The magnitude of two complex numbers multiplied is equal to the product of their individual magnitudes.
Understanding Complex Numbers and Magnitudes
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. The magnitude of a complex number, also known as its modulus, represents its distance from the origin in the complex plane.
Multiplying Complex Numbers
When multiplying two complex numbers, you multiply each term of the first number by each term of the second number, remembering that i² = -1. This results in a new complex number.
Magnitude of the Product
The magnitude of the product of two complex numbers is found by:
- Calculating the magnitudes of each individual complex number:
- For a complex number a + bi, the magnitude is calculated as √(a² + b²).
- Multiplying the two magnitudes together:
- The product of the magnitudes of the two complex numbers is the magnitude of their product.
Example
Let's consider two complex numbers:
- z₁ = 3 + 4i
- z₂ = 2 - i
- Calculate the magnitudes:
- |z₁| = √(3² + 4²) = 5
- |z₂| = √(2² + (-1)²) = √5
- Multiply the magnitudes:
- |z₁| |z₂| = 5 √5 = 5√5
Therefore, the magnitude of the product of z₁ and z₂ is 5√5.
Practical Insights
This concept is important in various fields, including:
- Electrical Engineering: Complex numbers are used to represent alternating currents and voltages, and their magnitudes represent the amplitude of these signals.
- Signal Processing: Complex numbers are used to analyze and manipulate signals, and their magnitudes represent the strength of the signals.
- Quantum Mechanics: Complex numbers are used to describe the wave function of particles, and their magnitudes represent the probability of finding the particle in a given state.
