Complex Conjugate of a Number

Compute the complex conjugate by changing the sign of the imaginary part

Conjugate Calculator

Complex Conjugate

The complex conjugate \(\overline{z}\) is obtained by changing the sign of the imaginary part. Geometrically this corresponds to a reflection across the real axis.

Complex number z = a + bi
+
i
Calculation Result
\(\overline{z}\) =

Conjugate - Properties

Definition
\[\text{If } z = a + bi \text{, then } \overline{z} = a - bi\]

Only the sign of the imaginary part changes!

Operation

Sign change

Imaginary part only
Geometry

Reflection

Across the real axis
Product
\[z \cdot \overline{z} = |z|^2\]

The product is always real!

Important Properties
  • \(\overline{\overline{z}} = z\) (Double conjugation)
  • \(z + \overline{z} = 2\text{Re}(z)\) (Real part)
  • \(z - \overline{z} = 2i\text{Im}(z)\) (Imaginary part)
  • \(|z| = |\overline{z}|\) (Same magnitude)
Algebraic Rules
  • \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)
  • \(\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}\)
  • \(\overline{z_1 / z_2} = \overline{z_1} / \overline{z_2}\)
  • \(\overline{z^n} = (\overline{z})^n\)

Definition of the Complex Conjugate

Every complex number has a complex conjugate obtained by changing the sign of the imaginary part. It is denoted by \(\overline{z}\).

Notation

For \(z = a + bi\):
\[\overline{z} = a - bi\] Alternative notations: \(z^*\), \(\text{conj}(z)\)

Geometric Meaning

Reflection across the real axis:
Real part stays the same
Imaginary part changes sign

Examples and Calculations

Example 1: Simple Conjugation

Given: \(z = 5 + 3i\)

Conjugate: \(\overline{z} = 5 - 3i\)

Real part 5 remains, imaginary part changes from +3 to -3

Example 2: Negative Imaginary Part

Given: \(z = 3 - 4i\)

Conjugate: \(\overline{z} = 3 + 4i\)

-4i becomes +4i

Example 3: Pure Real Number

Given: \(z = 7 + 0i\)

Conjugate: \(\overline{z} = 7 - 0i = 7\)

Real numbers are self-conjugate

Example 4: Pure Imaginary Number

Given: \(z = 0 + 6i\)

Conjugate: \(\overline{z} = 0 - 6i = -6i\)

Sign is reversed

Example 5: Product z·z̄

For z = 5 + 3i:

\(z \cdot \overline{z} = (5+3i)(5-3i)\)

\(= 25 - 15i + 15i - 9i^2\)

\(= 25 - 9(-1) = 25 + 9 = 34\)

Result: reelle Zahl!

Verification

Alternatively: \(|z|^2 = 5^2 + 3^2 = 25 + 9 = 34\) ✓
The formula \(z \cdot \overline{z} = |z|^2\) always holds!

Applications of the Complex Conjugate

1. Division of complex numbers
Application:
\[\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}\]

By multiplying numerator and denominator with \(\overline{z_2}\) the denominator becomes real!

2. Magnitude calculation
Formula:
\[|z| = \sqrt{z \cdot \overline{z}}\]

Alternative to formula \(|z| = \sqrt{a^2 + b^2}\)

3. Extracting real and imaginary parts
Formulas:
\[\text{Re}(z) = \frac{z + \overline{z}}{2}\] \[\text{Im}(z) = \frac{z - \overline{z}}{2i}\]
4. Quadratic equations
Property:
If \(z\) is a solution, then \(\overline{z}\) is also a solution for real coefficients.

Complex solutions always occur in pairs!

5. Electrical impedance
Physics:
For complex impedance \(Z\) the conjugate \(\overline{Z}\) corresponds to the impedance for negative frequencies.

Important for power calculations: \(P = U \cdot \overline{I}\)

6. Signal processing
Fourier transform:
The spectrum of real signals is Hermitian:
\(F(-\omega) = \overline{F(\omega)}\)

Complex Conjugates - Detailed Description

Geometric Interpretation

In the Gaussian number plane conjugation corresponds to a reflection across the real axis.

Visualization:
• Point \(z = a + bi\) lies above the real axis (when b > 0)
• Point \(\overline{z} = a - bi\) lies symmetrically below
• Both points have the same distance from the origin
• The real axis is the axis of symmetry

Important Property

The product of a number with its conjugate is always real and non-negative (or zero):

\[z \cdot \overline{z} = (a+bi)(a-bi) = a^2 + b^2 = |z|^2\]

This is the square of the magnitude!

Algebraic rules with conjugation

Conjugation is compatible with basic arithmetic operations:

Linearity and multiplicativity:
• \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\) (Addition)
• \(\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}\) (Subtraction)
• \(\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}\) (Multiplication)
• \(\overline{z_1 / z_2} = \overline{z_1} / \overline{z_2}\) (Division)

Involutive property

Conjugation is involutive, i.e. applying it twice returns the original number:

\[\overline{\overline{z}} = z\]

Two reflections = identity

Special cases

  • Real numbers: \(\overline{a} = a\) (self-conjugate)
  • Imaginary numbers: \(\overline{bi} = -bi\) (sign change)
  • Zero: \(\overline{0} = 0\)
  • Imaginary unit: \(\overline{i} = -i\)

More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •