Problem 1 - Contemporary Precalculus

Question

In Exercises \(1-8,\) plot the point in the complex plane corresponding to the number. $$3+2 i$$

Step-by-Step Solution

Answer

Question: Plot the complex number 3+2i on the complex plane. Answer: To plot the complex number \(3+2i\) on the complex plane, start at the origin (0,0), move 3 units to the right along the real axis, and then move 2 units up along the imaginary axis. The point you reach represents the complex number \(3+2i\).

Step 1: Understand the complex plane representation of complex numbers

The complex plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis. A complex number can be represented as a point in the complex plane where the coordinates of the point are represented by the real part and the imaginary part of the complex number. For example, a point P represents a complex number \(x + yi\) if the coordinates of P are (x, y). In our case, the complex number is \(3+2i\). The real part is 3, and the imaginary part is 2.

Step 2: Plot the point on the complex plane

Find the point in complex plane with coordinates (3, 2). This point represents the complex number \(3+2i\). To plot the point (3, 2) on the complex plane, follow these steps: 1. Start at the origin (0,0). 2. Move 3 units to the right along the real axis (the horizontal axis). 3. Move 2 units up along the imaginary axis (the vertical axis). 4. Mark the point where you stopped. This point represents the complex number \(3+2i\).

Key Concepts

Complex Numbers

Understanding complex numbers is foundational to many areas of mathematics, including algebra and calculus. A complex number typically takes the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying the equation \(i^2 = -1\). The real part of the complex number is denoted by \(a\), and the imaginary part is denoted by \(b\). For instance, in the complex number \(3+2i\), 3 is the real part and 2 is the imaginary part. Complex numbers can perform operations such as addition, subtraction, multiplication, and division, similar to real numbers, but with the additional aspect of involving the imaginary unit.

It is essential to approach complex numbers with an open mind because, despite their name, they make solving certain equations and representing multidimensional values quite simple once you're familiar with the concept.

Imaginary Axis

The imaginary axis is a vertical line in the complex plane that is used to graph the imaginary part of complex numbers. Much like the y-axis in the traditional coordinate system, the imaginary axis intersects the real axis at the origin point, creating a center point for our complex coordinate system.

When plotting a complex number like \(3+2i\), it involves locating the point corresponding to the imaginary part, which is 2 in our case, along this axis. This axis is orthogonal to the real axis, ensuring that real and imaginary parts are independent and can be treated as perpendicular dimensions in a plane. This is integral to the visualization and understanding of complex numbers as it allows for a clear distinction between the different components of a complex number.

Imaginary Values on the Axis

As you move up the imaginary axis, values become more positive, and as you move down, they become more negative. This property maintains a clear numerical progression similar to the y-axis in cartesian coordinates.

Real Axis

The real axis in the complex plane is much like the x-axis in a standard Cartesian coordinate system; it's a horizontal line that represents the real part of complex numbers. When plotting the complex number \(3+2i\), you first locate the real part, which is 3, on the real axis.

This axis, in combination with the imaginary axis, forms the complex plane's foundational grid system. The real axis is crucial for placing the complex number accurately in the plane, as it gives the exact horizontal position corresponding to the real component of the number.

Real Values on the Axis

The values on the real axis increase as you move to the right and decrease as you move to the left. This positive to negative progression is part of what makes the complex plane plot practical for representing all kinds of complex numbers.

Coordinate System

The coordinate system used to plot complex numbers is essentially a two-dimensional grid, formed by the intersection of the real and imaginary axes. This is akin to the Cartesian coordinate system and is alternatively called the Argand diagram or complex plane.

To locate a point in this system, we use an ordered pair (a, b), where 'a' is the value on the real axis and 'b' is the value on the imaginary axis. For the complex number \(3+2i\), its coordinates would be (3, 2). This system is fundamental for graphing complex numbers and understanding their spatial relations. Visually, it provides a way to interpret the magnitude and direction of complex numbers, and to perform operations that may be more abstract if handled algebraically.

Plotting Points and Understanding Distance

When plotting points, the distance from the origin can also represent the magnitude of the complex number, with the angle from the real axis indicating the so-called 'argument', a concept linked to the direction of the number in polar form representation.