Complex Number Calculator

Explore a complex number’s modulus, conjugate and argument on a live Argand diagram with correct quadrant handling.

Calculation inputs

Precision & interpretation

The conjugate a − bi reflects the point across the real axis. At z = 0 the modulus is 0 but the argument is undefined, so no direction is assigned.

Calculation results

Complex number z3 + 4iRectangular form · exact decimal inputs

Modulus |z|5

Principal argument53.13010235°

Conjugate3 − 4i

Step-by-step solution

Choose the formula
$|z| = √(a²+b²); θ = atan2(b,a)$
Substitute your inputs
$z = 3 + 4i$
Calculate in order
$|z| = √((3)² + (4)²) ≈ 5$
Write the result
$z = 3 + 4i; θ = 53.13010235°$

Argand plane

Equal axis scales. The segment from the origin has length |z|; zero has no defined argument.

The argument uses (−π, π], equivalent to (−180°, 180°]. Negative zero is normalized.

Exact value & calculation details

3 + 4i

Quantity	Value
Real part	3
Imaginary coefficient	4
Modulus	5
Argument	53.13010235°

Understand the calculation

Concept & meaning

The real part is the horizontal coordinate and the imaginary coefficient is the vertical coordinate on the Argand plane. The distance from the origin is the modulus.

Formula & special cases

The argument uses atan2(b,a), which preserves the quadrant and handles a = 0. A plain arctangent of b/a would lose that information. The principal argument is shown in (−π, π], or (−180°, 180°].

Precision & limitations

The conjugate a − bi reflects the point across the real axis. At z = 0 the modulus is 0 but the argument is undefined, so no direction is assigned.

| z | = \sqrt{a^{2} + b^{2}}

Worked example from your inputs

Rectangular form · exact decimal inputs = 3 + 4i

Frequently asked questions

Why use atan2 rather than arctan(b/a)?

atan2 uses both signs and works on the imaginary axis, so it identifies the correct quadrant.

What is the argument of zero?

It is undefined. The origin has no unique direction.

Can I show radians?

Yes. Changing the angle unit changes its display and polar form without changing the complex number.

What does the conjugate mean?

It changes the sign of the imaginary component and reflects the point across the real axis.

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What does the Complex Number calculate?

Explore a complex number’s modulus, conjugate and argument on a live Argand diagram with correct quadrant handling.

Which values should I enter?

Use the labeled fields above. Decimals: up to 6 places. Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

z = a + bi; |z| = √(a²+b²); θ = atan2(b,a)

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$OpenStax official logo$ OpenStax · Complex numbersModulus, argument and polar representationRead the reference $OpenStax official logo$ OpenStax · CoordinatesDistance, midpoint and Cartesian coordinatesRead the reference

References explain the mathematics; they do not imply institutional endorsement.