About

Misidentifying the complex conjugate of z = a + bi propagates errors through signal processing chains, quantum state calculations, and impedance matching circuits. The conjugate operation flips the sign of the imaginary component: z = a − bi. This calculator computes z, the modulus |z|, the argument arg(z), the polar representation, and verifies the identity z ⋅ z = |z|². It accepts standard algebraic notation including edge cases such as pure reals, pure imaginaries, and zero.

This tool assumes inputs in the Cartesian form a + bi where a, b ∈ R. The argument is computed via atan2(b, a) and is undefined for z = 0. Precision is limited to IEEE 754 double-precision floating point (~15 significant digits).

Formulas

The complex conjugate of a number z = a + bi is obtained by negating the imaginary part:

z = a − bi

The modulus (absolute value) is the distance from the origin in the complex plane:

|z| = √a² + b²

The argument is the angle from the positive real axis:

arg(z) = atan2(b, a)

The polar form expresses the number using modulus and argument:

z = r(cos θ + i sin θ)

A fundamental identity verified by this calculator:

z ⋅ z = a² + b² = |z|²

Where a = real part (Re(z)), b = imaginary part (Im(z)), r = |z| (modulus), θ = arg(z) (argument in radians), and i = √−1 (imaginary unit).

Reference Data

Complex Number z	Conjugate z	Modulus \|z\|	Argument (deg)	z ⋅ z
3 + 4i	3 − 4i	5	53.13°	25
1 + i	1 − i	1.4142	45°	2
−2 + 3i	−2 − 3i	3.6056	123.69°	13
5	5	5	0°	25
−7	−7	7	180°	49
4i	−4i	4	90°	16
−3i	3i	3	−90°	9
0.5 − 0.5i	0.5 + 0.5i	0.7071	−45°	0.5
−1 − i	−1 + i	1.4142	−135°	2
6 − 8i	6 + 8i	10	−53.13°	100
2 + 7i	2 − 7i	7.2801	74.05°	53
10 + 0i	10	10	0°	100
0 + 0i	0	0	undefined	0
−5 + 12i	−5 − 12i	13	112.62°	169
1.5 + 2.5i	1.5 − 2.5i	2.9155	59.04°	8.5

Frequently Asked Questions

When b = 0, the number is purely real. The conjugate of a real number equals itself: z = z. The modulus reduces to |a| and the argument is 0° for positive reals or 180° for negative reals.

For z = bi where a = 0, the argument is 90° (π/2 rad) if b > 0, or −90° (−π/2 rad) if b < 0. The atan2 function handles this correctly, unlike atan which would produce division by zero.

The argument arg(z) represents the angle of the vector from the origin. At the origin (a = 0, b = 0), there is no direction. The atan2(0, 0) call returns 0 in most implementations, but this is a convention, not a mathematical definition. This calculator reports it as undefined.

Yes. Conjugation is an involution: z = z. It distributes over addition and multiplication: z₁ + z₂ = z₁ + z₂. It is a field automorphism of C that fixes R.

IEEE 754 double-precision provides ~15 significant decimal digits. For inputs like 1e15 + 1i, the modulus calculation √(1e15)² + 1 may lose the contribution of 1. This calculator uses Math.hypot, which is designed to avoid overflow and underflow in intermediate computations.

The parser accepts: standard form 3+4i, negative imaginary 3−4i, pure real 5, pure imaginary 4i, bare imaginary unit i or −i, decimals 1.5+2.3i, and leading negatives −3−7i. Spaces are stripped. The letter i must appear after the imaginary coefficient.