Problem 40 - Advanced Engineering Mathematics

Question

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: Find the magnitude of the first complex number

The first complex number is \( \frac{1}{2} - \frac{1}{4} i \). To find its magnitude, use the formula \( |z| = \sqrt{a^2 + b^2} \), where \( z = a + bi \). Here, \( a = \frac{1}{2} \) and \( b = -\frac{1}{4} \). Therefore, the magnitude is: \[ |z_1| = \sqrt{ \left( \frac{1}{2} \right)^2 + \left( -\frac{1}{4} \right)^2 } = \sqrt{ \frac{1}{4} + \frac{1}{16} } = \sqrt{ \frac{5}{16} } = \frac{\sqrt{5}}{4} \]

Step 2: Find the magnitude of the second complex number

The second complex number is \( \frac{2}{3} + \frac{1}{6} i \). Using the same magnitude formula, where \( a = \frac{2}{3} \) and \( b = \frac{1}{6} \), calculate the magnitude: \[ |z_2| = \sqrt{ \left( \frac{2}{3} \right)^2 + \left( \frac{1}{6} \right)^2 } = \sqrt{ \frac{4}{9} + \frac{1}{36} } = \sqrt{ \frac{16}{36} + \frac{1}{36} } = \sqrt{ \frac{17}{36} } = \frac{\sqrt{17}}{6} \]

Step 3: Compare the magnitudes

Compare the magnitudes \( \frac{\sqrt{5}}{4} \) and \( \frac{\sqrt{17}}{6} \). Approximate \( \sqrt{5} \approx 2.236 \) and \( \sqrt{17} \approx 4.123 \), so \( \frac{\sqrt{5}}{4} \approx 0.559 \) and \( \frac{\sqrt{17}}{6} \approx 0.687 \). Therefore, \( \frac{\sqrt{5}}{4} < \frac{\sqrt{17}}{6} \), meaning the first complex number is closer to the origin.

Key Concepts

Magnitude of Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part and are often represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The magnitude, or modulus, of a complex number gives us an idea of its "size" or "length" on the complex plane. It tells us how far away the complex number is from the origin \((0,0)\).

To calculate the magnitude of a complex number \( z = a + bi \), we use the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

This formula is derived from the Pythagorean theorem. It essentially measures the hypotenuse of a right triangle where \( a \) and \( b \) are the two legs.

The magnitude always returns a non-negative number and provides a straightforward way to understand the size or distance of a complex number from the origin on the complex plane.

Distance from the Origin

The distance from the origin for a point represented by a complex number is synonymous with measuring the magnitude, as complex numbers are plotted on the complex plane. The real part \( a \) is placed on the horizontal axis, while the imaginary part \( b \) lies on the vertical axis.

The origin in the complex plane is like the starting point, located at \((0,0)\). When we calculate how far a complex number \( a + bi \) is from the origin, we are essentially finding its magnitude, \( |z| = \sqrt{a^2 + b^2} \).

For example, if we have a complex number \( \frac{1}{2} - \frac{1}{4}i \), calculating its distance from the origin:

• Find \( a = \frac{1}{2} \) and \( b = -\frac{1}{4} \).
• Apply in the magnitude formula \( |z_1| = \sqrt{ (\frac{1}{2})^2 + (-\frac{1}{4})^2 } = \frac{\sqrt{5}}{4} \).

This process tells us exactly how far \( \frac{1}{2} - \frac{1}{4}i \) is from the origin.

Complex Number Comparison

When comparing complex numbers, especially in terms of their proximity to the origin, we focus on their magnitudes. In our example, two complex numbers, \( \frac{1}{2} - \frac{1}{4} i \) and \( \frac{2}{3} + \frac{1}{6} i \), are being compared.

To determine which is closer:

• Calculate the magnitude of each number, as explained previously.
• Compare the magnitudes: a smaller value indicates the complex number that is closer to the origin.

For instance, the magnitudes we found were \( \frac{\sqrt{5}}{4} \approx 0.559 \) and \( \frac{\sqrt{17}}{6} \approx 0.687 \), showing that the first complex number \( \frac{1}{2} - \frac{1}{4} i \) is closer to the origin than the second one.

This method of comparison is crucial in various mathematical and engineering applications where understanding the behavior of complex systems in relation to a central point (origin) plays a vital role.