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Complex conjugate

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Algebra and Trigonometry

Definition

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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5 Must Know Facts For Your Next Test

  1. If $z = a + bi$, then its complex conjugate is denoted as $\overline{z} = a - bi$.
  2. Multiplying a complex number by its conjugate results in a real number: $(a+bi)(a-bi) = a^2 + b^2$.
  3. The modulus of a complex number can be found using its conjugate: $|z| = \sqrt{z \cdot \overline{z}}$.
  4. Complex conjugates are used to rationalize denominators in fractions with complex numbers.
  5. 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?
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