Problem 19 - Algebra and Trigonometry

Question

Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$ z_{1}=2-i, \quad z_{2}=2+i $$

Step-by-Step Solution

Answer

Plot z_1 at (2, -1), z_2 at (2, 1), z_1+z_2 at (4, 0), z_1z_2 at (5, 0) on the complex plane.

Step 1: Plot the Complex Numbers

First, recognize that the complex numbers are in the form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For \(z_1 = 2 - i\), the real part is 2 and the imaginary part is -1. Plot this point at (2, -1) on the complex plane. For \(z_2 = 2 + i\), plot this point at (2, 1) on the complex plane.

Step 2: Add the Complex Numbers

Calculate \(z_1 + z_2\). This is done by adding the real parts and the imaginary parts separately: \(z_1 + z_2 = (2 + 2) + (-i + i) = 4 + 0i = 4\). Plot this result at (4, 0) on the complex plane.

Step 3: Multiply the Complex Numbers

Find \(z_1 \times z_2\). Use the formula for multiplying complex numbers: \((a+bi)(c+di) = (ac-bd) + (ad+bc)i\). Thus, \(z_1 z_2 = (2)(2) - (-1)(1) + (2)(1) + (2)(-1)i = 4 + 1 + i \times 0 = 5\). Plot this result at (5, 0) on the complex plane.

Step 4: Interpret the Graph

Check your plots: \(z_1\) should be at (2, -1), \(z_2\) at (2, 1), \(z_1 + z_2\) at (4, 0), and \(z_1 z_2\) at (5, 0). Confirm that the operations on the complex numbers are reflected by their positions on the plane.

Key Concepts

Complex Plane

The complex plane is a visual tool used in mathematics to illustrate complex numbers, which are numbers made up of a real part and an imaginary part. You can think of it as a two-dimensional graph similar to the Cartesian coordinate system.
Instead of having an x-axis and y-axis, the complex plane uses the horizontal axis for the real part and the vertical axis for the imaginary part of a complex number.

• Each point on the complex plane represents a unique complex number.
• A complex number in the form of \( z = a + bi \) can be plotted as the point \((a, b)\) on this plane.

Plotting complex numbers like \( z_1 = 2 - i \) involves identifying 2 along the real axis and -1 along the imaginary. Similarly, \( z_2 = 2 + i \) translates to 2 along the real axis and 1 along the imaginary. This visualization helps in understanding their behavior under various operations.

Addition of Complex Numbers

Adding complex numbers is straightforward once you understand that each number is a combination of a real part and an imaginary part.
When we add two complex numbers, like \( z_1 = 2 - i \) and \( z_2 = 2 + i \), we simply add the real parts together and the imaginary parts together:

• Real Part: \( 2 + 2 = 4 \)
• Imaginary Part: \( -i + i = 0 \)

Thus, their sum \( z_1 + z_2 \) results in the complex number \( 4 + 0i \), or simply 4.
In terms of the complex plane, plot \((4, 0)\), which lies directly on the real axis, showing that the imaginary parts have effectively canceled each other out. This method illustrates how addition essentially involves combining vectors on the complex plane.

Multiplication of Complex Numbers

Multiplying complex numbers can be initially more complex than addition. However, it follows a clear pattern inspired by the distributive property used in algebra. Let's explore how multiplication works using \( z_1 = 2 - i \) and \( z_2 = 2 + i \):
To multiply, we expand the product using:

• \((2)(2) - (-1)(1) \) for the new real part, which is \(4 + 1 = 5\)
• \((2)(1) + (2)(-1)\) for the new imaginary part, resulting in \(0i\)

So, \( z_1 \times z_2 = 5 \+ 0i \).
On the complex plane, this product appears at \((5, 0)\), again on the real axis. This transformation on the plane explains how the multiplication of complex numbers can be visualized as scaling and rotating vectors, reflecting the sophisticated yet logical nature of these operations.