Properties of Complex Number Modulus

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.

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.

Explanation:

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

Square of Modulus

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

Explanation:

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

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.

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).

Triangle Inequality

The modulus of the sum of two complex numbers is less than or equal to the sum of their individual moduli.

Explanation:

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:

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

Therefore,

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

Exercise

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

Answer Key

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

   Calculate each modulus:

   $$|z_1| = |6 + 8i| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$

   Then:

   $$|z_1 \times z_2| = |z_1| \times |z_2| = 10 \times \sqrt{10} = 10\sqrt{10}$$

2. Given $z = 5 - 12i$.

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

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

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

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

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