Problem 50 - Algebra 2 and Trigonometry

Question

In \(44-51,\) locate the point that corresponds to each of the given complex numbers. $$ -3+0 i $$

Step-by-Step Solution

Answer

The point for \(-3 + 0i\) on the complex plane is \((-3, 0)\).

Step 1: Understand the Complex Number

The given complex number is \(-3 + 0i\). In a complex number of the form \(a + bi\), \(a\) is the real part and \(b\) is the imaginary part. Here, \(-3\) is the real part and \(0\) is the imaginary part.

Step 2: Interpret the Point on the Complex Plane

The complex plane is a coordinate system where the horizontal axis (real axis) represents the real part of the complex number, and the vertical axis (imaginary axis) represents the imaginary part. The point \(-3 + 0i\) corresponds to the point \((-3, 0)\) in this plane.

Step 3: Plot the Point

To locate the point \(-3 + 0i\) on the complex plane, start at the origin (0,0). Since the real part is -3, move 3 units left on the real axis. The imaginary part is 0, so there is no movement up or down the imaginary axis. The point \((-3, 0)\) is the location for \(-3 + 0i\).

Key Concepts

Complex Plane

The complex plane is a pivotal tool when working with complex numbers, offering a visual representation akin to the Cartesian coordinate system. Unlike the usual graphs used for real numbers, in the complex plane, the horizontal axis represents real numbers, while the vertical axis is for imaginary numbers. This unique setup allows each complex number to be plotted as a point in two-dimensional space.

To further understand:

• The real part of a complex number, such as -3 in -3 + 0i, determines the position along the horizontal axis.
• The imaginary part, in this case 0, dictates the vertical positioning.

For our complex number -3 + 0i, the point lands on the horizontal axis at (-3, 0). This visual tool transforms abstract complex numbers into manageable points, aiding comprehension and further mathematical exploration.

Real and Imaginary Numbers

Real and imaginary numbers are fundamental in forming complex numbers. A complex number is expressed as a + bi, with a and b representing the real and imaginary parts, respectively. Real numbers, such as -3 in -3 + 0i, exist along the number line and are familiar to us through daily arithmetic operations.

Imaginary numbers, however, might not feel as intuitive. They involve an imaginary unit, i, defined by the property that i^2 = -1. Imaginary numbers like 0i, signify no contribution to the imaginary axis.

Understanding real and imaginary numbers allows you to navigate the complexities of the imaginary unit. When these two components converge, they create points plotted on the complex plane, guiding you to deeper insights in mathematics.

Coordinate System

A coordinate system is essential for graphing complex numbers, providing a method to represent them visually. In a two-dimensional space, each point on the plane is defined by two values: one from the horizontal axis and one from the vertical axis.

In the complex plane:

• The horizontal axis (x-axis) is for real numbers.
• The vertical axis (y-axis) handles imaginary numbers.

For -3 + 0i, the coordinate (-3, 0) means moving 3 units left (reflecting the -3 of the real part) and no movement up or down (since the imaginary part is 0).

Utilizing the coordinate system in the complex plane allows us to map complex numbers precisely, transforming abstract mathematical expressions into tangible points, making them easier to understand and analyze.