Problem 78 - Precalculus

Question

Find the argument of \(z=b i,\) where \(b\) is a positive real number.

Step-by-Step Solution

Answer

The argument of \( z = bi \) is \( \frac{\pi}{2} \) radians.

Step 1: Identify the Complex Number Format

The complex number is given as \( z = bi \), where \( b \) is a positive real number. Here, the real part is zero and the imaginary part is \( b \).

Step 2: Determine the Argument of a Complex Number

The argument of a complex number \( z = a + bi \) is given by \( \arg(z) \). For a complex number on the imaginary axis, with zero real component and a positive imaginary component, the angle from the positive x-axis (real axis) is \( \frac{\pi}{2} \) radians.

Step 3: Confirm the Argument Based on the Imaginary Component

Since \( b \) is positive, \( z \) lies along the positive imaginary axis. Therefore, the argument of \( z \) is \( \frac{\pi}{2} \) radians.

Key Concepts

Argument of a Complex Number

The argument of a complex number is an essential concept in mathematics. It tells us the angle that the vector representing the number makes with the positive real axis on a complex plane. A complex number is usually written in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
When representing this number graphically, its position is determined by both \( a \) and \( b \). However, the argument is particularly interested in the direction of this vector.

• If both \(a\) and \(b\) are positive, the argument lies within the first quadrant.
• If \(a\) is negative and \(b\) is positive, the angle appears in the second quadrant.
• Negative values of both \(a\) and \(b\) correspond to the third quadrant.
• Positive \(a\) and negative \(b\) place the angle in the fourth quadrant.

In the case where \(a\) is zero and \(b > 0\), like our example with \( z = bi \), the argument becomes simply \( \frac{\pi}{2} \), since this angle is a straight line pointing directly upward along the imaginary axis. This helps in understanding the direction of the complex number on the plane.

Imaginary Axis

In the complex plane, numbers are plotted based on their real and imaginary components. The imaginary axis is the vertical line on this plane, representing the imaginary part of a complex number. Any complex number can be visualized in this 2-dimensional space, where:

• The real axis extends horizontally and represents the real component.
• The imaginary axis runs vertically and accounts for the imaginary component.

For a complex number like \( z = bi \) with a zero real component, its entire value is plotted along the imaginary axis. This means it does not shift left or right; it only moves up or down depending on whether \( b \) is positive or negative.
When dealing with purely imaginary numbers, understanding the imaginary axis is crucial, as it directly affects the argument and how these numbers are perceived geometrically.

Positive Imaginary Component

In complex numbers, the imaginary component is what gives a number its position along the imaginary axis. A positive imaginary component means that the number lies above the origin in the complex plane. This can be expressed as any number \(z\) where the format is \(z = bi\) and \(b > 0\).
Such numbers are straightforward to work with because they have a clear geometric interpretation:

• They rise upward from the origin along the vertical line of the imaginary axis.
• They will have an argument of exactly \(\frac{\pi}{2}\), indicating a 90-degree angle in polar coordinates.

If the positive imaginary component is large, the point is far from the origin, but if it's small, it stays near the origin. Understanding this helps simplify visualizing and calculating the behavior of complex numbers, especially when determining their arguments.