Properties of Complex Number Modulus

Properties of Modulus Operations

Let $z_1$ and $z_2$ be complex numbers.

Modulus of a Number, its Negative, and its Conjugate

The modulus of a complex number is equal to the modulus of its negative, and also equal to the modulus of its conjugate.

|z_1| = |-z_1| = |\bar{z_1}|

Explanation:

Recall that if $z_1 = x + iy$ , then $-z_1 = -x - iy$ and $\bar{z_1} = x - iy$ .

$|z_1| = \sqrt{x^2 + y^2}$
$|-z_1| = \sqrt{(-x)^2 + (-y)^2} = \sqrt{x^2 + y^2}$
$|\bar{z_1}| = \sqrt{x^2 + (-y)^2} = \sqrt{x^2 + y^2}$

All three yield the same value.

Modulus of Difference

The modulus of the difference of two complex numbers is the same if the order is reversed.

|z_1 - z_2| = |z_2 - z_1|

Explanation:

This is a direct consequence of the first property. We know $z_1 - z_2 = -(z_2 - z_1)$ . Then:

|z_1 - z_2| = |-(z_2 - z_1)| = |z_2 - z_1|

Square of Modulus

The square of the modulus of a complex number is equal to the complex number multiplied by its conjugate.

|z_1|^2 = z_1 \times \bar{z_1}

Explanation:

If $z_1 = x + iy$ , then $\bar{z_1} = x - iy$ .

z_1 \times \bar{z_1} = (x + iy)(x - iy) = x^2 - (iy)^2 = x^2 - i^2y^2 = x^2 - (-1)y^2 = x^2 + y^2

We also know that $|z_1| = \sqrt{x^2 + y^2}$ , so $|z_1|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2$ .

Thus, both sides are equal.

Modulus of Product

The modulus of the product of two complex numbers is equal to the product of their individual moduli.

|z_1 \times z_2| = |z_1| \times |z_2|

Modulus of Quotient

The modulus of the quotient of two complex numbers is equal to the quotient of their individual moduli (provided the denominator is non-zero).

\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}, \quad \text{for } z_2 \neq 0

Geometrically, if we consider $z_1$ , $z_2$ , and $z_1 + z_2$ as sides of a triangle on the complex plane, this property states that the length of one side ( $|z_1 + z_2|$ ) cannot be greater than the sum of the lengths of the other two sides ( $|z_1| + |z_2|$ ).

Using Modulus Properties

Suppose we are given the complex number $z = \frac{1 - 2i}{3 + 4i}$ . Find $|z|$ !

Solution:

We can view $z = \frac{z_1}{z_2}$ with $z_1 = 1 - 2i$ and $z_2 = 3 + 4i$ .

Using the Modulus of Quotient property:

|z| = \left| \frac{1 - 2i}{3 + 4i} \right| = \frac{|1 - 2i|}{|3 + 4i|}

Now we calculate the moduli of $z_1$ and $z_2$ :

|z_1| = |1 - 2i| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}

|z_2| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Therefore,

|z| = \frac{\sqrt{5}}{5}

This is much easier than first multiplying by the conjugate of the denominator and then calculating the modulus.

Exercise

If $z_1 = 6 + 8i$ and $z_2 = 3 - i$ , calculate $|z_1 \times z_2|$ using the modulus properties.
If $z = 5 - 12i$ , prove that $|z|^2 = z \times \bar{z}$ .

Answer Key

We use the property $|z_1 \times z_2| = |z_1| \times |z_2|$ .

Calculate each modulus:

$|z_1| = |6 + 8i| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$
$|z_2| = |3 - i| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$

Then:

$|z_1 \times z_2| = |z_1| \times |z_2| = 10 \times \sqrt{10} = 10\sqrt{10}$
Given $z = 5 - 12i$ .

Calculate the left side ( $|z|^2$ ):

$|z| = |5 - 12i| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$
$|z|^2 = 13^2 = 169$

Calculate the right side ( $z \times \bar{z}$ ):

The conjugate of $z$ is $\bar{z} = 5 + 12i$ .

$z \times \bar{z} = (5 - 12i)(5 + 12i) = 5^2 - (12i)^2 = 25 - 144i^2 = 25 - 144(-1) = 25 + 144 = 169$

Since the left side (169) equals the right side (169), the statement $|z|^2 = z \times \bar{z}$ is proven.

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