What Is Root 3 I In Mod-arg Form

What is √3i in Mod-Arg Form? A Comprehensive Guide

This article will explain how to express the complex number √3i in its polar form, also known as modulus-argument form (mod-arg form). Understanding this conversion is crucial for various complex number operations, particularly multiplication and division. We'll break down the process step-by-step, ensuring a clear understanding of the underlying concepts.

Understanding Complex Numbers and Mod-Arg Form

A complex number is a number that can be expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part, and i is the imaginary unit (√-1). The mod-arg form provides an alternative representation: r(cos θ + i sin θ), or more compactly, r cis θ. Here:

• r represents the modulus (or magnitude) of the complex number – the distance from the origin to the point representing the complex number on the complex plane.
• θ represents the argument (or phase) of the complex number – the angle the line connecting the origin to the complex number makes with the positive real axis, measured counterclockwise.

Converting √3i to Mod-Arg Form

Let's convert √3i into mod-arg form. Notice that the real part is 0, and the imaginary part is √3. We can represent this as 0 + √3i.

1. Finding the Modulus (r)

The modulus, r, is calculated using the Pythagorean theorem:

r = √(a² + b²)

In our case, a = 0 and b = √3. Therefore:

r = √(0² + (√3)²) = √3

2. Finding the Argument (θ)

The argument, θ, is determined by the position of the complex number on the complex plane. Since the real part is 0 and the imaginary part is positive, the complex number lies on the positive imaginary axis. Therefore, the argument is:

θ = π/2 radians or 90°

3. Expressing in Mod-Arg Form

Now that we have the modulus (r = √3) and the argument (θ = π/2), we can express √3i in mod-arg form:

√3i = √3(cos(π/2) + i sin(π/2)) or √3 cis(π/2)

In Summary:

The complex number √3i, when expressed in modulus-argument form, is √3(cos(π/2) + i sin(π/2)) or √3 cis(π/2). This representation simplifies calculations involving multiplication and division of complex numbers. Remember to always consider the quadrant of the complex number when determining the argument to ensure accuracy. This approach provides a geometric interpretation that enhances understanding beyond simple algebraic manipulation. Understanding this conversion is key to mastering complex number manipulation and various applications in mathematics, engineering, and physics.