Complex number calculator guide
A complex number combines a real part and an imaginary coefficient in the form a + bi, where i squared equals -1. Addition and subtraction combine matching parts, while multiplication uses distribution and i squared = -1.
Division multiplies by the denominator conjugate, producing the real denominator c squared + d squared. The calculator also reports the modulus and principal argument of complex-valued results.
How to calculate with complex numbers
- Choose an operation: Select arithmetic, modulus, or conjugate. Unary operations use only the first number.
- Enter rectangular components: For a + bi, enter a in the real field and b in the imaginary-coefficient field.
- Calculate: Select Calculate to produce the complex or scalar result.
- Inspect polar information: For complex-valued results, use the modulus and principal argument as an additional check.
Formula and variables
The conjugate c - di removes the imaginary part from the division denominator. The modulus is sqrt(a^2 + b^2), the distance from the origin in the complex plane.
(a+bi)(c+di) = (ac-bd) + (ad+bc)i; (a+bi)/(c+di) = [(ac+bd) + (bc-ad)i]/(c^2+d^2)- a, c — Real parts
- Horizontal components in the complex plane.
- b, d — Imaginary coefficients
- Vertical components multiplying i.
- i — Imaginary unit
- A number satisfying i^2 = -1.
- |z| — Modulus
- Distance of z from zero, sqrt(a^2 + b^2).
Multiply two complex numbers
Multiply 3 + 4i by 5 - 2i.
- First number
- 3 + 4i
- Second number
- 5 - 2i
- Real part = 3(5) - 4(-2) = 23
- Imaginary part = 3(-2) + 4(5) = 14
Result: (3 + 4i)(5 - 2i) = 23 + 14i.
Distribution produces four terms, and the i-squared term contributes to the real part.
Understanding your results
Rectangular result
The real and imaginary components are displayed in a + bi form with the sign normalized.
Modulus and argument
The modulus is nonnegative. The principal argument uses atan2 and is reported in degrees; it is undefined for 0 + 0i.
Assumptions
- Inputs are finite real coefficients of complex numbers in rectangular form.
- The principal argument is reported in the conventional range from -180 to 180 degrees.
- Division requires a nonzero second complex number.
Limitations
- The calculator does not perform symbolic algebra, roots, powers, logarithms, or polar-form arithmetic.
- Floating-point results can contain small rounding error.
- Only one binary or unary operation is evaluated at a time.
- The displayed argument is one representative; adding integer multiples of 360 degrees gives coterminal angles.
Common mistakes
- Forgetting that i squared equals -1 during multiplication.
- Dividing real and imaginary components separately.
- Changing both parts when forming a conjugate instead of only the imaginary sign.
- Calling the imaginary coefficient bi instead of b in an input field.
- Assuming the argument of zero is defined.
Practical use cases
Algebra practice
Check rectangular-form arithmetic and conjugate steps.
Signals and circuits
Combine phasor-like values when a rectangular representation is appropriate.
Planning and decision guide
Choose rectangular or polar form
Rectangular form is convenient for addition and subtraction. Polar form is often more convenient for repeated multiplication, division, powers, and roots; use a dedicated polar tool for those workflows.
Frequently asked questions
What is the conjugate of a + bi?
It is a - bi: the real part stays the same and the imaginary sign changes.
How is a complex modulus calculated?
For z = a + bi, the modulus is sqrt(a^2 + b^2).
Why can I not divide by 0 + 0i?
Its squared modulus is zero, so the complex-division denominator is zero.
Is the principal argument unique?
The displayed principal argument is unique within the selected range, but coterminal angles differ by multiples of 360 degrees.
What does i represent?
The imaginary unit i is defined by i^2 = -1.
Sources and review
- Complex Numbers — OpenStax Algebra and Trigonometry 2e. Accessed 2026-08-29.
- Polar Form of Complex Numbers — OpenStax Algebra and Trigonometry 2e. Accessed 2026-08-29.
Reviewed 2026-08-29.