91Ó°ÊÓ

To understand the modulus of a complex number, think of it like finding the length of a vector in the complex plane. A complex number is written in the form of \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The modulus is always non-negative and is calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

For example, in the complex number \(z = bi\), the real part \(a\) is 0, and the imaginary part is \(b\). Applying the modulus formula simplifies to:

• \(|z| = \sqrt{0^2 + b^2}\)

This essentially means that for a complex number located only on the imaginary axis, its modulus is the absolute value of its imaginary part.

Imaginary numbers are a fascinating concept. They are numbers that can be written in the form \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This construction allows us to work with numbers that extend beyond the traditional real number line.When the real part \(a = 0\), such as in \(z = bi\), we have a purely imaginary number. Imaginary numbers are essential in fields of engineering and physics, providing a way to represent oscillations, phases, and other phenomena. It is important to handle these numbers carefully, especially when moving to operations like finding the modulus or adding/subtracting them from real numbers.

The term "absolute value" often comes up when discussing the modulus of a complex number. In essence, the absolute value of a number is its distance from zero on the number line, regardless of direction. For real numbers, the absolute value of any number \(b\) can be represented as:

• \(|b| = \sqrt{b^2}\)

This concept is directly related to the modulus of a complex number. When we compute \(|z|\) for a complex number \(z = bi\), and realize it is equal to \(|b|\), we see they share the same principle. This principle states that even though \(b\) could be negative, the modulus, or the absolute value, will always be non-negative. In the given exercise where \(b\) is negative, the modulus equates to \(-b\), ensuring positivity. Understanding absolute value helps reinforce why the modulus of complex numbers behaves the way it does.