AcademyComplex Arithmetic
Academy
Conjugate
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Complex Conjugate
The complex conjugate reflects a complex number across the real axis in the complex plane.
Definition
For \(z = a + bi\), the conjugate is:
Definition
\[z^* = a - bi\]
The imaginary part changes sign while the real part stays the same.
Key Properties
Product
\[z \cdot z^* = a^2 + b^2 = |z|^2\]
Sum
\[z + z^* = 2a = 2\text{Re}(z)\]
Difference
\[z - z^* = 2bi = 2i\text{Im}(z)\]
Conjugate Rules
SumConj
\[(z_1 + z_2)^* = z_1^* + z_2^*\]
ProductConj
\[(z_1 z_2)^* = z_1^* z_2^*\]
QuotientConj
\[\left(\frac{z_1}{z_2}\right)^* = \frac{z_1^*}{z_2^*}\]
Using Conjugate to Divide
The conjugate is essential for making the denominator real:
DivideUsing
\[\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{z_1 z_2^*}{|z_2|^2}\]
In Polar Form
If \(z = re^{i\theta}\), then:
PolarConj
\[z^* = re^{-i\theta}\]
The modulus stays the same, but the angle is negated.