Problem 15 - Algebra and Trigonometry

Question

Sketch the complex number \(z,\) and also sketch \(2 z,-z\) and \(\frac{1}{2} z\) on the same complex plane. $$ z=1+i $$

Step-by-Step Solution

Answer

Plot points (1,1), (2,2), (-1,-1), and (0.5,0.5) on the complex plane.

Step 1: Identify the Complex Number

The given complex number is \( z = 1 + i \). This means that on the complex plane, \( z \) has a real part of 1 and an imaginary part of 1. We can represent this on the complex plane as the point \( (1,1) \).

Step 2: Sketch the Complex Number \( z \) on the Plane

Plot the point \( (1,1) \) on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Thus, \( z = 1+i \) is represented by the point in the first quadrant one unit right and one unit up from the origin.

Step 3: Calculate and Sketch \( 2z \)

Calculate \( 2z \) where \( z = 1+i \). This gives \( 2z = 2(1+i) = 2 + 2i \). Plot the corresponding point \( (2,2) \) on the complex plane. This point is two units to the right and two units up from the origin.

Step 4: Calculate and Sketch \( -z \)

Calculate \( -z \) where \( z = 1+i \). This gives \( -z = -(1+i) = -1 - i \). Plot the point \( (-1,-1) \) on the complex plane. This point is located one unit left and one unit down from the origin.

Step 5: Calculate and Sketch \( \frac{1}{2}z \)

Calculate \( \frac{1}{2}z \) where \( z = 1+i \). This gives \( \frac{1}{2}z = \frac{1}{2}(1+i) = \frac{1}{2} + \frac{1}{2}i \). Plot the point \( \left(\frac{1}{2}, \frac{1}{2}\right) \) on the complex plane. This is located midway between the origin and the point \( (1,1) \).

Key Concepts

Complex Plane

The complex plane is a fascinating and useful tool in mathematics, providing a visual way to represent complex numbers. In this plane, we use two axes: the horizontal axis is called the "real axis," and the vertical axis is known as the "imaginary axis." By doing so, each complex number can be pinpointed on this plane as a unique point.

Think of the complex plane as an extended version of the traditional Cartesian plane you use for graphing functions. Each point on the complex plane corresponds to a complex number, where the real part dictates the position along the horizontal axis, and the imaginary part determines the position along the vertical axis. This setup makes it easier to visualize and understand the behavior of complex numbers and their operations.

Real Part

The real part of a complex number is an essential component, helping us identify its position on the complex plane. Any complex number can be written in the form of \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.

In our exercise, the given complex number is \(z = 1 + i\). Here, the real part is \(1\).

• Position on the Horizontal Axis: The real part represents how far along the horizontal axis the number will be plotted.
• Understanding Operations: When performing operations such as addition, multiplication, or scaling with complex numbers, the changes in the real part can help us predict how a certain number will move along the horizontal axis on the plane.

Imaginary Part

Just like the real part, the imaginary part is crucial in representing complex numbers on the complex plane. For a complex number \(a + bi\), \(b\) is the imaginary part.

For our example \(z = 1 + i\), the imaginary part is also \(1\).

• Position on the Vertical Axis: The imaginary part indicates the movement along the vertical axis.
• Multidimensional Representation: It aids in showing numbers not just as line segments but also as vectors, providing a more comprehensive view of complex number operations.

Understanding the imaginary part allows students to see how operations affect the up and down "movement" of complex numbers on the complex plane. This perspective helps in visualizing additions or subtractions of imaginary parts as vector-like operations.

Plotting Points

Plotting points on the complex plane is a straightforward process once you are familiar with the real and imaginary parts of complex numbers. Each complex number corresponds to a unique point by using its real part as the x-coordinate and its imaginary part as the y-coordinate.

Let's break down the steps using our examples:

• Original Complex Number \(z = 1 + i\): Plot point at \((1, 1)\), as the real part is 1 (x-coordinate) and the imaginary part is 1 (y-coordinate).
• Scaled Number \(2z\): Multiply both parts of the complex number by 2, giving \(2 + 2i\), then plot at \((2, 2)\).
• Negative Number \(-z\): Reverse both parts, giving \(-1 - i\), then plot at \((-1, -1)\).
• Half Number \(\frac{1}{2}z\): Divide both parts by 2, resulting in \(\frac{1}{2} + \frac{1}{2}i\), then plot at \(\left(\frac{1}{2}, \frac{1}{2}\right)\).

Understanding how to plot complex numbers accurately is key in visualizing mathematical problems involving complex numbers. It enhances comprehension and provides a geometric perspective that aids in further mathematical explorations.