Complex Number Conjugate

What is a Complex Number Conjugate?

Every complex number $z = x + iy$ has a "pair" called the conjugate. The conjugate of $z$ is written with the symbol $\bar{z}$ .

Complex Numbers

Complex Number Conjugate

What is a Complex Number Conjugate?

Every complex number $z = x + iy$ has a "pair" called the conjugate. The conjugate of $z$ is written with the symbol $\bar{z}$ .

Getting the conjugate is very easy: just change the sign of the imaginary part.

Formal Definition

If $z = x + iy$ is a complex number, with $x$ as the real part and $y$ as the imaginary part, then its conjugate is:

\bar{z} = x - iy

This means the real part ( $x$ ) stays the same, while the sign of the imaginary part ( $y$ ) is flipped (positive becomes negative, negative becomes positive).

Examples of Finding the Conjugate

Let's look at some examples:

If $z = 2 + i$

Here, $x=2$ and $y=1$ .

Then its conjugate is $\bar{z} = 2 - i$ . (The sign of the imaginary part $+1$ becomes $-1$ )
If $z = 2$

We can write $z = 2 + 0i$ . Here, $x=2$ and $y=0$ .
If $z = 3 - 2i$

Here, $x=3$ and $y=-2$ .

Then its conjugate is . (The sign of the imaginary part becomes )
If $z = 3i$

We can write $z = 0 + 3i$ . Here, $x=0$ and $y=3$ .

Visualization of the Conjugate

Geometrically, the conjugate $\bar{z}$ is the reflection of $z$ across the real axis ( $x$ -axis) in the complex plane.

Visualization of

z = 3+2i

and its Conjugate

\bar{z} = 3-2i

Notice how

z

and

\bar{z}

are like reflections across the real axis.

Complex Number Congruence

Is it possible for a complex number $z=x+iy$ to be equal to its conjugate $\bar{z}=x-iy$ ? If so, what is the condition?

Answer:

Yes, it's possible. For $z = \bar{z}$ , then:

x+iy = x-iy

This can only happen if $iy = -iy$ , which means $2iy = 0$ .

Since $i \neq 0$ , it must be that $y=0$ .

So, a complex number is equal to its conjugate if and only if its imaginary part is zero, or in other words, if the complex number is a real number.

Properties of Conjugate Operations

The conjugate operation has several interesting properties that are useful in calculations. Let $z, z_1,$ and $z_2$ be any complex numbers.

Sum and Difference

The conjugate of the sum (or difference) of two complex numbers is equal to the sum (or difference) of their conjugates.

\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}

Product and Quotient

The conjugate of the product (or quotient) of two complex numbers is equal to the product (or quotient) of their conjugates.

\overline{z_1 \times z_2} = \bar{z_1} \times \bar{z_2}

Inverse

The conjugate of the inverse of a complex number is equal to the inverse of its conjugate.

\overline{z^{-1}} = (\bar{z})^{-1}

Double Conjugate

Taking the conjugate twice returns the complex number to its original form.

\overline{(\bar{z})} = z

Relationship with Real and Imaginary Parts

Adding and subtracting a complex number with its conjugate yields interesting relationships with its real and imaginary parts:

z + \bar{z} = 2 \text{Re}(z)

Multiplication by Conjugate

Multiplying a complex number by its conjugate yields the square of its modulus (a non-negative real number).

z \times \bar{z} = (\text{Re}(z))^2 + (\text{Im}(z))^2 = |z|^2

Exercise

Find the conjugate of each of the following complex numbers!

$2+i^2$
$1+\frac{1}{i}$
$1+2i$

Answer Key

First, simplify the complex number:

$z = 2+i^2 = 2+(-1) = 1$ . Since $z=1$ is a real number ( $1+0i$ ),

its conjugate is $\bar{z} = 1$ .
Simplify first:

Remember that

$\frac{1}{i} = \frac{1}{i} \times \frac{-i}{-i} = \frac{-i}{-i^2} = \frac{-i}{-(-1)} = \frac{-i}{1} = -i$
$z = 1+2i$ .

Directly use the definition: $\bar{z} = 1 - 2i$ .

Then its conjugate is $\bar{z} = 2 - 0i = 2$ . (The imaginary part is $0$ , its sign doesn't change)

The conjugate of a real number is the real number itself.

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

-2

+2

Then its conjugate is $\bar{z} = 0 - 3i = -3i$ . (The sign of the imaginary part $+3$ becomes $-3$ )

The conjugate of a purely imaginary number is its negative.

So, $z = 1 + \frac{1}{i} = 1 - i$ .

Its conjugate is $\bar{z} = 1 - (-i) = 1 + i$ .