The complex conjugate of a complex number $a + bi$ is $a - bi$. Conjugation changes the sign of the imaginary part while keeping the real part unchanged.
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If $z = a + bi$, then its complex conjugate is denoted as $\overline{z} = a - bi$.
Multiplying a complex number by its conjugate results in a real number: $(a+bi)(a-bi) = a^2 + b^2$.
The modulus of a complex number can be found using its conjugate: $|z| = \sqrt{z \cdot \overline{z}}$.
Complex conjugates are used to rationalize denominators in fractions with complex numbers.
For any two complex numbers $z_1$ and $z_2$, $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$ and $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$.
Review Questions
What is the complex conjugate of $4 - 3i$?
How do you find the modulus of a complex number using its conjugate?
Why is multiplying a complex number by its conjugate useful?
Related terms
Imaginary Unit: $i$, defined as $\sqrt{-1}$, satisfying $i^2 = -1$.
Modulus: The modulus of a complex number $a + bi$ is given by $|a + bi| = \sqrt{a^2 + b^2}$.
Rationalizing the Denominator: A process involving multiplication by the conjugate to eliminate imaginary parts from the denominator.