# Complex Number & Phasor Calculator Add, subtract, multiply, divide, and find roots of complex numbers, with automatic rectangular-to-polar phasor conversion, using this free calculator. --- - **Canonical URL:** https://dothecalculation.com/calculators/complex-number-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Complex Number & Phasor Calculator Add, subtract, multiply, divide, and find the roots of complex numbers entered in rectangular form, with automatic polar (magnitude/angle) conversion. - Rectangular (a+bi) input for both operands - Automatic magnitude and phase angle output - De Moivre's complex roots finder ## Introduction to Complex Numbers and Vector Fields A complex number is a number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying the equation \(i^2 = -1\). Complex numbers extend the one-dimensional real number line into a two-dimensional complex plane (called the Argand diagram), where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. This two-dimensional representation makes complex numbers exceptionally useful in geometry, physics, and engineering, particularly for modeling rotating vectors and alternating current (AC) circuits. In electrical engineering, complex numbers are used as phasors to represent sinusoids, allowing circuit analysis involving resistors, capacitors, and inductors to be solved using algebraic equations rather than differential equations. The magnitude of the complex number represents the amplitude of the signal, while the phase angle represents the time shift or phase. This calculator performs arithmetic operations (addition, subtraction, multiplication, and division) on two complex numbers entered in rectangular form, and separately finds the \(n\)-th roots of the first number using De Moivre's Theorem. Every result is automatically converted to polar form (magnitude and angle) alongside the rectangular answer, with the full step-by-step working shown. ## How to Use This Calculator Enter \(Z_1 = a + bi\) and, for arithmetic operations, \(Z_2 = c + di\) in rectangular form. Choose add, subtract, multiply, divide, or n-th roots (which only uses \(Z_1\) and a root degree from 2 to 6). The calculator returns the rectangular result, its magnitude and phase angle, an Argand-plane plot, and the step-by-step working. ## Rectangular and Polar Coordinate Conversions A complex number \(Z\) can be represented in two primary formats: 1. **Rectangular Form**: $$Z = a + bi$$ 2. **Polar Form**: $$Z = r(\cos \theta + i \sin \theta) = r e^{i\theta} = r \angle \theta$$ Where \(r\) is the magnitude (absolute value) and \(\theta\) is the argument (phase angle). The formulas to convert between these forms are: $$r = |Z| = \sqrt{a^2 + b^2}, \quad \theta = \text{atan2}(b, a)$$ $$a = r \cos \theta, \quad b = r \sin \theta$$ This calculator performs these conversions automatically, displaying the argument in both degrees and radians. ## Worked Case Study: Multi-Step Complex Division Let us divide \(Z_1 = 3 + 4i\) by \(Z_2 = 1 - 2i\) in rectangular form. To divide complex numbers, we multiply both the numerator and the denominator by the complex conjugate of the denominator (\(Z_2^* = 1 + 2i\)) to clear the imaginary part from the divisor: $$\frac{3 + 4i}{1 - 2i} = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$$ First, expand the numerator using FOIL: $$(3 + 4i)(1 + 2i) = 3(1) + 3(2i) + 4i(1) + 4i(2i) = 3 + 6i + 4i + 8i^2$$ Substitute \(i^2 = -1\): $$3 + 10i - 8 = -5 + 10i$$ Next, expand the denominator: $$(1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 5$$ Now, divide the terms: $$\frac{-5 + 10i}{5} = -1 + 2i$$ The quotient is \(-1 + 2i\). Converting this result to polar form gives a magnitude of \(r = \sqrt{(-1)^2 + 2^2} = \sqrt{5} \approx 2.236\) and a phase angle of \(\theta = \text{atan2}(2, -1) \approx 116.57°\). ## De Moivre's Theorem and Roots of Complex Numbers De Moivre's Theorem states that for any real number \(x\) and integer \(n\): $$(\cos x + i \sin x)^n = \cos(nx) + i \sin(nx)$$ In polar coordinates, this simplifies raising a complex number to a power: $$Z^n = (r \angle \theta)^n = r^n \angle (n\theta)$$ To find the \(n\)-th roots of a complex number \(Z = r \angle \theta\), we find \(n\) distinct values using the formula: $$W_k = r^{1/n} \angle \left( \frac{\theta + 2k\pi}{n} \right) \quad \text{for } k = 0, 1, \dots, n-1$$ These roots are symmetrically spaced on a circle in the complex plane, representing the vertices of a regular polygon. ## Related Calculators For real-number polynomial roots (including cases where a quadratic's discriminant is negative and produces a complex pair), use the [quadratic solver](/calculators/quadratic-solver). To work with vectors in real coordinate space instead of the complex plane, see the [vector math calculator](/calculators/vector-math-calculator). ## Frequently asked questions ### What is a complex number? A complex number is a number that can be written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i² = -1. ### What is the imaginary unit i? The imaginary unit i is defined as the mathematical constant whose square is -1: i² = -1. ### How do you add or subtract complex numbers? Add or subtract their real parts and imaginary parts separately: (a+bi) ± (c+di) = (a±c) + (b±d)i. ### How do you multiply complex numbers? Multiply them using the distributive law (FOIL) and substitute i² = -1: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. ### What is a complex conjugate? The complex conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate always yields a real number: a² + b². ### How do you convert from rectangular to polar form? Calculate the magnitude r = √(a² + b²) and the phase angle θ = atan2(b, a). ### What is polar form of a complex number? Polar form represents a complex number by its distance from the origin (magnitude r) and its angle from the positive real axis (argument θ), written as r∠θ or re^(iθ). ### What is a phasor? A phasor is a complex number used in engineering to represent a sinusoidal signal, capturing its amplitude and phase angle. ### What is De Moivre's Theorem? De Moivre's Theorem states that raising a complex number to a power n corresponds to raising its magnitude to n and multiplying its angle by n: Zⁿ = rⁿ∠(nθ). ### How many n-th roots does a complex number have? A complex number has exactly n distinct n-th roots, which are spaced symmetrically on a circle of radius r^(1/n) in the complex plane. ### Can you divide complex numbers in polar form? Yes, division is easy in polar form: divide the magnitudes and subtract the angles: (r₁∠θ₁) / (r₂∠θ₂) = (r₁/r₂)∠(θ₁ − θ₂). ### What is an Argand diagram? An Argand diagram is a geometric plot of complex numbers as points or vectors in a two-dimensional plane, where the x-axis represents the real part and the y-axis represents the imaginary part. ## Related concepts - **Argand Diagram** — A coordinate grid plotting complex numbers as vectors on real (horizontal) and imaginary (vertical) axes. - **Euler's Formula** — The fundamental formula connecting trigonometry and complex exponentials: e^(iθ) = cos θ + i sin θ. - **Phasor Analysis** — An engineering technique using complex numbers to analyze alternating current (AC) circuit components. ## Related guides - [Scientific Notation Guide: Powers of Ten Made Practical](https://dothecalculation.com/blog/math/scientific-notation-basics) — Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator. ## Related calculators - [Rounding Calculator](https://dothecalculation.com/calculators/rounding-calculator) — Use our free rounding calculator to round numbers to any place value. 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Rounding errors, numerical tolerances, or algorithm constraints might apply near poles, boundary conditions, or complex coordinate spaces. Always verify critical computations independently._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/complex-number-calculator). Quote freely with attribution and a link to this page._