Problem 39 - Advanced Engineering Mathematics

Question

In Problems 39 and 40 , determine which complex number is closer to the origin. $$ 10+8 i, \quad 11-6 i $$

Step-by-Step Solution

Answer

The complex number \( 11 - 6i \) is closer to the origin.

Step 1: Understand the Concept

To determine which complex number is closer to the origin, we need to calculate the magnitude of each complex number. The one with the smaller magnitude is closer to the origin.

Step 2: Calculate the Magnitude of the First Complex Number

The magnitude of a complex number \( a + bi \) is calculated using the formula \( \sqrt{a^2 + b^2} \). For the complex number \( 10 + 8i \), the magnitude is \( \sqrt{10^2 + 8^2} \). Calculate this as follows:\[ \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} = 12.81 \]

Step 3: Calculate the Magnitude of the Second Complex Number

Now calculate the magnitude of the complex number \( 11 - 6i \) using the same formula. The magnitude is \( \sqrt{11^2 + (-6)^2} \). Calculate this as follows:\[ \sqrt{11^2 + (-6)^2} = \sqrt{121 + 36} = \sqrt{157} = 12.53 \]

Step 4: Compare the Magnitudes

Compare the magnitudes of the two complex numbers. The first complex number \( 10 + 8i \) has a magnitude of 12.81, and the second \( 11 - 6i \) has a magnitude of 12.53.

Step 5: Determine Closer Number to the Origin

Since 12.53 (magnitude of \( 11 - 6i \)) is less than 12.81 (magnitude of \( 10 + 8i \)), the complex number \( 11 -6i \) is closer to the origin.

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number is a measure of its distance from the origin in the complex plane. It's similar to finding the length of a vector that originates from the point (0, 0). The formula to calculate the magnitude of a complex number, which is commonly written as \( a + bi \), is \( \sqrt{a^2 + b^2} \). This formula derives from the Pythagorean theorem, making complex numbers easy to visualize as vectors.

• For \( 10 + 8i \), calculate as \( \sqrt{10^2 + 8^2} = \sqrt{164} = 12.81 \).
• For \( 11 - 6i \), calculate as \( \sqrt{11^2 + (-6)^2} = \sqrt{157} = 12.53 \).

These values give us a "length" to each complex number, helping to figure out which is closer to the origin.

Distance from Origin

Distance from the origin to any point in the complex plane can be determined using the magnitude of the corresponding complex number. The term "origin" refers to the point (0, 0), which is where both real and imaginary components are zero in the plane.Using the magnitude:- A smaller magnitude indicates that the complex number is closer to the origin.- For example, the magnitude of \( 11 - 6i \) is less than \( 10 + 8i \), meaning \( 11 - 6i \) is closer to the origin.Understanding these distances allows for appropriate measures and placements of complex numbers on the plane, aiding in visualization and spatial reasoning.

Complex Number Comparison

Comparing complex numbers involves examining their magnitudes rather than their real or imaginary parts individually. The magnitude gives a complete sense of how far each number is from the origin, regardless of direction.Steps to compare:1. **Calculate magnitudes**: Each complex number must first have its magnitude calculated.2. **Compare the values**: A number with a smaller magnitude is closer to the origin.So, when comparing \( 10 + 8i \) and \( 11 - 6i \), we find that \( 11 - 6i \) has a smaller magnitude of 12.53 compared to 12.81 of \( 10 + 8i \). Therefore, \( 11 - 6i \) is closer to the origin. This process simplifies the understanding and comparison of complex numbers without needing their graphical representation.