The modulus of a complex number is its distance from the origin in the complex plane. It's a non-negative real number that represents the magnitude or absolute value of the complex number.
Understanding Complex Numbers
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
- Real Part: The real part of a complex number is denoted by a.
- Imaginary Part: The imaginary part of a complex number is denoted by b.
Calculating the Modulus
The modulus of a complex number z = a + bi is calculated using the Pythagorean theorem:
|z| = √(a² + b²)Examples
- Example 1: The complex number z = 3 + 4i has a modulus of:
- |z| = √(3² + 4²) = √(9 + 16) = √25 = 5
- Example 2: The complex number z = -2 - i has a modulus of:
- |z| = √((-2)² + (-1)²) = √(4 + 1) = √5
Practical Insights
The modulus of a complex number has applications in various fields, including:
- Electrical Engineering: Representing alternating current (AC) signals.
- Physics: Describing wave phenomena.
- Mathematics: Solving equations and analyzing functions.
Conclusion
The modulus of a complex number is a fundamental concept in mathematics and has practical applications in various fields. It helps us understand the magnitude or absolute value of a complex number, which is crucial for many calculations and analyses.
