Problem 38 - Advanced Engineering Mathematics

Question

$$\left|\frac{1}{2}-\frac{1}{4} i\right|=\frac{\sqrt{5}}{4} \text { and }\left|\frac{2}{3}+\frac{1}{6} i\right|=\frac{\sqrt{17}}{6}, \text { since } \frac{\sqrt{5}}{4}<\frac{\sqrt{17}}{6}, \frac{1}{2}-\frac{1}{4} i \text { is closer to the origin. }$$

Step-by-Step Solution

Answer

\( \frac{1}{2} - \frac{1}{4} i \) is closer to the origin with the magnitude \( \frac{\sqrt{5}}{4} \).

Step 1: Understand the Meaning of Magnitude

The magnitude of a complex number, expressed as \( a + bi \), is the distance of that number from the origin in the complex plane, calculated using the formula \( \sqrt{a^2 + b^2} \). This result represents the "length" of the vector in the Argand diagram.

Step 2: Calculate the Magnitude of \( \frac{1}{2} - \frac{1}{4} i \)

The magnitude for the complex number \( \frac{1}{2} - \frac{1}{4} i \) can be calculated as follows:\[\text{Magnitude} = \sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{4}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{16}} = \sqrt{\frac{5}{16}} = \frac{\sqrt{5}}{4}\]

Step 3: Calculate the Magnitude of \( \frac{2}{3} + \frac{1}{6} i \)

Similarly, for the complex number \( \frac{2}{3} + \frac{1}{6} i \), the magnitude is calculated as:\[\text{Magnitude} = \sqrt{\left(\frac{2}{3}\right)^2 + \left(\frac{1}{6}\right)^2} = \sqrt{\frac{4}{9} + \frac{1}{36}} = \sqrt{\frac{17}{36}} = \frac{\sqrt{17}}{6}\]

Step 4: Compare the Magnitudes

Comparing the magnitudes \( \frac{\sqrt{5}}{4} \) and \( \frac{\sqrt{17}}{6} \), we need to establish which is smaller to determine the complex number closer to the origin. Since \( \frac{\sqrt{5}}{4} < \frac{\sqrt{17}}{6} \), \( \frac{1}{2} - \frac{1}{4} i \) is closer to the origin.

Key Concepts

Magnitude of Complex Numbers

The magnitude of complex numbers is a fundamental concept in understanding their position in the complex plane. In mathematics, a complex number is expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The magnitude provides a way to quantify how "large" a complex number is with respect to its distance from the origin in the complex plane.
To find the magnitude, you use the formula \( \sqrt{a^2 + b^2} \). This formula is derived from the Pythagorean theorem, enabling us to find the "length" of the hypotenuse of a right triangle formed by the real and imaginary parts. For example, for the complex number \( \frac{1}{2} - \frac{1}{4} i \), the magnitude is calculated as:
\[\sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{4}\right)^2} = \frac{\sqrt{5}}{4}\]
This calculation tells us that the "size" or "distance from the origin" of this complex number on the complex plane is \( \frac{\sqrt{5}}{4} \). Understanding this concept is crucial when analyzing complex numbers geometrically.

Argand Diagram

The Argand diagram is a graphical representation of complex numbers, where each number corresponds to a unique point in a two-dimensional plane. This plane is similar to the Cartesian plane, but it is primarily used to showcase complex numbers, making it a "complex plane."
On an Argand diagram:

• The horizontal axis represents the real part of a complex number.
• The vertical axis represents the imaginary part.

Imagine plotting the complex number \( \frac{2}{3} + \frac{1}{6} i \). You'd move \(\frac{2}{3}\) units along the real axis and \(\frac{1}{6}\) units along the imaginary axis, thus positioning this number in the complex plane. The beauty of the Argand diagram is that it allows students to visualize complex numbers as points or vectors, which helps in understanding their magnitudes and the distance between them.

Distance from the Origin

The distance from the origin in the complex plane refers to the magnitude of a complex number. Just like in the original exercise, determining which complex number is closer to the origin involves calculating their magnitudes and comparing them.
Let’s consider two complex numbers: \( \frac{1}{2} - \frac{1}{4} i \) and \( \frac{2}{3} + \frac{1}{6} i \). By calculating their respective magnitudes:

• \( \frac{1}{2} - \frac{1}{4} i \) has a magnitude of \( \frac{\sqrt{5}}{4} \).
• \( \frac{2}{3} + \frac{1}{6} i \) has a magnitude of \( \frac{\sqrt{17}}{6} \).

The smaller the magnitude, the closer the complex number is to the origin. In this case, since \( \frac{\sqrt{5}}{4} < \frac{\sqrt{17}}{6} \), \( \frac{1}{2} - \frac{1}{4} i \) is closer to the origin.
Comparing distances from the origin provides a way to rank complex numbers according to their proximity to the center of the complex plane.

Complex Plane

The complex plane, integral to the study of complex numbers, is a two-dimensional plane structured so that each point corresponds to a unique complex number. It merges the geometric nature with the arithmetic nature of complex numbers, making it an invaluable tool for complex analysis.
In the complex plane:

• The x-axis is often referred to as the "real axis".
• The y-axis is known as the "imaginary axis".

When you plot a complex number like \( a + bi \), it translates into moving \( a \) units on the real axis and \( b \) units on the imaginary axis. This creates a vector from the origin to the point \( (a, b) \).
The complex plane allows for a vivid interpretation of complex number operations, such as addition, subtraction, and especially finding magnitudes, as these operations can be visualized as transformations or movements within this plane.
By using the complex plane, complex numbers find applications not just in theoretical mathematics but also in physics and engineering, where visualization aids in grasping the dynamics of real-world problems.