01 / 21

Using Conjugates with
Complex Numbers

Master the conjugate of a complex number and learn how to use it to find moduli and quotients with confidence.

Algebra 2

What Is a Complex Number?

A complex number has the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The symbol \( i \) satisfies \( i^2 = -1 \).

Real Part
The value \( a \) on the horizontal axis of the complex plane.
Imaginary Part
The value \( b \) on the vertical axis, multiplied by \( i \).

Visualizing on the Complex Plane

Re Im 1 2 3 -1 -2 1 2 -1 \( z = 3 + 2i \) \( \bar{z} = 3 - 2i \) real axis reflect

Defining the Conjugate

If \( z = a + bi \), then the complex conjugate is:

You simply negate the imaginary part. The real part stays the same.

\( z = 5 + 3i \)
\( \bar{z} = 5 - 3i \)
\( z = -2 - 7i \)
\( \bar{z} = -2 + 7i \)
\( z = 4i \)
\( \bar{z} = -4i \)

Why Do Conjugates Matter?

Conjugates unlock two essential operations with complex numbers:

1
Finding the modulus. Multiplying \( z \cdot \bar{z} \) always gives a real number: \( |z|^2 \).
2
Dividing complex numbers. Multiplying numerator and denominator by the conjugate of the denominator eliminates the imaginary part in the denominator.

Core Identity: \( z \cdot \bar{z} = |z|^2 \)

Re Im \( z = 3 + 4i \) \( a = 3 \) \( b = 4 \) \( |z| = 5 \) \( |z| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \)

The modulus \( |z| = \sqrt{a^2 + b^2} \) measures the distance from the origin to \( z \) in the complex plane.

Dividing with Conjugates

To divide \( \dfrac{z_1}{z_2} \), multiply both numerator and denominator by the conjugate of the denominator:

The denominator becomes a real number \( c^2 + d^2 \), so the result is in standard \( a + bi \) form.

Key Distinctions

Operation Notation Result Type
Conjugate \( \bar{z} = a - bi \) Complex
Modulus \( |z| = \sqrt{a^2 + b^2} \) Real (non-negative)
Modulus squared \( z \cdot \bar{z} = a^2 + b^2 \) Real (non-negative)
Quotient \( \dfrac{z_1}{z_2} = \dfrac{z_1 \bar{z_2}}{|z_2|^2} \) Complex

How to Apply Conjugates

1
Identify the conjugate. For \( z = a + bi \), write \( \bar{z} = a - bi \).
2
For the modulus: Compute \( z \cdot \bar{z} = a^2 + b^2 \), then \( |z| = \sqrt{a^2 + b^2} \).
3
For division: Multiply numerator and denominator by the conjugate of the denominator.
4
Expand and simplify. Use \( i^2 = -1 \) to combine like terms into \( a + bi \) form.

Example 1: Finding the Modulus

Find \( |z| \) for \( z = 5 - 12i \).

1
Write the conjugate: \( \bar{z} = 5 + 12i \)
2
Multiply: \( z \cdot \bar{z} = (5 - 12i)(5 + 12i) = 25 + 144 = 169 \)
3
Take the square root: \( |z| = \sqrt{169} = 13 \)

Example 2: Dividing Complex Numbers

Simplify \( \dfrac{2 + 3i}{1 - i} \).

1
Conjugate of denominator: \( \overline{1 - i} = 1 + i \)
2
Multiply top and bottom: \( \dfrac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)} \)
3
Expand numerator: \( 2 + 2i + 3i + 3i^2 = 2 + 5i - 3 = -1 + 5i \)
4
Simplify denominator: \( 1 + 1 = 2 \)

Example 3: A Tougher Quotient

Simplify \( \dfrac{4 - i}{3 + 2i} \).

1
Conjugate of denominator: \( \overline{3 + 2i} = 3 - 2i \)
2
Numerator: \( (4 - i)(3 - 2i) = 12 - 8i - 3i + 2i^2 = 12 - 11i - 2 = 10 - 11i \)
3
Denominator: \( (3 + 2i)(3 - 2i) = 9 + 4 = 13 \)

Useful Properties

Sum
\( \overline{z_1 + z_2} = \bar{z_1} + \bar{z_2} \)
Product
\( \overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2} \)
Double Conjugate
\( \overline{\bar{z}} = z \)

Extracting Real and Imaginary Parts

You can recover the real and imaginary parts of any complex number using conjugates:

Real Part
\( \text{Re}(z) = \dfrac{z + \bar{z}}{2} \)
Imaginary Part
\( \text{Im}(z) = \dfrac{z - \bar{z}}{2i} \)

Common Mistakes

✕ Negating Both Parts
Writing \( \overline{3+2i} = -3 - 2i \). Only the imaginary part changes sign: the correct conjugate is \( 3 - 2i \).
✕ Forgetting \( i^2 = -1 \)
When expanding \( (a+bi)(c-di) \), the term \( bdi^2 \) becomes \( -bd \), not \( +bd \). Missing the sign flip is a common error.
✕ Wrong Conjugate for Division
Multiplying by the conjugate of the numerator instead of the denominator. Always use the denominator's conjugate.

Summary of Conjugate Uses

Goal Method Formula
Find conjugate Negate imaginary part \( \overline{a+bi} = a - bi \)
Find modulus Multiply by conjugate, take root \( |z| = \sqrt{z \cdot \bar{z}} \)
Divide complex numbers Multiply by conjugate of denominator \( \dfrac{z_1}{z_2} = \dfrac{z_1 \bar{z_2}}{|z_2|^2} \)
Verify real-ness Check \( z = \bar{z} \) \( z \in \mathbb{R} \iff \bar{z} = z \)

Question 1 of 5

What is the conjugate of \( 7 - 3i \)?

Question 2 of 5

What is the modulus of \( z = 6 + 8i \)?

Question 3 of 5

Simplify: \( \dfrac{1 + i}{1 - i} \)

Question 4 of 5

What is the product \( (4 + 5i)(4 - 5i) \)?

Question 5 of 5

Simplify \( \dfrac{3 - 2i}{4 + i} \) into \( a + bi \) form.