Problem 17 - 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. $$-1.8$$

Step-by-Step Solution

Answer

-1.8 is located between -2 and -1 on the number line, closer to -1.

Step 1: Drawing a number line

Firstly, draw a horizontal line and mark it off from -5 to 5. Each tick on the line should correspond to an integer.

Step 2: Locating the real number

Now, locate and mark -1.8 on the number line. Given that -1.8 is a negative decimal number and it is more than -2 and less than -1, it should fall between -2 and -1 on the number line.

Step 3: Graphing the real number

Lastly, graph or plot -1.8 on the number line. This can be done by drawing a small circle or dot above the location of -1.8.

Key Concepts

Understanding the Number Line

The number line is a fundamental concept that allows us to visually represent the order and magnitude of real numbers. To create a number line, you generally draw a horizontal line with tick marks at regular intervals, usually corresponding to integers, which are the whole numbers and their opposites. Each point on the line corresponds to a real number, and the distance between the tick marks represents a unit of one.

To plot positive numbers, you move to the right from the origin (which is zero), and for negative numbers, you move to the left. The number line extends infinitely in both directions, but for practical purposes, we often limit the scope to a particular range, like from (-5) to (5) as in our exercise.

When drawing a number line, ensure that:

• The line is straight and long enough to include all the numbers you wish to represent.
• The ticks are evenly spaced.
• Each tick is clearly labeled with the corresponding number, especially the integers.
• The line is properly centered with an arrow on both ends to indicate that it extends indefinitely.

As a visual aid, the number line serves as a powerful tool to help students better understand the relationships between numbers, including whole numbers, fractions, and decimals.

Graphing Negative Decimals

Negative decimals are simply decimal numbers that fall below zero on the number line. To graph a negative decimal such as (-1.8), you must understand that it lies between two integers - in this case, (-2) and (-1). This can sometimes be tricky because decimals require us to think in parts of a whole.

When graphing negative decimals:

• Identify the two integers between which your decimal number falls. For (-1.8), those integers are (-2) and (-1).
• Estimate the position of the decimal. Since (-1.8) is close to (-2), it should be plotted closer to the (-2) mark on the number line.
• Make a small mark or dot above the estimated position on the number line to graph the decimal.

Understanding how to visualize these decimal positions helps in not just graphing them, but also in attaining a better grasp of the concept of magnitude in relation to zeros, which is vital for mathematical operations like addition, subtraction, multiplication, and division.

Integers: The Basics

Integers are a special group of numbers that include all the whole numbers and their negatives, plus zero. They do not include fractions or decimals. In our given exercise, the number line shows integers ranging from (-5) to (5), which includes zero as the middle point, or the neutral element.

Some key properties of integers include:

• They can be positive, negative, or zero.
• Positive integers are to the right of zero on a number line, while negative integers are to the left.
• The difference between any two adjacent integers is always one.
• When adding or subtracting integers, they follow specific rules that can lead to positive or negative outcomes, or even zero, which can be visualized on a number line.

Integers are crucial in mathematics because they represent discrete units and can be used in counting, ordering, and basic arithmetic. When working with integers, especially in algebra, it's important to understand their properties and how they relate to each other on the number line.