The imaginary unit solves x² = −1 when reals cannot
Denoted i (or j in electrical engineering), the imaginary unit is a constant satisfying x² = −1. Pairing it with the reals under + and × builds the complex numbers a + bi. Both i and −i square to −1; every nonzero complex number has two square roots.
Early-modern mathematicians counted as numbers only quantities that measurement or simple arithmetic could produce, and they eyed even negatives with suspicion, so a root of a negative looked meaningless. Descartes is usually credited with the label imaginary, Isaac Newton was using it by 1670, and Leonhard Euler introduced the letter i. Despite the name, the construction is perfectly sound: ordinary algebra carries over if you treat i as an unknown and swap every i squared for minus 1.
Doing so makes the powers cycle. The cube of i is minus i, the fourth power is 1, the fifth returns to i, and the pattern repeats through 1, i, minus 1 and minus i forever, with i to the zero equal to 1 as for any nonzero number. In rectangular form i is 0 plus 1i; in polar form it has magnitude 1 and an angle of a right angle, written e to the power pi i over 2. On the complex plane it sits one unit up the vertical axis, at right angles to 1, so multiplying by a unit-size complex number rotates points around the origin while adding one slides them.
Curiously, nothing distinguishes i from minus i except the label. The two roots are each other's negatives and reciprocals, and neither is positive or negative in the real sense. Formally, the complex field has two automorphisms fixing every real number, the identity and complex conjugation, so it is unique only up to a non-unique isomorphism.
Other disguises work too. The 2-by-2 matrix with 0 and minus 1 in its top row and 1 and 0 below squares to minus the identity and can stand in for i; any traceless real 2-by-2 matrix with determinant one serves equally. Algebraists obtain the complex numbers as real polynomials modulo x squared plus 1, and geometric algebra uses a unit bivector, which likewise squares to minus 1. Careless use of radicals causes trouble, as the bogus chain concluding that minus 1 equals 1 demonstrates.
Source: Imaginary unit
