2i)³ term by term and check that you get −11 − 2i.
