Conjugates of Complex Numbers

Changing the Sign of the Imaginary Part

Every complex number $z = x + iy$ has a conjugate, denoted by $\bar{z}$.

It is obtained by reversing the sign of the imaginary part. The real part keeps its value and its sign, so only the term multiplied by $i$ changes.

Definition of the Complex Conjugate

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).

Applying the Conjugate Rule to Different Forms

Apply the definition to several forms of complex numbers:

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.

Equality with the Conjugate

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?

Derivation:

Start with $z = \bar{z}$:

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

Let $z, z_1,$ and $z_2$ be complex numbers. Conjugation obeys the following rules. Each rule has the same source: applying the operation twice returns the original number, and the sign change moves through sums, products, and quotients without touching the real parts.

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 and its conjugate isolates 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 complex number. Two of the three contain a power of $i$ or a division by $i$, so simplify the expression first and read the conjugate from the simplified form.

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

Worked Solutions

1. 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:

   Rationalize the denominator by multiplying by $-i$:

   $$\begin{aligned} \frac{1}{i} &= \frac{1}{i} \times \frac{-i}{-i} \\ &= \frac{-i}{-i^2} \\ &= \frac{-i}{-(-1)} \\ &= \frac{-i}{1} \\ &= -i \end{aligned}$$

   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: $\bar{z} = 1 - 2i$.