The modulus of a complex number represents its distance from the origin on the complex plane. You can find it using the Pythagorean theorem.
Finding the Modulus
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Represent the complex number: A complex number is typically written as a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
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Square the real and imaginary parts: Square the real part (a) and the imaginary part (b).
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Sum the squares: Add the squared values from step 2.
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Take the square root: Find the square root of the sum from step 3. This gives you the modulus of the complex number.
Formula
The modulus of a complex number a + bi is denoted by |a + bi| and calculated as:
| a + bi | = √(a² + b²)
Examples
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Example 1: Find the modulus of the complex number 3 + 4i.
| 3 + 4i | = √(3² + 4²) = √(9 + 16) = √25 = 5
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Example 2: Find the modulus of the complex number -2 - i.
| -2 - i | = √((-2)² + (-1)²) = √(4 + 1) = √5
Practical Insights
The modulus of a complex number is useful in various applications, including:
- Representing the magnitude of a complex quantity: For example, in electrical engineering, the modulus of a complex impedance represents the resistance to the flow of current.
- Calculating the distance between two complex numbers: The modulus of the difference between two complex numbers gives the distance between them on the complex plane.
- Solving complex equations: The modulus of a complex number can be used to simplify complex equations and find solutions.
