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:

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:

1. 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 $-1$)

2. If $z = 2$

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

   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.

3. If $z = 3 - 2i$

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

   Then its conjugate is $\bar{z} = 3 - (-2i) = 3 + 2i$. (The sign of the imaginary part $-2$ becomes $+2$)

4. If $z = 3i$

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

   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.

Visualization of the Conjugate

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

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:

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.

Product and Quotient

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

Inverse

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

Double Conjugate

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

Relationship with Real and Imaginary Parts

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

Multiplication by Conjugate

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

Exercise

Find the conjugate of each of the following complex numbers!

1. $2+i^2$
2. $1+\frac{1}{i}$
3. $1+2i$

Answer Key

1. First, simplify the complex number:

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

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

2. Simplify first:

   Remember that

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

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

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

3. $z = 1+2i$.

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