Complex Conjugate and Modulus

1) Identify the complex number

A complex number is usually written as [ z=a+bi ] where (a) is the real part and (b) is the imaginary coefficient.

2) Find the conjugate

The conjugate changes the sign of the imaginary part: [ \overline{z}=a-bi ] So if (z=4-3i), then (\overline{z}=4+3i).

3) Find the modulus

The modulus measures the distance from the origin in the complex plane: [ |z|=\sqrt{a^2+b^2} ] For (z=4-3i), [ |z|=\sqrt{4^2+(-3)^2}=\sqrt{25}=5. ]

4) Simplify carefully

Keep radicals simplified and write the conjugate with the correct sign change. If the modulus appears squared, use [ |z|^2=a^2+b^2. ]

5) Quick check

• The conjugate should have the same real part and opposite imaginary sign.
• The modulus should always be a nonnegative real number.
• For a good check, verify that (z\overline{z}=|z|^2).