Imaginary number

An imaginary number is a complex number with zero real part, expressible as a real multiple of the imaginary unit i, where i² = −1.

Imaginary unit: The number i defined by i² = −1, used to represent imaginary numbers. Multiplying i by a real number produces an imaginary number.
Complex multiplication: Multiplication of numbers a + bi using i² = −1 and ordinary distributive arithmetic. It explains products such as i times i, which equal −1.
Alternating current: Electric current that periodically reverses direction, commonly varying sinusoidally. Complex numbers encode phase and amplitude in sinusoidal circuit analysis.
Pure imaginary number: An imaginary number of the form bi, where b is a nonzero real number. This narrower usage excludes zero, which some definitions include among imaginary numbers.
Real number: A number on the continuous number line, including rational and irrational numbers. Imaginary numbers use real numbers as their coefficients.
Complex conjugate: The complex number obtained by changing the sign of a number’s imaginary part. Conjugation reverses an imaginary number across the real axis.
Phasor: A complex-number representation of a sinusoidal quantity’s amplitude and phase. Its imaginary component helps express phase shifts compactly.
Quaternion: A number system extending complex numbers with three imaginary units and noncommutative multiplication. Quaternions generalize the use of imaginary units beyond a single i.
Complex number: A number expressible as a + bi, where a and b are real numbers and i² = −1. Imaginary numbers are precisely the complex numbers whose real component is zero.
Complex modulus: The nonnegative distance of a complex number from the origin in the complex plane. An imaginary number’s modulus is the absolute value of its real coefficient.

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