Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$-3+2 i$$

Step-by-Step Solution

Answer

The vector for \(-3 + 2i\) is drawn from \((0, 0)\) to \((-3, 2)\) in the complex plane.

Step 1: Understanding the Complex Number

A complex number is in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For the given number \(-3 + 2i\), the real part \(a\) is \(-3\) and the imaginary part \(b\) is \(2\).

Step 2: Identify the Axes

The complex plane is a two-dimensional plane where the horizontal axis represents the real part of the complex number (the real axis) and the vertical axis represents the imaginary part (the imaginary axis).

Step 3: Plot the Real Component

On the complex plane, starting from the origin \((0,0)\), move \(-3\) units along the horizontal (real) axis to the left because the real part is \(-3\).

Step 4: Plot the Imaginary Component

From the point \((-3,0)\), move 2 units upward along the vertical (imaginary) axis because the imaginary part is \(2\). The point now is located at \((-3, 2)\).

Step 5: Draw the Vector

Draw a vector (an arrow) starting from the origin \((0,0)\) to the point \((-3, 2)\) to represent the complex number \(-3 + 2i\) on the complex plane.

Key Concepts

Complex Plane

The complex plane is a vital concept when dealing with complex numbers. It acts as a framework that allows us to visualize these numbers, much like how a graph allows us to visualize coordinates. Imagine a flat surface that includes two axes: a horizontal axis called the real axis and a vertical axis called the imaginary axis.

• The horizontal (real) axis is where we plot the real part of a complex number.
• The vertical (imaginary) axis is used for the imaginary part.

By utilizing the complex plane, each complex number can be pinpointed as a specific point or vector. This visualization helps in better understanding how complex numbers behave.
The complex plane enables us to move beyond the limitations of the traditional number line by incorporating the imaginary dimension, which adds depth to mathematical calculations.

Real and Imaginary Parts

Every complex number is made up of two components: the real part and the imaginary part. These are combined in the form of \( a + bi \). Let's break this down:

• Real Part: Represented by \( a \), which is a real number. In our example, \(-3\), is the real part and it indicates a position on the horizontal axis.
• Imaginary Part: Given by \( bi \), where \( i \) is the imaginary unit (which satisfies \( i^2 = -1 \)), and \( b \) is a real number. Here, \( 2i \) represents the imaginary part, showing a movement on the vertical axis.

Understanding these parts is essential because each influences how a complex number is plotted and manipulated. The real part dictates horizontal movement on the complex plane, while the imaginary part dictates vertical movement.

Vector Representation

When we plot a complex number in the complex plane, we can also represent it as a vector. A vector is essentially a directed line segment that starts from one point and ends at another, depicting both direction and magnitude. For complex numbers:

• The origin \((0,0)\) is your starting point on the complex plane.
• From the origin, the complex number \(-3 + 2i\) can be represented by moving \(-3\) units along the real axis and then \(+2\) units along the imaginary axis.

This movement terminates at the point \((-3, 2)\), which becomes the endpoint of your vector. Drawing the vector from the origin to \((-3, 2)\) visually represents the complex number \(-3 + 2i\).
Vectors are crucial because they encapsulate both magnitude and direction, allowing complex numbers to be manipulated similar to forces acting in physics.