Problem 40 - Advanced Engineering Mathematics

Question

In Problems 39 and 40, determine which complex number is closer to the origin. $$ \frac{1}{2}-\frac{1}{4} i, \quad \frac{2}{3}+\frac{1}{6} i $$

Step-by-Step Solution

Answer

The complex number \( \frac{1}{2} - \frac{1}{4}i \) is closer to the origin.

Step 1: Understand the Problem

We are given two complex numbers and need to determine which one is closer to the origin. The complex numbers are \( \frac{1}{2} - \frac{1}{4}i \) and \( \frac{2}{3} + \frac{1}{6}i \).

Step 2: Distance Formula for Complex Numbers

The distance of a complex number \( a + bi \) from the origin is given by \( \sqrt{a^2 + b^2} \). We need to calculate this distance for both complex numbers.

Step 3: Calculate Distance for the First Complex Number

For the complex number \( \frac{1}{2} - \frac{1}{4}i \), the distance is \( \sqrt{ \left( \frac{1}{2} \right)^2 + \left( \frac{-1}{4} \right)^2 } = \sqrt{ \frac{1}{4} + \frac{1}{16} } = \sqrt{ \frac{4}{16} + \frac{1}{16} } = \sqrt{ \frac{5}{16} } = \frac{\sqrt{5}}{4}.\)

Step 4: Calculate Distance for the Second Complex Number

For the complex number \( \frac{2}{3} + \frac{1}{6}i \), the distance is \( \sqrt{ \left( \frac{2}{3} \right)^2 + \left( \frac{1}{6} \right)^2 } = \sqrt{ \frac{4}{9} + \frac{1}{36} } = \sqrt{ \frac{16}{36} + \frac{1}{36} } = \sqrt{ \frac{17}{36} } = \frac{\sqrt{17}}{6}.\)

Step 5: Compare the Distances

Now we compare the two distances. We have \( \frac{\sqrt{5}}{4} \approx 0.559 \) and \( \frac{\sqrt{17}}{6} \approx 0.689 \). Since \( 0.559 < 0.689 \), the first complex number is closer to the origin.

Key Concepts

Distance Formula

The distance formula is a fundamental concept used when dealing with complex numbers or points on a plane. In the realm of complex numbers, each number can be viewed as a point in a 2-dimensional space, known as the complex plane. Here, the horizontal axis is the real part of the complex number, while the vertical axis represents the imaginary part.

To find the distance of a complex number from the origin, which is the point (0,0), we apply the formula: \[ d = \sqrt{a^2 + b^2} \] Here,

• \(a\) is the real part of the complex number,
• \(b\) is the imaginary part.

This formula essentially calculates the distance by treating the real and imaginary parts as a coordinate in a Cartesian plane. It's much like finding the length of the hypotenuse of a right triangle, where \(a\) and \(b\) are the lengths of the other two sides.

By consistently applying this formula, you can evaluate how far any complex number lies from the origin.

Origin in Complex Plane

Understanding the concept of the origin in the complex plane can simplify many tasks and calculations involving complex numbers. The origin is denoted as (0,0), meaning both the real and imaginary parts are zero.

• It serves as a reference point, akin to the center of a graph, where no numeric component extends horizontally or vertically.
• When we're asked to determine the proximity of a complex number to the origin, our aim is to gauge how close this point is to (0,0).

Visualizing complex numbers as positions in this plane helps to better understand their relationships and operations. This visualization shows the importance of the origin as the baseline from which distances are calculated. Essentially, the concept of "closeness" or "distance" in the complex plane translates to how near or far a complex number is from this pivotal point.

Comparison of Distances

When comparing the distances of different complex numbers from the origin, you rely heavily on numerical computations. Once the distances are calculated using the distance formula, comparing them is straightforward.

Consider two complex numbers: \( z_1 \) and \( z_2 \).

• If the distance of \( z_1 \) from the origin is smaller than that of \( z_2 \), \( z_1 \) is closer.
• Otherwise, \( z_2 \) is closer if its distance is lesser.

In the example provided, we computed the distances: one resulted in approximately 0.559 and the other in 0.689. Since 0.559 is less than 0.689, the first complex number is closer to the origin.

This kind of comparison is crucial in various mathematical and engineering applications where precise decisions based on position or proximity are required. By efficiently employing the distance formula and making accurate comparisons, we can make significant insights into the properties of complex numbers in the plane.