Problem 19 - Introductory Algebra for College Students

Question

Start by drawing a number line that shows integers from \(-5\) to \(5 .\) Then graph each real number on your number line. $$-\frac{16}{5}$$

Step-by-Step Solution

Answer

\(-\frac{16}{5}\) is graphed on the number line between \(-4\) and \(-3\), but closer to \(-3\).

Step 1: Understanding the task

The task is asking to draw a number line and then to graph the real number \(-\frac{16}{5}\) on that line. The first thing to do is to understand what \(-\frac{16}{5}\) represents as a decimal, which simplifies to \(-3.2\)

Step 2: Drawing the number line

Draw a linear graph with 11 points (representing integers from -5 to 5) evenly spaced. Mark each point with the corresponding integer value.

Step 3: Plotting the real number

\(-\frac{16}{5}\) or \(-3.2\) falls between \(-4\) and \(-3\). As it's closer to \(-3\), mark a point between \(-4\) and \(-3\) but closer to \(-3\). This point represents \(-\frac{16}{5}\) or \(-3.2\).

Key Concepts

Integers

Integers are numbers that don't have any fractional or decimal component. They include the whole numbers and their negatives.

• Whole numbers are natural counting numbers starting from 0, 1, 2, 3, and so on.
• Negative numbers are the just the opposite of whole numbers, like -1, -2, -3, etc.
• All these numbers together, both positive and negative including zero, form the set of integers.

When working with integers on a number line, you can think of them as evenly spaced points. Each integer corresponds to a specific location on this line.
This consistent spacing makes integers ideal for basic counting tasks. It's important to remember that zero is neither positive nor negative—it sits right in the middle.

Real Numbers

Real numbers are an extensive class of numbers that include pretty much everything we can measure on a number line:

• All integers (like -2, 0, 7)
• Fractions like \(-\frac{1}{2}\)
• Decimals, whether they come from fractions like \(-3.2\) or irrational numbers like \(\pi\)

Real numbers can be found anywhere on the number line. They provide a continuous spectrum of values. What's important about real numbers is openness—they are not just points like integers but cover everything in between.
For example, given any two real numbers, there's always another real number in between them.
This is the key to understanding the concept of density of real numbers on a number line.

Graphing

Graphing on a number line is a visual representation of numbers that helps you understand their order and relative positions. Here's how you can graph efficiently:

• Draw a straight horizontal line and mark it with evenly spaced points to represent integers.
• To plot a real number like \(-\frac{16}{5}\), first convert it to a decimal format, \(-3.2\) in this case.
• Locate \(-3.2\) by identifying its position between two integers, here between \-3\ and \-4\.

The key aspect of graphing real numbers is identifying how close they are to the nearest integers. \(-3.2\) is closer to \(-3\) than \(-4\), which helps you place it accurately on the line.
Graphing on a number line enhances understanding because of its visual impact—it's easier to compare and locate numbers at a glance.
It's perfect for organizing numbers in a logical and sequential way.