A pair of simultaneous equations that resist any real-number solution is used to motivate complex numbers: algebraic manipulation reduces the system to a quadratic with a negative discriminant, forcing a square root of a negative number. This historical impasse mirrors Gerolamo Cardano’s 16th-century encounters with such roots while solving cubics, which René Descartes derided as “imaginary.” Over time mathematicians accepted these quantities: Euler introduced the symbol i for √−1, Argand and Wessel interpreted them geometrically as points on a plane with a real axis and an orthogonal imaginary axis, Hamilton formalized them as ordered pairs (a,b) with algebraic addition and multiplication rules, and Gauss leveraged the system to prove the fundamental theorem of algebra. Multiplication acquires a clear geometric meaning: lengths multiply and angles add, so multiplying by i effects a 90° rotation.
Applying this framework to the original problem, −108 is written as 36×(−3), yielding solutions that include ±3i and thus two complex conjugates whose imaginary parts cancel when multiplied, producing the expected real product and sum. The broader point is that complex numbers extend the one-dimensional real line to a two-dimensional plane where addition is vector addition and multiplication combines rotation and scaling, with trigonometric representations giving further insight.
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