Complex conjugate

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 $a - bi$.

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?

Related terms

Imaginary Unit:

$i$, defined as $\sqrt{-1}$, used to express imaginary numbers.

Modulus:

The magnitude or absolute value of a complex number, given by $\sqrt{a^2 + b^2}$ for $z = a + bi$.

Real Part: The component $a$ in the complex number $a+bi$, representing its projection on the real axis.