Understanding the Modulus and Argument of Complex Number z i

Introduction

A complex number is an essential concept in mathematics, often used in various fields such as signal processing, control systems, and physics. A complex number is generally written in the form z a bi, where a and b are real numbers, and i represents the imaginary unit that satisfies i^2 -1. This article will explore the modulus and argument of the complex number z i.

Modulus of z i

The modulus of a complex number z a bi, denoted by mod(z), is defined as:

[ mod(z) sqrt{a^2 b^2} ]

For the complex number z i, we can determine its modulus as follows:

Here, a 0 and b 1.

Therefore, the modulus is:

[ mod(i) sqrt{0^2 1^2} sqrt{1} 1 ]

Argument of z i

The argument of a complex number, denoted by arg(z)', is the angle that the vector representing the complex number makes with the positive real axis in the complex plane. This angle is usually measured in radians.

The argument is given by:

[ arg(z) tan^{-1}left(frac{b}{a}right) ]

For z i where a 0 and b 1, we have:

The point (0, 1) lies on the positive imaginary axis, which is 90^u00b0 or (frac{u03C0}{2}) radians from the real axis.

Thus, for z i, the argument is:

[ arg(i) frac{u03C0}{2} text{ radians} ] or [90^u00b0 ]

Summary

Modulus of z i: [mod(i) 1] Argument of z i: [arg(i) frac{u03C0}{2} text{ radians}] or [90^u00b0]

By understanding the modulus and argument of complex numbers, you can effectively describe the position of any complex number in the complex plane and perform various mathematical operations involving complex numbers.