Conjugate Complex Numbers

Reviewed by the Helping with Math Team · Updated 14 October 2026

Introduction

Each new kind of number solves equations the old kind could not. x + 5 = 2 has no answer in the natural numbers, so we need the integers (x = −3). 2x = 3 needs the rational numbers (x = 3/2). x² = 2 needs irrational numbers (x = √2), and together these make the real numbers.

But x² + 1 = 0 has no real answer, because no real number squared is negative. Mathematicians invented the imaginary unit i, with i² = −1 (Euler introduced the letter i). Numbers of the form a + bi, with a and b real, are called complex numbers. Here a is the real part, Re(z), and b is the imaginary part, Im(z).

Definition

The complex conjugate of z = a + bi is the complex number with the same real part and the opposite imaginary part: z̄ = a − bi. On the complex plane, z̄ is the mirror image of z in the horizontal (real) axis.

The word “conjugate” is used the same way elsewhere in algebra: a + √b and a − √b are conjugates, and multiplying them removes the square root, just as multiplying complex conjugates removes i.

How do we write Conjugate of a Complex Number?

If z = a + bi, then its conjugate is z̄ = a − bi. Some books write z* instead of z̄. To find a conjugate, change the sign in front of i (replace i by −i).

• z = 3 + 4i → z̄ = 3 − 4i
• z = −2 − 5i → z̄ = −2 + 5i
• z = 7 (real) → z̄ = 7
• z = 6i (imaginary) → z̄ = −6i

Properties of Conjugate of Complex Numbers

Let z = a + bi, z₁ and z₂ be complex numbers. Then:

1. Conjugate twice gives back z. The conjugate of z̄ = a − bi is a + bi = z.
2. Sum: z + z̄ = (a + bi) + (a − bi) = 2a = 2 Re(z), a real number.
3. Difference: z − z̄ = (a + bi) − (a − bi) = 2bi = 2i Im(z).
4. Real numbers: z = z̄ exactly when b = 0, so z is real.
5. Imaginary numbers: z̄ = −z (equivalently z + z̄ = 0) exactly when a = 0, so z is purely imaginary.
6. Conjugate of a sum: the conjugate of z₁ + z₂ equals z̄₁ + z̄₂.
7. Conjugate of a product: the conjugate of z₁z₂ equals z̄₁ z̄₂.
8. Conjugate of a quotient: the conjugate of z₁/z₂ equals z̄₁/z̄₂ (z₂ ≠ 0).
9. Product with the conjugate: zz̄ = a² + b² = |z|², which is real and never negative.

Check with z₁ = 1 + 2i and z₂ = 3 − i: z₁z₂ = 3 − i + 6i − 2i² = 5 + 5i, whose conjugate is 5 − 5i. And z̄₁z̄₂ = (1 − 2i)(3 + i) = 3 + i − 6i − 2i² = 5 − 5i. They match.

Argand Diagram of a Conjugate of a Complex Number

The complex number z = x + yi is drawn as the point P(x, y) on the complex plane, also called the Argand plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis, at right angles (perpendicular).

1. A real number (y = 0) lies on the real axis.
2. A purely imaginary number (x = 0) lies on the imaginary axis.
3. The distance from the origin to P is the modulus, |z| = √(x² + y²), the absolute value of z.
4. The angle from the positive real axis to OP is the argument, arg(z).

The conjugate z̄ = x − yi is the point (x, −y): the reflection of z in the real axis. It has the same modulus as z and the opposite argument.

Real part a

3

Imaginary part b

4

z = 3 + 4i, z̄ = 3 − 4i. z + z̄ = 6. z × z̄ = 9 + 16 = 25. |z| = 5. arg z = 53.1°, arg z̄ = −53.1°.

Argument or Amplitude of a Complex number for different signs of real and imaginary parts

Let θ be the acute angle with tan θ = |y/x|. The argument of z = x + yi (taking −π < arg z ≤ π) depends on the quadrant of P:

Example: z = −1 + i is in the second quadrant with tan θ = 1, so θ = π/4 and arg z = π − π/4 = 3π/4 (135°). Its conjugate −1 − i has argument −3π/4.

Multiplication of Complex Conjugate

For z₁ = a + bi and z₂ = c + di, multiply out like two binomials and use i² = −1:

(a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i

When the second number is the conjugate of the first (c = a, d = −b), the imaginary parts cancel:

(a + bi)(a − bi) = a² − (bi)² = a² − b²i² = a² + b²

Example: (3 + 4i)(3 − 4i) = 9 + 16 = 25, and |3 + 4i| = √25 = 5.

Dividing complex numbers. This is the main use of the conjugate. Multiply the top and bottom by the conjugate of the bottom, which makes the bottom real:

(2 + 3i)/(1 − i) = (2 + 3i)(1 + i)/((1 − i)(1 + i)) = (2 + 2i + 3i + 3i²)/(1 + 1) = (−1 + 5i)/2 = −1/2 + (5/2)i

Key Facts and Summary

1. The conjugate of z = a + bi is z̄ = a − bi (also written z*).
2. z̄ is the reflection of z in the real axis: same modulus, opposite argument.
3. z + z̄ = 2 Re(z) and z − z̄ = 2i Im(z).
4. zz̄ = a² + b² = |z|², always a real number.
5. The conjugate of a sum, product or quotient is the sum, product or quotient of the conjugates.
6. To divide by a complex number, multiply the top and bottom by its conjugate.
7. z is real when z = z̄, and purely imaginary when z̄ = −z.

Related worksheets: complex fractions, natural numbers and subtracting whole numbers and decimals.

Complex conjugate FAQs

Practice with worksheets