Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$6-5 i$$

Step-by-Step Solution

Answer

Plot the point (6, -5) from the origin in the complex plane.

Step 1: Identify the Components of the Complex Number

The complex number given is \(6 - 5i\). Here, \(6\) is the real part, and \(-5\) is the imaginary part of the complex number.

Step 2: Understand the Complex Plane

The complex plane consists of a horizontal axis (the real axis) and a vertical axis (the imaginary axis). A complex number \(a + bi\) is represented as a point \( (a, b) \) on this plane.

Step 3: Plot the Real Part

Plot the real part of the complex number on the horizontal axis. For \(6 - 5i\), this is \(6\). Mark the point 6 on the real axis.

Step 4: Plot the Imaginary Part

Plot the imaginary part on the vertical axis. For \(6 - 5i\), this is \(-5\). Move to the point -5 on the imaginary axis.

Step 5: Draw the Vector

Start from the origin \((0,0)\). Draw a line (vector) from the origin to the point \((6, -5)\) on the complex plane. This represents the complex number \(6 - 5i\) as a vector.

Key Concepts

Complex Plane

The complex plane is like a playground for complex numbers, drawn on a flat, two-dimensional surface. Think of it like a coordinate system, split into a horizontal axis and a vertical axis. The horizontal axis is called the real axis, and the vertical one is known as the imaginary axis.

Each complex number can be uniquely represented as a point on this plane. For instance, in our exercise, the complex number is \(6 - 5i\). Here, the number \(6\) is on the real axis, while \(-5\) exists on the imaginary axis.

This setup allows us to visualize complex numbers not just as mere combinations of numbers, but as directional arrows from the origin \((0, 0)\) to their positions on the plane. This representation brings great insights into understanding the magnitude and direction of complex numbers easily.

Real and Imaginary Parts

Every complex number has two essential parts: the real part and the imaginary part. If you take a close look at a complex number like \(a + bi\), \(a\) is the real part, while \(b\) is the coefficient of the imaginary unit \(i\).

In our example, the number is \(6 - 5i\). Let's break it down:

• The real part is \(6\).
• The imaginary part is \(-5\) (remember the negative sign indicates direction on the imaginary axis).

These parts are plotted on their respective axes. The real part positions the complex number horizontally from the origin, while the imaginary part shifts it vertically. Understanding these pieces is crucial because every complex number can be imagined as a vector starting from the origin, described by these two coordinates: real and imaginary.

Graphing Vectors

Graphing vectors helps us see complex numbers as journeys on the complex plane. Vectors have both a direction and a length.

To graph the complex number \(6 - 5i\) as a vector, start at the origin \((0,0)\).

• First, move horizontally to \(6\), capturing the real part.
• Then, move vertically downwards to \(-5\), respecting the sign of the imaginary part.

This movement creates a path that bridges the origin to the point \((6, -5)\). The line you draw here is the vector. This vector not only visually represents the complex number but also gives you a feel for its direction in the coordinate system.

The beauty of this approach is that it simplifies complex operations, like adding or multiplying complex numbers, by translating them into geometric actions with vectors. Complex numbers thus become more intuitive and approachable when seen as navigational quests on the complex plane.