Table of Contents
What is Absolute Value?
The absolute value of a complex number \(a + bi\) is the distance of the number from the origin (0,0) in the complex plane and is denoted by \(|a + bi|\). The formula for the absolute value of a complex number is given by:
\(|a + bi|\) = \(\sqrt{a^2 + b^2}\)
Here, a is the real part, b is the imaginary part, and \(\sqrt{ }\) denotes the square root.
Absolute Value Definition
The absolute value of a number is a mathematical concept that represents the distance of that number from zero on the number line, regardless of its direction. It provides a measure of the magnitude or size of a real or complex number. For a real number aa, the absolute value is denoted by \(|a|\), and for a complex number a + bia+bi, the absolute value is denoted by \(|a + bi|\).
Absolute Value Symbol
The symbol used to represent the absolute value of a number is vertical bars or pipes surrounding the number. For a real number a, it is written as \(|a|\), and for a complex number \( a + bi\), it is written as \( |a + bi|\).
Real Numbers:
For real numbers, the absolute value is straightforward. For example:
- |5| = 5 because 5 units away from zero in the positive direction.
- −7 is 7 units away from zero in the negative direction, so |-7| = 7.
Complex Numbers
For complex numbers a + bi, where a and b are real numbers, the absolute value is calculated using the formula:
\(|a + bi| \) = \(\sqrt{a^2 + b^2}\). This formula represents the distance of the complex number from the origin in the complex plane.
Properties:
- The absolute value is always a non-negative real number.
- |a| = a if a is non-negative or zero; otherwise, |a| = -a if a is negative.
- The absolute value of the product of two numbers is the product of their absolute values: \(|ab| = |a|\) \(\cdot |b∣\).
