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Complex Conjugate — Definition, Formula & Examples

Complex Conjugate

The complex conjugate of a + bi  is a – bi, and similarly the complex conjugate of a – bi  is a + bi. This consists of changing the sign of the imaginary part of a complex number. The real part is left unchanged.

Complex conjugates are indicated using a horizontal line over the number or variable. For example, Math equation showing the complex conjugate notation: 3 + 7i = 3 − 7i, with a horizontal bar over the left side..

Note: Complex conjugates are similar to, but not the same as, conjugates.

Expression Lowercase italic letter z, representing a complex number variable used in complex conjugate notation. Complex Conjugate z-bar (z with a horizontal overline), representing the complex conjugate notation of variable z
5 – 2i
4i + 1
–5i
12
Math expression showing 2 + √6, an example of a complex conjugate expression.
5 + 2i
–4i + 1
5i
12
Math expression showing 2 + √6, an example of a complex conjugate expression.

 

See also

Complex number formulas

Key Formula

If z=a+bi, then zˉ=a−bi\text{If } z = a + bi, \text{ then } \bar{z} = a - bi
Where:
  • zz = A complex number
  • zˉ\bar{z} = The complex conjugate of z
  • aa = The real part of the complex number
  • bb = The coefficient of the imaginary part
  • ii = The imaginary unit, where i² = −1

Worked Example

Problem: Find the complex conjugate of z=3+4iz = 3 + 4i, then multiply zz by its conjugate.
Step 1: Identify the real part and the imaginary part. Here, the real part is 3 and the imaginary part is 4i.
z=3+4i⇒a=3,  b=4z = 3 + 4i \quad \Rightarrow \quad a = 3,\; b = 4
Step 2: Change the sign of the imaginary part to form the conjugate.
zˉ=3−4i\bar{z} = 3 - 4i
Step 3: Multiply z by its conjugate using the distributive property (FOIL).
z⋅zˉ=(3+4i)(3−4i)z \cdot \bar{z} = (3 + 4i)(3 - 4i)
Step 4: Apply the difference-of-squares pattern: (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2.
=32+42=9+16=25= 3^2 + 4^2 = 9 + 16 = 25
Answer: The complex conjugate of 3+4i3 + 4i is 3−4i3 - 4i. Their product is 2525, a real number.

Another Example

This example shows the most common practical use of the complex conjugate: multiplying the numerator and denominator of a fraction to eliminate the imaginary part from the denominator (rationalizing).

Problem: Use the complex conjugate to simplify the fraction 2+i3−2i\dfrac{2 + i}{3 - 2i}.
Step 1: Identify the conjugate of the denominator. The denominator is 3−2i3 - 2i, so its conjugate is 3+2i3 + 2i.
3−2i‾=3+2i\overline{3 - 2i} = 3 + 2i
Step 2: Multiply both the numerator and denominator by this conjugate.
2+i3−2i⋅3+2i3+2i\frac{2 + i}{3 - 2i} \cdot \frac{3 + 2i}{3 + 2i}
Step 3: Expand the numerator using FOIL.
(2+i)(3+2i)=6+4i+3i+2i2=6+7i−2=4+7i( 2 + i)(3 + 2i) = 6 + 4i + 3i + 2i^2 = 6 + 7i - 2 = 4 + 7i
Step 4: Simplify the denominator using the difference-of-squares pattern.
(3−2i)(3+2i)=9+4=13(3 - 2i)(3 + 2i) = 9 + 4 = 13
Step 5: Write the result in standard form a+bia + bi.
4+7i13=413+713 i\frac{4 + 7i}{13} = \frac{4}{13} + \frac{7}{13}\,i
Answer: 2+i3−2i=413+713 i\dfrac{2 + i}{3 - 2i} = \dfrac{4}{13} + \dfrac{7}{13}\,i

Frequently Asked Questions

What is the difference between a conjugate and a complex conjugate?
A conjugate (sometimes called a radical conjugate) changes the sign between two terms in a binomial, such as turning a+ba + \sqrt{b} into a−ba - \sqrt{b}. A complex conjugate specifically changes the sign of the imaginary part of a complex number. So while 3+53 + \sqrt{5} and 3−53 - \sqrt{5} are conjugates, 3+5i3 + 5i and 3−5i3 - 5i are complex conjugates. The underlying idea—flipping one sign to create a useful product—is the same, but the terms apply to different types of expressions.
What happens when you multiply a complex number by its conjugate?
The result is always a non-negative real number. Specifically, (a+bi)(a−bi)=a2+b2(a + bi)(a - bi) = a^2 + b^2. The imaginary parts cancel out completely. This property is why conjugates are used to rationalize denominators that contain complex numbers.
What is the complex conjugate of a real number?
A real number is its own complex conjugate. For example, the conjugate of 77 is just 77, because you can write 77 as 7+0i7 + 0i, and changing the sign of 0i0i gives 7−0i=77 - 0i = 7. Similarly, the conjugate of −3-3 is −3-3.

Complex Conjugate vs. Conjugate (Radical Conjugate)

Complex ConjugateConjugate (Radical Conjugate)
DefinitionChanges the sign of the imaginary part: a + bi → a − biChanges the sign between two terms: a + √b → a − √b
Applies toComplex numbers involving iExpressions involving radicals or mixed terms
Product resulta² + b² (sum of squares, always real)a² − b (difference, eliminates the radical)
Primary useRationalizing complex denominators; finding modulusRationalizing radical denominators

Why It Matters

Complex conjugates appear whenever you need to divide complex numbers, since multiplying by the conjugate removes ii from the denominator. They are also essential for finding the modulus (absolute value) of a complex number: ∣z∣=z⋅zˉ|z| = \sqrt{z \cdot \bar{z}}. In later courses such as signal processing, quantum mechanics, and differential equations, conjugate pairs arise constantly when analyzing polynomial roots and wave behavior.

Common Mistakes

Mistake: Changing the sign of both the real and imaginary parts, writing the conjugate of 3+4i3 + 4i as −3−4i-3 - 4i.
Correction: Only the sign of the imaginary part changes. The real part stays the same. The correct conjugate of 3+4i3 + 4i is 3−4i3 - 4i.
Mistake: Forgetting that i2=−1i^2 = -1 when multiplying a complex number by its conjugate, leading to a2−b2a^2 - b^2 instead of a2+b2a^2 + b^2.
Correction: When you expand (a+bi)(a−bi)(a + bi)(a - bi), you get a2−(bi)2=a2−b2i2=a2−b2(−1)=a2+b2a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 - b^2(-1) = a^2 + b^2. The negative from i2i^2 flips the subtraction to addition.

Related Terms