You cannot take the square root of a negative number. It is undefined and mathematically impossible.

When Rafael Bombelli published his Algebra in Bologna in 1572, he included rules for calculating with quantities he called "plus of minus" and "minus of minus," the square roots of negative numbers. He did not call them impossible or mark them as forbidden territory. He gave them consistent arithmetic rules, demonstrated how to manipulate them, and showed that calculations passing through them could yield real, verifiable answers. Most of his contemporaries found the notation baffling and the results unconvincing.

René Descartes, in his 1637 La Géométrie, introduced the term "imaginary" for roots of negative numbers, and the name remained. Descartes was not using the word neutrally. He was marking these quantities off from the "real" numbers, indicating that they had no genuine existence, that they might serve calculation but did not represent anything that could be said to be real. The label shaped how these quantities were taught and received for the next three centuries.

The formal establishment of complex numbers as mathematically legitimate came through Leonhard Euler and Carl Friedrich Gauss. Euler introduced the notation i for the square root of negative one in 1777 and used it without qualification throughout his work. Gauss proved the fundamental theorem of algebra in his 1799 doctoral dissertation, showing that every polynomial equation has as many roots, real or complex, as its degree. The proof required complex numbers at its foundation. Gauss also demonstrated that complex numbers could be represented geometrically as points on a plane, with real numbers along one axis and imaginary multiples along the perpendicular. A number like 3 + 4i was not a phantom but a position in two-dimensional space, as concrete as any coordinate on a map, and it could be added, multiplied, and transformed according to rules that corresponded to rotations and scalings of the plane.

The 20th century confirmed the physical necessity of complex numbers. James Clerk Maxwell's formulation of electromagnetic theory in the 1860s used complex quantities to represent sinusoidal waves, and electrical engineers adopted complex impedance as a standard computational tool by the early 1900s. When Werner Heisenberg and Erwin Schrödinger developed quantum mechanics in 1925 and 1926, complex numbers were not a notational convenience. The wave function describing a quantum particle is irreducibly complex-valued. Schrödinger's equation includes i explicitly in its fundamental statement. A 2021 paper in Nature, reporting results from two independent experimental teams, provided direct evidence that quantum mechanics cannot be reformulated using only real numbers: the complex structure is physically necessary, not a shorthand that could be replaced without loss.

Mathematics education generally lagged behind the physics by decades. Secondary school algebra through the late 20th century reached complex numbers only as an addendum following the quadratic formula: numbers of the form a + bi, mentioned briefly, rarely placed in any application context, almost never connected to the engineering and physics where they had been indispensable for a century. Teachers introduced i as the square root of negative one and moved on. The label "imaginary" persisted on textbook pages alongside content that had long made the label false, with no indication that Maxwell's equations, circuit analysis, Fourier transforms, signal processing, and the fundamental equations of quantum mechanics all relied on the same quantities Descartes had marked as non-real.