Complex Number Calculator
Add, subtract, multiply and divide complex numbers step by step, and find the modulus, argument, conjugate and polar form on the complex plane.
Try an example
Result of (3 + 4i) + (1 − 2i)
Complex Number Properties of the result
- Rectangular form
- 4 + 2i
- Real part, Re(z)
- 4
- Imaginary part, Im(z)
- 2
- Modulus |z|
- √20 = 2√5 ≈ 4.4721
- Argument θ
- atan2(2, 4) = ≈ 26.5651°≈ 0.4636 rad
- Conjugate z̄
- 4 − 2i
- Polar form
- 2√5(cos 26.5651° + i sin 26.5651°)
- Exponential form
- 2√5e^(i·0.4636)exponent in radians
Step-by-Step Solution
Step 1 — Add real parts and imaginary parts separately
Real parts combine with real parts, and imaginary parts combine with imaginary parts.
(a + bi) + (c + di) = (a + c) + (b + d)i
Step 2 — Substitute the values
(3 + 4i) + (1 − 2i)
= (3 + 1) + (4 − 2)i
Step 3 — Write in a + bi form
= 4 + 2i
Complex plane (Argand diagram)
Complex number formulas
| Addition | (a + bi) + (c + di) = (a + c) + (b + d)i |
|---|---|
| Subtraction | (a + bi) − (c + di) = (a − c) + (b − d)i |
| Multiplication | (a + bi)(c + di) = (ac − bd) + (ad + bc)i |
| Division | (a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²), c + di ≠ 0 |
| Modulus | |z| = √(a² + b²) |
| Argument | arg(z) = θ = atan2(b, a), −180° < θ ≤ 180° |
| Conjugate | z̄ = a − bi, z · z̄ = a² + b² |
| Reciprocal | 1/z = (a − bi) / (a² + b²) |
| Polar form | z = r(cos θ + i sin θ), r = |z| |
| Exponential form | z = re^(iθ) (θ in radians) |
| Back to rectangular | a = r cos θ, b = r sin θ |
Here z = a + bi with real part a and imaginary part b, the second number is c + di, and i is the imaginary unit with i² = −1.
Complex numbers often appear as roots of equations: the Quadratic Formula Calculator and the Polynomial Root Calculator return complex roots in a + bi form. A complex number behaves like a 2D vector for addition and length, which the Vector Calculator covers. For sines, cosines and other functions of the angle, use the Scientific Calculator.
This complex number calculator works with numbers of the form z = a + bi, where a is the real part, b is the imaginary part and i² = −1. Enter the real and imaginary parts of two complex numbers, choose add, subtract, multiply or divide, and the result appears in rectangular form with every step of the working shown.
The calculator also finds the modulus (magnitude), argument, conjugate, polar form and exponential form of the result, in degrees or radians, and plots the numbers on the complex plane. Switch to One number mode to convert a single complex number to polar form or to find its reciprocal, square and cube.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Addition: (3 + 4i) + (1 − 2i) | 4 + 2i | (a + c) + (b + d)i = (3 + 1) + (4 − 2)i = 4 + 2i. |
| Subtraction: (5 + 3i) − (2 + 7i) | 3 − 4i | (a − c) + (b − d)i = (5 − 2) + (3 − 7)i = 3 − 4i. |
| Multiplication: (2 + 3i)(4 − i) | 11 + 10i | Expand: 8 − 2i + 12i − 3i². Replace i² with −1: −3i² = 3. Combine: (8 + 3) + (−2 + 12)i = 11 + 10i. |
| Division: (4 + 2i) ÷ (1 + i) | 3 − i | Multiply top and bottom by the conjugate 1 − i. Denominator: 1² + 1² = 2. Numerator: (4·1 + 2·1) + (2·1 − 4·1)i = 6 − 2i. Divide: 6/2 − (2/2)i = 3 − i. Check: (1 + i)(3 − i) = 3 − i + 3i − i² = 4 + 2i. |
| Modulus: |3 + 4i| | 5 | |z| = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5. |
| Polar form of 3 + 4i | 5(cos 53.13° + i sin 53.13°) | r = √(3² + 4²) = 5. θ = atan2(4, 3) ≈ 53.13° ≈ 0.9273 rad (Quadrant I). So 3 + 4i = 5(cos 53.13° + i sin 53.13°) = 5e^(0.9273i). |
What Is a Complex Number?
A complex number is written z = a + bi. The real number a is the real part, Re(z), and the real number b is the imaginary part, Im(z). The imaginary unit i is defined by i² = −1, so i is a square root of −1.
Complex numbers extend the real numbers. The equation x² + 1 = 0 has no real solution, because no real number squared is negative, but in the complex numbers it has two solutions, x = i and x = −i. Every real number is also a complex number with imaginary part 0: 5 is 5 + 0i. A number with real part 0, such as 5i, is called purely imaginary.
Complex Number Forms: Rectangular, Polar and Exponential
Rectangular (Cartesian) form, a + bi, gives the real and imaginary parts directly. It is the easiest form for addition and subtraction, since you just combine matching parts.
Polar (trigonometric) form, r(cos θ + i sin θ), describes the same number by its distance r = |z| from the origin and its angle θ from the positive real axis. It is useful for multiplication, division and powers: to multiply, multiply the moduli and add the angles. For example, 3 + 4i = 5(cos 53.13° + i sin 53.13°).
Exponential form, re^(iθ), comes from Euler's formula e^(iθ) = cos θ + i sin θ, with θ in radians. It is the standard notation in advanced mathematics, physics, signal processing and electrical engineering, where rotating quantities are written as e^(iωt). Engineers often write the polar form as r∠θ and use j instead of i.
Rectangular Form to Polar Form
To convert z = a + bi to polar form, find the modulus r = √(a² + b²) and the argument θ = atan2(b, a). Using atan2 rather than arctan(b/a) matters: it looks at the signs of both a and b and puts the angle in the right quadrant. For −1 + i, arctan(1/−1) gives −45°, but the point is in Quadrant II and the correct argument is 135°.
To go back, use a = r cos θ and b = r sin θ. Always check which angle unit you are working in: 53.13° and 0.9273 radians are the same angle, and mixing the two is a common source of wrong answers.
Modulus, Argument and Conjugate
The modulus |z| = √(a² + b²) is the distance from 0 to the point (a, b), so |3 + 4i| = 5. The argument arg(z) is the angle of that point, measured counterclockwise from the positive real axis. The conjugate of a + bi is a − bi, its mirror image across the real axis.
Multiplying a number by its conjugate always gives a real result: (a + bi)(a − bi) = a² + b² = |z|². This is exactly why the conjugate is used to divide complex numbers.
When to Use This Calculator
Use it to check algebra and precalculus homework on complex arithmetic, to verify division and conversions done by hand, and to see how a complex number sits on the complex plane.
Complex numbers are also routine in technical work. In AC circuit analysis, impedances such as Z = R + jX are complex, and voltages and currents are handled as phasors in polar form. Signal processing uses them in the Fourier transform to describe the amplitude and phase of each frequency. In control systems, the poles of a transfer function are complex numbers whose position in the complex plane determines stability. Physics uses them in wave equations and quantum mechanics.
How to Use the Complex Number Calculator
- Enter the real part (a) and imaginary part (b) of the first complex number. The preview shows it as a + bi, for example 3 + 4i or 3 − 4i. Use a minus sign for a negative imaginary part.
- Enter the real part (c) and imaginary part (d) of the second complex number, z₂ = c + di.
- Select Add, Subtract, Multiply or Divide. The result updates as you type.
- Read the result in rectangular form (a + bi), then follow the step-by-step working for the operation you chose.
- Check the Complex Number Properties panel for the modulus, argument, conjugate and polar form of the result. Switch the angle unit between degrees and radians if needed.
- Use the complex plane (Argand diagram) to see each number as a point and a vector from the origin, or switch to One number mode to convert a single complex number to polar and exponential form.
Frequently Asked Questions
What is a complex number?
A number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. a is the real part and b is the imaginary part. For 3 − 4i, the real part is 3 and the imaginary part is −4.
What does i mean in a complex number?
i is the imaginary unit, defined so that i² = −1. Its powers repeat in a cycle of four: i, −1, −i, 1. Electrical engineers often write j instead, because i is used for current.
How do you add complex numbers?
Add the real parts and add the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i. For example, (3 + 4i) + (1 − 2i) = 4 + 2i.
How do you subtract complex numbers?
Subtract the real parts and the imaginary parts separately, taking care with signs: (5 + 3i) − (2 + 7i) = (5 − 2) + (3 − 7)i = 3 − 4i.
How do you multiply complex numbers?
Expand like two binomials, then replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For example, (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 8 + 3 + 10i = 11 + 10i.
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. For (4 + 2i)/(1 + i), multiply by (1 − i): the numerator becomes 6 − 2i and the denominator becomes 1² + 1² = 2, so the answer is 3 − i. You can check it: (1 + i)(3 − i) = 4 + 2i.
What is the modulus of a complex number?
The modulus |a + bi| = √(a² + b²) is the distance from the origin to the point (a, b) on the complex plane. It is also called the magnitude or absolute value. |3 + 4i| = √(9 + 16) = 5.
What is the conjugate of a complex number?
The conjugate of a + bi is a − bi: the imaginary part changes sign. The conjugate of 3 + 4i is 3 − 4i. A number times its conjugate is always the real number a² + b².
How do you find the argument of a complex number?
Use θ = atan2(b, a), the angle from the positive real axis to the point (a, b). For 3 + 4i, θ ≈ 53.13° (0.9273 rad). atan2 accounts for the quadrant, which arctan(b/a) alone does not: the argument of −1 − i is −135°, not 45°.
How do you convert a complex number to polar form?
Find r = √(a² + b²) and θ = atan2(b, a), then write z = r(cos θ + i sin θ), or re^(iθ) with θ in radians. For 1 + i, r = √2 and θ = 45° = π/4, so 1 + i = √2(cos 45° + i sin 45°) = √2·e^(iπ/4).
Can a complex number have a zero real part?
Yes. When a = 0 the number is purely imaginary, such as 5i, and it lies on the imaginary axis. When b = 0 the number is real, such as 5, so every real number is also a complex number.
What happens when you divide by zero?
Division by 0 + 0i is undefined, just as division by 0 is for real numbers: the denominator c² + d² would be 0. The calculator shows an error instead of a result. 0 also has no reciprocal and no defined argument.
Related Calculators
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 quadratic equations.
Polynomial Root Calculator
Find all real and complex roots of a polynomial, with exact answers, multiplicities, factored form, a graph, and step-by-step working.
Vector Calculator
Add, subtract and scale 2D and 3D vectors, find magnitude, direction and unit vectors, and calculate dot products, cross products and the angle between vectors, step by step.
Scientific Calculator
Advanced math calculator supporting trigonometry, logs, exponentials, and roots.
Matrix Calculator
Calculate determinant, trace, and inverse values for a 2x2 matrix.
Fraction Calculator
Add, subtract, multiply, and divide fractions.
Last updated: September 27, 2026.