Why were complex numbers gradually accepted as legitimate solutions to equations?

The acceptance of complex numbers as legitimate mathematical entities was a gradual process driven by their utility and, eventually, their geometric interpretation.

Initially, mathematicians were reluctant to accept numbers involving the square root of negative numbers, as they lacked a direct connection to the physical world. While 16th-century mathematicians like Girolamo Cardano and Rafael Bombelli encountered these values when solving cubic equations, they often viewed them as "useless" or conceptually uncomfortable.12

Acceptance grew as mathematicians realized that algebra and analysis could be extended to the complex domain in a natural and powerful way. By the 18th century, they were established as sensible concepts, and their utility in solving problems became so significant that philosophical concerns were largely set aside.1

Final doubts were dispelled in the 19th century through geometric and formal interpretations:
* Geometric Interpretation: Caspar Wessel and Jean-Robert Argand demonstrated that complex numbers could be represented as directed segments on a plane, a concept popularized by Carl Friedrich Gauss.34
* Formal Interpretation: William Rowan Hamilton defined complex numbers as pairs of real numbers, providing a rigorous arithmetic foundation that did not require an essentialist definition.3