Complex Number Calculator

How to use the Complex Number Calculator

1. Choose an Operation. Add, subtract, multiply and divide use two numbers. Conjugate, modulus, argument, polar form, power and roots use only the first number, z₁.
2. Enter z₁ as a Real part (a) and an Imaginary part (b). For 5 − 2i, enter 5 and −2. For a pure imaginary number such as 4i, enter 0 and 4. A blank part counts as 0.
3. For a two-number operation, enter z₂'s Real part (c) and Imaginary part (d). For a power or root, enter n.
4. Press Calculate. The answer appears in rectangular form (a + bi), polar form (r ∠ θ), exponential form (r·e^(iθ)) and trigonometric form, with the modulus, the angle and the steps. For roots, every root is listed in a table.

Complex number formulas

A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1.

Adding and subtracting work part by part. Multiplying is like multiplying two binomials, with i² replaced by −1. Dividing works by multiplying the top and bottom by the conjugate of the denominator, which turns the denominator into the real number c² + d². The modulus and argument describe the same point as a distance and an angle; the principal argument is the one between −180° and 180° (−π and π radians). Powers and roots are easiest in polar form: raise r to the power and multiply the angle, or take the root of r and divide the angle.

Example: multiply, divide and convert 3 + 4i

Let z₁ = 3 + 4i and z₂ = 1 − 2i.

Multiply. (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i.

Divide. Multiply the top and bottom by the conjugate of z₂, which is 1 + 2i:

1. Top: (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i
2. Bottom: 1² + 2² = 5
3. Result: (−5 + 10i) ÷ 5 = −1 + 2i

Check by multiplying back: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = 3 + 4i.

Polar form. r = √(3² + 4²) = √25 = 5, and θ = atan2(4, 3) ≈ 53.130102°, which is 0.927295 radians. So 3 + 4i = 5 ∠ 53.13° = 5·e^(0.927295i).

Example from an AC circuit

Electronics uses the same math but usually writes j instead of i. Ohm's law still holds for alternating current when voltage, current and impedance are all written as complex numbers: I = E ÷ Z. Suppose a 120-volt source drives a load with an impedance of Z = 3 + 4j ohms. Choose Divide, enter z₁ = 120 + 0i and z₂ = 3 + 4i:

I = 120 ÷ (3 + 4j) = 120(3 − 4j) ÷ 25 = 14.4 − 19.2j amps

In polar form that is 24 ∠ −53.13°: a 24-amp current whose angle is 53.13° behind the voltage.

How to read the result

• Rectangular form, a + bi, gives the real and imaginary parts. It is the easiest form for adding and subtracting.
• Polar form, r ∠ θ, gives the distance and the angle in degrees. Multiplying in polar form is simple: multiply the r values and add the angles.
• Exponential form, r·e^(iθ), is the same point with θ in radians, using Euler's formula e^(iθ) = cos θ + i sin θ.
• Argument is shown in degrees, radians and as a multiple of π (0.5π is 90°). The argument mode also gives the angle from 0° to 360°, since −90° and 270° point the same way.
• Values are rounded to 6 decimal places. Leftovers from rounding, such as 1.2 × 10^−16 from cos 90°, are shown as 0.

Assumptions and limitations

• Angles are principal values between −180° and 180°. Some textbooks prefer 0° to 360°; both describe the same number.
• Powers must be whole numbers from −1,000 to 1,000, and roots from 2 to 100. 0⁰ is taken as 1.
• Division by 0 + 0i, the argument of 0, and 0 raised to a negative power are undefined. The calculator says so instead of showing a number.
• The math uses standard double-precision numbers, accurate to about 15 significant digits. Whole-number powers are computed by repeated multiplication, so (1 + i)⁸ comes out as exactly 16.
• The principal root (k = 0) is the one most software returns, but all n roots are equally valid solutions of wⁿ = z. For everyday real-number arithmetic, the Basic Calculator is simpler.

Common mistakes

• Forgetting that i² = −1. In (3 + 4i)(1 − 2i), the −8i² term becomes +8, not −8.
• Choosing the wrong quadrant. tan⁻¹(b ÷ a) alone gives 45° for both 1 + i and −1 − i. atan2(b, a) uses the signs of both parts and correctly gives 45° and −135°.
• Dividing part by part. (a + bi) ÷ (c + di) is not (a ÷ c) + (b ÷ d)i. Multiply by the conjugate instead.
• Mixing degrees and radians. e^(iθ) needs θ in radians, so 53.13° becomes 0.927295. Converting between the two also comes up in the Area of a Sector Calculator.
• Putting the sign in the wrong field. For 5 − 2i, the imaginary part is −2, not 2.

Formulas and content last reviewed on October 13, 2026. Last updated October 13, 2026. Results are estimates for general information only. Spotted a problem? Tell us and we will check it.