In mathematics, the absolute value of a complex number is a measure of its distance from the origin in the complex plane. It is also known as the modulus or magnitude of the complex number. The absolute value of a complex number is always a non-negative real number.
To understand the concept of absolute value of a complex number, one should have knowledge of complex numbers, the complex plane, and the distance formula.
The formula to calculate the absolute value of a complex number z = a + bi, where a and b are real numbers, is given by:
|z| = √(a^2 + b^2)
To apply the formula, substitute the real and imaginary parts of the complex number into the formula and perform the necessary calculations. The result will be the absolute value of the complex number.
The symbol used to represent the absolute value of a complex number is two vertical bars surrounding the complex number. For example, |z| represents the absolute value of the complex number z.
There are a few methods to find the absolute value of a complex number:
Example 1: Find the absolute value of the complex number z = 3 + 4i.
Solution: Using the formula, we substitute a = 3 and b = 4 into the formula:
|z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
Therefore, the absolute value of the complex number z = 3 + 4i is 5.
Q: Can the absolute value of a complex number be negative? A: No, the absolute value of a complex number is always a non-negative real number.
Q: What is the geometric interpretation of the absolute value of a complex number? A: Geometrically, the absolute value of a complex number represents its distance from the origin in the complex plane.
Q: How is the absolute value of a complex number related to its conjugate? A: The absolute value of a complex number is equal to the square root of the product of the complex number and its conjugate.