Imaginary unit
The imaginary unit is the number i defined by i² = −1. It extends the real numbers to complex numbers of the form a + bi.
Square root: A number whose square equals a specified number. The defining equation for i makes it a square root of −1.
Powers of i: The powers of i repeat in the cycle i, −1, −i, 1. Repeated multiplication by the imaginary unit follows a four-step cycle.
Gerolamo Cardano: An Italian mathematician whose 1545 book Ars Magna presented methods for solving cubic equations. His cubic formulas exposed calculations involving square roots of negative numbers.
Alternating current: Electric current that periodically reverses direction, commonly varying sinusoidally over time. Complex numbers simplify calculations for sinusoidal voltage and current.
Real number: A number on the continuous number line, including rational and irrational numbers. No real number squares to −1, unlike the imaginary unit.
Polynomial: An algebraic expression built from variables, coefficients, and nonnegative integer powers. The equation x² + 1 = 0 has no real root but has roots involving i.
Complex multiplication: Multiplication of complex numbers combines their magnitudes and angles, or their real and imaginary components. Multiplying by i turns each complex number through a quarter-turn.
Rafael Bombelli: An Italian mathematician who developed consistent rules for manipulating imaginary quantities in the sixteenth century. His treatment made expressions involving i usable in cubic equations.
Phasor: A complex number representing the amplitude and phase of a sinusoidal quantity. Its imaginary component encodes phase relationships in circuit analysis.
Hyperbolic number: A number of the form a + bj, where j² = 1 and j is not the real number 1. It extends the reals using a unit whose square is positive rather than negative.