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Imaginary (Complex) Number Calculator

Add, subtract, multiply, or divide two complex numbers (a + bi) and display results in rectangular form, modulus, or argument instantly.

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Inputs

Real part of z₁ (a)

Imaginary part of z₁ (b)

Real part of z₂ (c)

Imaginary part of z₂ (d)

Operation

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What Is a Complex Number?

A complex number takes the rectangular form z = a + bi, where a is the real part and b is the imaginary part. The symbol i denotes the imaginary unit, defined as i = √(−1), so i² = −1. Every real number is a special case of a complex number with b = 0, making the complex number system a complete superset of the reals. The complex plane maps a along the horizontal axis and b along the vertical axis, giving every complex number a unique geometric position.

Core Formulas

Modulus (Absolute Value)

The modulus of z = a + bi measures its distance from the origin in the complex plane: |z| = √(a² + b²). This follows directly from the Pythagorean theorem. For z = 3 + 4i, |z| = √(9 + 16) = √25 = 5. For z = 5 + 12i, |z| = √(25 + 144) = √169 = 13. The modulus is always a non-negative real number and equals the amplitude of the corresponding phasor in AC circuit analysis.

Argument (Phase Angle)

The argument θ = atan2(b, a) gives the counterclockwise angle from the positive real axis to the vector representing z, expressed in radians. For z = 1 + i, θ = atan2(1, 1) = π/4 ≈ 0.7854 rad (45°). For z = −1 + 0i, θ = π radians (180°). For z = 0 + 1i, θ = π/2 ≈ 1.5708 rad (90°). The argument is undefined when z = 0.

Arithmetic Operations

Addition and Subtraction

Addition and subtraction combine real and imaginary parts independently. Given z₁ = a + bi and z₂ = c + di:

Addition: z₁ + z₂ = (a + c) + (b + d)i
Subtraction: z₁ − z₂ = (a − c) + (b − d)i

Example: (3 + 2i) + (1 + 5i) = 4 + 7i. Example: (3 + 2i) − (1 + 5i) = 2 − 3i. Addition in rectangular form is straightforward and is the preferred method when summing impedances in AC circuits.

Multiplication

Apply the distributive (FOIL) property and substitute i² = −1: z₁ · z₂ = (ac − bd) + (ad + bc)i. Example: (2 + 3i)(4 − i) = (2·4 − 3·(−1)) + (2·(−1) + 3·4)i = (8 + 3) + (−2 + 12)i = 11 + 10i. In polar form, multiplication becomes elegant: |z₁ · z₂| = |z₁| · |z₂| and arg(z₁ · z₂) = arg(z₁) + arg(z₂), making polar representation the preferred choice for multiplying phasors.

Division

Multiply numerator and denominator by the complex conjugate z₂* = c − di to eliminate i from the denominator: z₁ / z₂ = (ac + bd)/(c² + d²) + (bc − ad)/(c² + d²) · i. Example: (3 + 2i)/(1 + 5i) — multiply by (1 − 5i)/(1 − 5i) to obtain [(3 + 10) + (2 − 15)i]/(1 + 25) = (13 − 13i)/26 = 0.5 − 0.5i. Division requires z₂ ≠ 0 (c and d cannot both equal zero).

Polar and Exponential Forms

Any complex number converts to polar form as z = |z| · (cos θ + i sin θ), or equivalently to Euler's exponential form z = |z| · e^(iθ). According to the Complex Number Primer by Paul Dawkins at Lamar University, the polar form is especially powerful for raising complex numbers to integer powers and extracting nth roots via De Moivre's theorem. For z = 2 + 2i: |z| = √8 ≈ 2.828 and θ = π/4, so z = 2√2 · e^(iπ/4).

Real-World Applications

Complex numbers model physical phenomena across science and engineering:

Electrical engineering: Impedance Z = R + jX (engineers use j instead of i to avoid confusion with current) combines resistance R and reactance X. A 50 Ω resistor in series with a 75 Ω inductive reactance gives Z = 50 + 75j Ω, with |Z| ≈ 90.1 Ω.
Signal processing: The Fast Fourier Transform (FFT) decomposes time-domain signals into complex frequency components, enabling MP3 audio compression and JPEG image encoding.
Quantum mechanics: Wave functions ψ are complex-valued; the squared modulus |ψ|² gives the probability density of finding a particle at a given location.
Control systems: Pole-zero placement in the complex s-plane determines whether a feedback control system is stable, oscillatory, or divergent.

How to Use This Imaginary Number Calculator

Enter the real parts a and c and the imaginary parts b and d for z₁ = a + bi and z₂ = c + di. Select the arithmetic operation (+, −, ×, ÷) and choose the desired output format: full rectangular result, modulus |z|, or argument arg(z) in radians. As the TCC Engineering reference on complex number representation confirms, engineers routinely switch between rectangular and polar form — rectangular for addition and subtraction, polar for multiplication and division. This imaginary number calculator handles both representations instantly, with no manual conversion required.

Reference

Frequently asked questions

What is an imaginary number and how does it differ from a complex number?

An imaginary number is a multiple of i = √(−1), such as 3i or −7.5i — it has no real part. A complex number a + bi includes both a real part a and an imaginary part b. Every pure imaginary number is a complex number with a = 0, but not every complex number is purely imaginary. For example, 4 + 3i is complex but not purely imaginary, while 0 + 5i = 5i qualifies as purely imaginary.

How do you multiply two complex numbers step by step?

Multiply complex numbers using the FOIL method and the identity i² = −1. For z₁ = a + bi and z₂ = c + di, the formula is z₁ · z₂ = (ac − bd) + (ad + bc)i. Example: (2 + 3i)(4 − i) = (2·4 − 3·(−1)) + (2·(−1) + 3·4)i = (8 + 3) + (−2 + 12)i = 11 + 10i. Alternatively, in polar form, multiply the moduli and add the arguments to obtain the same result.

What does the modulus of a complex number represent?

The modulus |z| of a complex number z = a + bi is its Euclidean distance from the origin in the complex plane, calculated as |z| = √(a² + b²). It is always a non-negative real number. For z = 5 + 12i, |z| = √(25 + 144) = √169 = 13. In electrical engineering, the modulus equals the peak amplitude of the corresponding AC phasor, and in quantum mechanics it relates directly to probability density when squared.

How do you divide complex numbers using the conjugate method?

Divide complex numbers by multiplying both numerator and denominator by the conjugate of the denominator. For z₁/z₂ where z₂ = c + di, multiply by (c − di)/(c − di) to get z₁/z₂ = (ac + bd)/(c² + d²) + (bc − ad)/(c² + d²) · i. Example: (6 + 2i)/(1 + 3i) = [(6 + 6) + (2 − 18)i]/(1 + 9) = (12 − 16i)/10 = 1.2 − 1.6i. The denominator z₂ must not equal zero.

What is the argument of a complex number and in what units is it expressed?

The argument arg(z) is the angle θ = atan2(b, a) that the complex number z = a + bi makes with the positive real axis, measured counterclockwise in radians. One full rotation equals 2π radians (360°). For z = 0 + 1i, arg(z) = π/2 ≈ 1.5708 rad (90°). For z = −1 + 0i, arg(z) = π ≈ 3.1416 rad (180°). The argument is undefined when z = 0, since no direction can be assigned to the origin.

Where are complex numbers used in real engineering and science?

Complex numbers appear throughout electrical engineering, physics, and computing. AC circuit analysis uses complex impedance Z = R + jX to combine resistance and reactance into a single quantity. Signal processing relies on complex Fourier transforms to decompose audio and video — the FFT algorithm underpins MP3 and JPEG compression standards. In quantum mechanics, the Schrödinger equation governs complex-valued wave functions. Control engineers place poles and zeros in the complex s-plane to design stable systems such as autopilots and industrial motor controllers.