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Complex conjugate

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College Algebra

Definition

A complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If the complex number is $a + bi$, its complex conjugate is $a - bi$.

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5 Must Know Facts For Your Next Test

  1. The complex conjugate of $z = a + bi$ 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 sum of a complex number and its conjugate is always real: $(a+bi) + (a-bi) = 2a$.
  4. The product of a complex number and its conjugate gives the modulus squared: $|z|^2 = z \cdot \overline{z}$.
  5. Complex conjugates are used to rationalize denominators in fractions involving complex numbers.

Review Questions

  • What is the complex conjugate of $3 - 4i$?
  • How do you find the product of a complex number and its conjugate?
  • Why are complex conjugates useful in simplifying expressions involving division?
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