Complex Number Calculator
Add, subtract, multiply and divide complex numbers a + bi, with the expansion using i² = −1 and conjugate-based division shown step by step. Also shows the modulus, argument, conjugate and polar form.
Try an example
Enter z₁ and z₂ in the form 3 + 4i to get the four arithmetic results plus |z₁|, argument, conjugate and polar form.
How to use
- 1
Enter complex numbers
Fill in z₁ and z₂ in forms like "3 + 4i", "-2i" or "5".
- 2
Read the arithmetic
Addition, subtraction, multiplication and division appear at once with the working shown.
- 3
Read the properties
The modulus |z|, argument, conjugate and polar form of z₁ are shown together.
Features
- All four arithmetic operations at once, with the i² = −1 expansion shown
- Division via the conjugate with working (dividing by zero reports an error)
- Modulus, argument (degrees and radians), conjugate and polar form together
- Inputs are never transmitted; all computation happens locally in your browser
Use cases
Check complex homework
Verify arithmetic with complex numbers from algebra and precalculus courses.
Quadratics with negative discriminant
Confirm sums and products of roots like x = −1 ± 2i against Vieta's formulas.
Convert to polar form
Find |z| and the argument, then assemble r(cos θ + i sin θ).
Details
A complex number is written a + bi (a and b real, i the imaginary unit with i² = −1) and corresponds to the point (a, b) in the plane. Addition and subtraction combine real and imaginary parts separately; multiplication expands with the distributive law and then applies i² = −1.
For division, multiply the numerator and denominator by the conjugate of the denominator (the number with the imaginary part negated). The denominator becomes the real number a² + b², making the division straightforward. For example, (1 + i) ÷ (1 − i) equals 2i ÷ 2 = i after multiplying both by (1 + i).
The modulus |z| = √(a² + b²) is the distance from the origin, and the argument arg(z) is the angle from the positive real axis; together they form the polar representation r(cos θ + i sin θ). The argument is undefined when z = 0. Keep the absolute value of each part at or below 1,000,000. Inputs are never transmitted; all computation happens locally in your browser.
FAQ
How do I divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. The denominator becomes the real number (c + di)(c − di) = c² + d², so you simply divide the real and imaginary parts by it.
What does i² = −1 mean?
It is the definition of the imaginary unit i: squaring i gives −1. Whenever i² appears in an expansion, replace it with −1. In (1 + 2i)(3 + 4i), the term 8i² becomes −8, giving −5 + 10i.
What do the modulus and argument represent?
Treating a + bi as the point (a, b) in the plane, the modulus |z| = √(a² + b²) is the distance from the origin and the argument arg(z) is the counter-clockwise angle from the positive real axis. Together they give the polar form r(cos θ + i sin θ).
Can I enter roots like x = −1 ± 2i from a quadratic?
Yes. Type them in directly, for example "-1+2i". You can then read the sum −2 and product 5 of the roots to check Vieta's formulas (sum = −b/a, product = c/a).
Related Tools
Quadratic Equation
Solve ax² + bx + c = 0 with real or complex roots, the discriminant and substitution into the quadratic formula.
Vector Calculator (Dot Product)
Compute the dot product, magnitudes and the angle between 2D or 3D vectors with fully substituted working. Includes the cosθ = (a·b) ÷ (|a||b|) step and perpendicular detection via a·b = 0 — handy for math and physics.
Trigonometry Calculator
Enter an angle to get sin, cos and tan at once. Special angles like 30°, 45° and 60° are shown as exact values (1/2, √3/2), with degree and radian support, pi expressions such as pi/6, plus the quadrant signs and the reference-angle working.
Matrix Determinant & Inverse
Compute the determinant and inverse of 2×2 and 3×3 matrices with cofactor-expansion working. Detects singular matrices (det = 0) and verifies the inverse — useful for math B and linear algebra students.
All processing happens in your browser. Your files are never uploaded.
Verified: Known values ((1+2i)(3+4i)=−5+10i, (1+i)÷(1−i)=i, |3+4i|=5, product of −1±2i equals 5), division by zero, format errors and the English page are covered by browser tests
Did you know?
Multiplying by the conjugate yields the squared modulus: (3+4i)(3−4i) = 9+16 = 25 = |3+4i|². This property is the key to complex division, turning the denominator into a real number.