Complex Number Calculator

About This Calculator

Complex numbers extend the real number line into a two-dimensional plane, allowing every polynomial equation to have roots. A complex number has the form $z = a + bi$ where $a$ is the real part, $b$ is the imaginary part, and $i$ satisfies $i^2 = -1$.

This calculator supports nine operations: addition, subtraction, multiplication, division, modulus, argument, conjugate, integer powers (De Moivre's theorem), and nth roots. Results are displayed in both rectangular $(a + bi)$ and polar $(r\angle\theta)$ form.

How to Use

1. Select an operation from the radio cards: Add, Subtract, Multiply, Divide, Modulus, Argument, Conjugate, Power, or nth Root.
2. Enter z₁ by filling in the real part (a) and imaginary part (b).
3. Enter z₂ (for binary operations only) by filling in c and d.
4. For Power, enter the integer exponent n. For nth Root, enter the root degree n (must be at least 2).
5. Click Calculate to see the result in rectangular and polar form, along with the modulus and argument.

Formulas

Representations

A complex number can be expressed in rectangular, polar, or exponential form:

where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z) = \operatorname{atan2}(b, a)$.

Addition and Subtraction

Multiplication

In polar form: multiply moduli and add arguments.

Division

In polar form: divide moduli and subtract arguments.

Modulus

Argument

Important: Use $\operatorname{atan2}(b, a)$, not $\arctan(b/a)$, because $\operatorname{atan2}$ correctly identifies all four quadrants.

Conjugate

Key identity: $z \cdot \bar{z} = |z|^2 = a^2 + b^2$.

Powers (De Moivre's Theorem)

Valid for all integers n (positive, negative, and zero). For the polar form advantage, trigonometric identity derivation, and applications, see the De Moivre Calculator guide.

nth Roots

for $k = 0, 1, \ldots, n-1$. This produces exactly $n$ distinct roots, equally spaced around a circle of radius $r^{1/n}$. For the geometric arrangement, roots of unity, and worked examples, see the nth Root Calculator guide.

Worked Examples

Example 1: Adding Phasor Signals

Two signals have phasor representations $V_1 = 3 + 4i$ and $V_2 = 1 - 2i$. Find the resultant phasor.

Modulus: $|4+2i| = \sqrt{16+4} = \sqrt{20} \approx 4.47$. Argument: $\theta \approx 26.57°$.

Example 2: Dividing Complex Numbers

Compute $(3+4i) \div (1+2i)$.

Example 3: Power via De Moivre

Compute $(1+i)^{10}$.

In polar form: $r = \sqrt{2}$, $\theta = \pi/4$.

Since $5\pi/2 = \pi/2 + 2\pi$, the result is $32(\cos(\pi/2) + i\sin(\pi/2)) = 32i$.

Example 4: Cube Roots of −8

Find all cube roots of $-8$.

Polar form: $r = 8$, $\theta = \pi$. Cube root of modulus: $r^{1/3} = 2$.

These three roots are equally spaced at 120° intervals on a circle of radius 2.

Example 5: AC Circuit Impedance

An RLC series circuit has impedance $Z = 100 + 40j\,\Omega$. Find the magnitude and phase angle.

The impedance magnitude is approximately 107.7 Ω with a phase angle of 21.8°.

Tips and Common Mistakes

• Use atan2, not atan: $\operatorname{atan2}(b, a)$ correctly identifies the quadrant for the argument. Plain $\arctan(b/a)$ is ambiguous for quadrants II and III.
• Radians vs. degrees: The argument is computed in radians by default. Multiply by $180/\pi$ to convert to degrees.
• Division requires the conjugate: When dividing, multiply both numerator and denominator by the conjugate of the denominator to rationalize it.
• All n roots exist: When computing nth roots, there are always exactly $n$ distinct values. Do not stop at the principal root.
• arg(0) is undefined: The argument of $0 + 0i$ has no defined value because there is no direction from the origin to itself.
• Precision matters: Floating-point arithmetic can introduce small rounding errors. Results like $1.23 \times 10^{-16}$ are effectively zero.
• Engineering convention: Electrical engineers use $j$ instead of $i$ because $i$ represents electric current.

Glossary

Complex Number

A number of the form $a + bi$ with real part $a$ and imaginary part $b$.

Real Part

The component $a$ in $z = a + bi$. Denoted $\operatorname{Re}(z)$.

Imaginary Part

The component $b$ in $z = a + bi$. Denoted $\operatorname{Im}(z)$.

Modulus (Absolute Value)

The distance from the origin: $|z| = \sqrt{a^2 + b^2}$. Always non-negative.

Argument (Phase Angle)

The angle $\theta = \operatorname{atan2}(b, a)$ from the positive real axis to the point $z$.

Polar Form

Expressing $z$ as $r(\cos\theta + i\sin\theta)$ or $r\angle\theta$.

Rectangular (Cartesian) Form

Expressing $z$ as $a + bi$.

Argand Diagram

A graphical representation of complex numbers on a 2D plane with real and imaginary axes.

De Moivre's Theorem

The formula $z^n = r^n \operatorname{cis}(n\theta)$ for integer powers of complex numbers.

nth Root

One of $n$ distinct values $w$ satisfying $w^n = z$.

Conjugate

The number $\bar{z} = a - bi$, obtained by flipping the sign of the imaginary part.

Frequently Asked Questions

What is a complex number?

A complex number has the form $z = a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$. Complex numbers extend the real number system and are essential in engineering, physics, and mathematics.

Why do engineers use j instead of i?

In electrical engineering, $i$ already represents electric current. To avoid confusion, engineers use $j$ for the imaginary unit. Mathematically, $j = i$ — they are the same concept.

What is the difference between modulus and argument?

The modulus $|z|$ is the distance from the origin (how far from zero). The argument $\arg(z)$ is the angle from the positive real axis (which direction from the origin). Together they define the polar form.

Why does the nth root of a complex number have n values?

The Fundamental Theorem of Algebra guarantees that the equation $w^n = z$ has exactly $n$ solutions in the complex numbers. These roots are equally spaced around a circle on the Argand diagram, separated by angles of $2\pi/n$.

What is Euler's formula?

Euler's formula states $e^{i\theta} = \cos\theta + i\sin\theta$. The special case $\theta = \pi$ gives Euler's identity: $e^{i\pi} + 1 = 0$, connecting five fundamental constants.

Can I compute negative powers?

Yes. For any nonzero $z$, $z^{-n} = 1/z^n$. De Moivre's theorem handles negative exponents naturally: $z^{-n} = r^{-n} \operatorname{cis}(-n\theta)$. The only restriction is $0^{-n}$, which is undefined.

References

1. NIST Digital Library of Mathematical Functions: Complex Variables — National Institute of Standards and Technology (2024)
2. NIST DLMF: Euler's Formula and Exponential Function — National Institute of Standards and Technology (2024)
3. Mathematics LibreTexts: The Polar Form of Complex Numbers — LibreTexts Mathematics (2023)
4. Mathematics LibreTexts: De Moivre's Theorem and Powers of Complex Numbers — LibreTexts Mathematics (2023)
5. Wikipedia: Euler's Formula — Wikipedia / Community peer review (2024)
6. Wikipedia: De Moivre's Formula — Wikipedia / Community peer review (2024)
7. Wikipedia: Complex Number — Wikipedia / Community peer review (2024)
8. Wikipedia: Fundamental Theorem of Algebra — Wikipedia / Community peer review (2024)

Disclaimer

This calculator is provided for educational and reference purposes. While we strive for accuracy, floating-point arithmetic inherently introduces small rounding errors. Results should not be used as the sole basis for engineering decisions in safety-critical applications. Always verify critical calculations independently.

The formulas implemented follow standard complex analysis conventions as documented by NIST, Wolfram MathWorld, and standard university-level textbooks. The principal argument convention used is $(-\pi, \pi]$.