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.
Conjugate - Properties
Definition
Only the sign of the imaginary part changes!
Operation
Sign change
Imaginary part onlyGeometry
Reflection
Across the real axisProduct
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
- \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)
- \(\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}\)
- \(\overline{z_1 / z_2} = \overline{z_1} / \overline{z_2}\)
- \(\overline{z^n} = (\overline{z})^n\)
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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
\[\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}\]
By multiplying numerator and denominator with \(\overline{z_2}\) the denominator becomes real!
2. Magnitude calculation
\[|z| = \sqrt{z \cdot \overline{z}}\]
Alternative to formula \(|z| = \sqrt{a^2 + b^2}\)
3. Extracting real and imaginary parts
\[\text{Re}(z) = \frac{z + \overline{z}}{2}\] \[\text{Im}(z) = \frac{z - \overline{z}}{2i}\]
4. Quadratic equations
If \(z\) is a solution, then \(\overline{z}\) is also a solution for real coefficients.
Complex solutions always occur in pairs!
5. Electrical impedance
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
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.
• 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):
This is the square of the magnitude!
Algebraic rules with conjugation
Conjugation is compatible with basic arithmetic operations:
• \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\) (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:
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: \(\overline{i} = -i\)
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