Modulus
Understanding the Modulus in Complex Numbers
The modulus of a complex number is a concept that extends the idea of absolute value from real numbers to complex numbers. It is a measure of the distance of a complex number from the origin in the complex plane.
Definition
For a complex number $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with the property $i^2 = -1$, the modulus of $z$ is denoted by $|z|$ and is defined as:
$$|z| = \sqrt{a^2 + b^2}$$
This formula is derived from the Pythagorean theorem, as the modulus represents the hypotenuse of a right-angled triangle with sides of lengths $a$ and $b$.
Properties of Modulus
The modulus has several important properties:
- Non-negative: $|z| \geq 0$
- Multiplicative: $|zw| = |z||w|$ for any complex numbers $z$ and $w$
- Conjugate: $|z| = |\bar{z}|$, where $\bar{z}$ is the conjugate of $z$
- Triangle Inequality: $|z + w| \leq |z| + |w|$
- Zero Modulus: $|z| = 0$ if and only if $z = 0$
Table of Differences and Important Points
| Property | Description | Formula | Example |
|---|---|---|---|
| Non-negative | The modulus is always a non-negative real number. | $ | z |
| Multiplicative | The modulus of a product is the product of the moduli. | $ | zw |
| Conjugate | The modulus of a complex number is equal to the modulus of its conjugate. | $ | z |
| Triangle Inequality | The modulus of a sum is less than or equal to the sum of the moduli. | $ | z + w |
| Zero Modulus | The modulus of a complex number is zero if and only if the number itself is zero. | $ | z |
Examples to Explain Important Points
Example 1: Non-negative Property
Consider the complex number $z = -3 + 4i$. Its modulus is calculated as:
$$|z| = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
Notice that despite the real part being negative, the modulus is non-negative.
Example 2: Multiplicative Property
Let $z = 1 + i$ and $w = 2 - i$. Then:
$$|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$$ $$|w| = \sqrt{2^2 + (-1)^2} = \sqrt{5}$$
Now, the modulus of the product $zw$ is:
$$|zw| = |(1 + i)(2 - i)| = |2 - i + 2i - i^2| = |3 + i| = \sqrt{3^2 + 1^2} = \sqrt{10}$$
And indeed, $\sqrt{2} \cdot \sqrt{5} = \sqrt{10}$, confirming the multiplicative property.
Example 3: Triangle Inequality
Take $z = 1 + i$ and $w = 2 - 2i$. We have:
$$|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$$ $$|w| = \sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$$
Now, consider the sum $z + w$:
$$|z + w| = |(1 + i) + (2 - 2i)| = |3 - i| = \sqrt{3^2 + (-1)^2} = \sqrt{10}$$
Comparing the sum of the moduli and the modulus of the sum:
$$|z| + |w| = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$$
$$|z + w| = \sqrt{10}$$
Since $\sqrt{10} < 3\sqrt{2}$, the triangle inequality holds.
Example 4: Zero Modulus
For the complex number $z = 0 + 0i$, the modulus is:
$$|z| = \sqrt{0^2 + 0^2} = 0$$
This is the only case where the modulus is zero, which corresponds to the point at the origin of the complex plane.
Understanding the modulus of complex numbers is crucial for various applications in mathematics, physics, and engineering, where complex numbers are used to represent oscillations, waves, and other phenomena.