Problem 17 - Introductory and Intermediate Algebra for College Students 4th

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

The real number \(-1.8\) should be represented on the number line slightly to the right of the integer \(-2\).

Step 1: Draw a Number Line

First, one should create a number line with integers ranging from \(-5\) to \(5\). Make sure there is enough space between each integer to fit the decimal number to be graphed.

Step 2: Locate the Integer Part

Next, locate the integer part of \(-1.8\) on the number line, which is \(-2\). This is the closest integer that's less than \(-1.8\).

Step 3: Locate the Decimal Part

The last part is to locate where \(-1.8\) would be on the line between \(-2\) and \(-1\). As the decimal part is \(0.2\), we place the real number \(-1.8\) slightly towards the right of \(-2\), representing the decimal part.

Key Concepts

Graphing Real Numbers

When dealing with real numbers, it is important to first understand the concept of the number line. This line is a visual representation of numbers that helps in understanding their positions and relationships. To graph real numbers, you begin by drawing this number line. Choose a range that includes the integers relevant to the numbers you're graphing. For the exercise where you need to graph i.e., -1.8 make sure that your number line spans the necessary integers, which in this case are from i.e., -5 to 5. Ensure the line is long enough with evenly spaced tick marks for each integer.

Real numbers include both rational and irrational numbers. Real numbers like -1.8 show a position between integers. You will need to identify and mark both the integer part and the fractional part.

• Integer Part: The integer part is the whole number part of the real number. It helps you initially place the real number on the number line.
• Fractional Part: The decimal or fractional portion shows the exact position between two integers.

Graphing these numbers helps in visually interpreting their magnitudes and relationships.

Locating Integers

To locate integers on a number line, observe the whole numbers without any fractional parts. Numbers like i.e., -2, -1, 0, 1, 2, are your guideposts. When graphing real numbers, understanding the location of these integers is crucial since they act as stable reference points.

Start by making sure your number line is properly labeled with integers that are evenly spaced. This uniformity provides an easy reference for accuracy. In our example of -1.8, locate the integer -2 first on the number line.

• Even Spacing: Ensure that the space between consecutive integers is consistent.
• Central Point: Identify the integer that most closely matches the real number's whole part, which for -1.8 is -2.

Using the integer as a base, we can then plot the exact real number position by considering its decimal part.

Number Line Representation

A number line is not only a tool for plotting numbers but also a way to visually represent mathematical concepts such as ordering and distance.

Each point on a number line corresponds to a unique real number, offering a clear visualization. To represent a number like i.e., -1.8 start by marking the integers i.e., -5 through 5. Then, identify and locate where the number falls between the relevant integers.

The number -1.8 lies between -2 and -1, closer to -2. This precise placement demonstrates how decimals and fractions manifest on the line.

• Decoy Markers: These are smaller tick marks between integers that help plot real numbers more accurately.
• Pictorial Clarity: This visual aid simplifies understanding relationships between numbers, and especially beneficial for negative numbers.

A well-drawn number line improves comprehension by offering visual context to numeric representations.