Properties of modulus
Properties of Modulus
The modulus of a complex number is a fundamental concept in complex analysis and has several important properties. A complex number is usually denoted by $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property $i^2 = -1$. The modulus of a complex number $z$, denoted by $|z|$, is defined as the distance of the point representing the complex number from the origin in the complex plane. Mathematically, it is given by:
$$|z| = \sqrt{a^2 + b^2}$$
Properties of Modulus
The modulus of a complex number has several properties that are useful in various mathematical analyses. Here are some of the key properties:
- Non-negativity: The modulus of a complex number is always non-negative.
- Multiplicativity: The modulus of the product of two complex numbers is the product of their moduli.
- Triangle Inequality: The modulus of the sum of two complex numbers is less than or equal to the sum of their moduli.
- Conjugate Property: The modulus of a complex number is equal to the modulus of its conjugate.
- Multiplicative Inverse: The modulus of the reciprocal of a non-zero complex number is the reciprocal of the modulus of the number.
Let's explore these properties in more detail, with formulas and examples.
Non-negativity
For any complex number $z$, the modulus $|z|$ is always greater than or equal to zero.
$$|z| \geq 0$$
Multiplicativity
For any two complex numbers $z_1$ and $z_2$, the modulus of their product is equal to the product of their moduli.
$$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$$
Triangle Inequality
For any two complex numbers $z_1$ and $z_2$, the modulus of their sum satisfies the triangle inequality.
$$|z_1 + z_2| \leq |z_1| + |z_2|$$
Conjugate Property
For any complex number $z$, the modulus of $z$ is equal to the modulus of its conjugate $\bar{z}$.
$$|z| = |\bar{z}|$$
Multiplicative Inverse
For any non-zero complex number $z$, the modulus of its multiplicative inverse is the reciprocal of the modulus of $z$.
$$\left|\frac{1}{z}\right| = \frac{1}{|z|}$$
Table of Properties
| Property | Description | Formula | Example |
|---|---|---|---|
| Non-negativity | Modulus is always non-negative | $ | z |
| Multiplicativity | Modulus of a product is the product of moduli | $ | z_1 \cdot z_2 |
| Triangle Inequality | Modulus of a sum is less than or equal to the sum of moduli | $ | z_1 + z_2 |
| Conjugate Property | Modulus of a number is equal to the modulus of its conjugate | $ | z |
| Multiplicative Inverse | Modulus of the reciprocal is the reciprocal of the modulus | $\left | \frac{1}{z}\right |
Examples
Example 1: Non-negativity
Consider the complex number $z = -3 + 4i$. The modulus of $z$ is:
$$|z| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
As expected, the modulus is non-negative.
Example 2: Multiplicativity
Let $z_1 = 1 + i$ and $z_2 = 2 - 3i$. Then:
$$|z_1 \cdot z_2| = |(1 + i) \cdot (2 - 3i)| = |2 - 3i + 2i - 3i^2| = |5 - i| = \sqrt{25 + 1} = \sqrt{26}$$
And:
$$|z_1| \cdot |z_2| = \sqrt{1^2 + 1^2} \cdot \sqrt{2^2 + (-3)^2} = \sqrt{2} \cdot \sqrt{13} = \sqrt{26}$$
Hence, the property is verified.
Example 3: Triangle Inequality
Consider $z_1 = 1 + i$ and $z_2 = 2 - 2i$. Then:
$$|z_1 + z_2| = |(1 + i) + (2 - 2i)| = |3 - i| = \sqrt{9 + 1} = \sqrt{10}$$
And:
$$|z_1| + |z_2| = \sqrt{1^2 + 1^2} + \sqrt{2^2 + (-2)^2} = \sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$$
Since $\sqrt{10} < 3\sqrt{2}$, the triangle inequality holds.
Example 4: Conjugate Property
For $z = 3 - 4i$, the conjugate is $\bar{z} = 3 + 4i$. The modulus of both is:
$$|z| = |\bar{z}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
Example 5: Multiplicative Inverse
For $z = 1 + i$, the multiplicative inverse is $\frac{1}{z} = \frac{1}{1 + i}$. The modulus of the inverse is:
$$\left|\frac{1}{z}\right| = \left|\frac{1}{1 + i}\right| = \left|\frac{1 - i}{(1 + i)(1 - i)}\right| = \left|\frac{1 - i}{1 + 1}\right| = \left|\frac{1 - i}{2}\right| = \frac{1}{\sqrt{2}}$$
And the reciprocal of the modulus of $z$ is:
$$\frac{1}{|z|} = \frac{1}{\sqrt{1^2 + 1^2}} = \frac{1}{\sqrt{2}}$$
Thus, the property is confirmed.
Understanding these properties of modulus is essential for solving complex number problems and is particularly useful in the fields of complex analysis, signal processing, and other areas of mathematics and engineering.