Modulus and Conjugate of a Complex Number

The Conjugate of a Complex Number

The conjugate of a complex number $z = a+ib$ is obtained by changing the sign of its imaginary part. It is denoted by $\bar{z}$.

$$\bar{z} = a - ib$$

Properties of the Conjugate:

• $\overline{(\bar{z})} = z$
• $z + \bar{z} = 2 \text{Re}(z)$
• $z - \bar{z} = 2i \text{Im}(z)$
• $z = \bar{z} \iff z$ is purely real.
• $z + \bar{z} = 0 \iff z$ is purely imaginary.
• $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$
• $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$
• $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$ (provided $z_2 \ne 0$)

The Modulus of a Complex Number

The modulus of a complex number $z = a+ib$ is its distance from the origin in the Argand plane. It is a non-negative real number denoted by $|z|$.

$$|z| = \sqrt{a^2 + b^2} = \sqrt{(\text{Re}(z))^2 + (\text{Im}(z))^2}$$

Properties of the Modulus:

• $|z| \ge 0$, and $|z|=0 \iff z=0$.
• $z \bar{z} = |z|^2$ (This is a very important property used for division and finding inverses).
• $|z| = |\bar{z}| = |-z| = |-\bar{z}|$
• $|z_1 z_2| = |z_1| |z_2|$
• $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$ (provided $z_2 \ne 0$)
• Triangle Inequality:
• $|z_1 + z_2| \le |z_1| + |z_2|$
• $|z_1 - z_2| \ge ||z_1| - |z_2||$

Example 1: Finding Modulus and Conjugate

Question: Find the modulus and conjugate of $z = -3 + 4i$.

Solution:

• Conjugate: We change the sign of the imaginary part. $\bar{z} = -3 - 4i$.
• Modulus: $|z| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Example 2: Using Properties to Find Modulus

Question: Find the modulus of $z = \frac{(1+i)(2-i)}{(1-3i)^2}$.

Solution: Using the properties $|z_1 z_2| = |z_1||z_2|$ and $|z_1/z_2| = |z_1|/|z_2|$:

$$|z| = \frac{|1+i| |2-i|}{|(1-3i)^2|} = \frac{|1+i| |2-i|}{|1-3i|^2}$$

Now, we calculate each modulus: $|1+i| = \sqrt{1^2+1^2} = \sqrt{2}$ $|2-i| = \sqrt{2^2+(-1)^2} = \sqrt{5}$ $|1-3i| = \sqrt{1^2+(-3)^2} = \sqrt{10}$

$$|z| = \frac{\sqrt{2} \cdot \sqrt{5}}{(\sqrt{10})^2} = \frac{\sqrt{10}}{10} = \frac{1}{\sqrt{10}}$$

Example 3: Finding the Multiplicative Inverse

Question: Find the multiplicative inverse of $z = 2-3i$.

Solution: The formula for the inverse is $z^{-1} = \frac{\bar{z}}{|z|^2}$.

• $\bar{z} = 2+3i$
• $|z|^2 = 2^2 + (-3)^2 = 4 + 9 = 13$.

$$z^{-1} = \frac{2+3i}{13} = \frac{2}{13} + i\frac{3}{13}$$

Example 4: Solving Equations with Conjugates

Question: If $3z + 2\bar{z} = 5+i$, find the complex number z.

Solution: Let $z = x+iy$. Then $\bar{z} = x-iy$. Substitute these into the equation:

$$3(x+iy) + 2(x-iy) = 5+i$$

$$(3x+3iy) + (2x-2iy) = 5+i$$

Group the real and imaginary parts:

$$(3x+2x) + i(3y-2y) = 5+i$$

$$5x + iy = 5+i$$

Equating the real and imaginary parts: $5x = 5 \implies x=1$. $y = 1$. Therefore, $z = 1+i$.