Problem 16 - 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. $$\frac{7}{3}$$

Step-by-Step Solution

Answer

The fraction \(\frac{7}{3}\), which is approximately 2.33, is marked one third of the way between integers 2 and 3 on the number line.

Step 1: Understand Fraction as Decimal

First, convert the fraction \(\frac{7}{3}\) to a decimal. This is done by dividing 7 by 3 which gives approximately 2.33.

Step 2: Draw the Number Line

Draw a straight horizontal line. This represents your number line. Mark off equal spaces along the line, representing one unit each. Label each space with consecutive integers. In this case, label them from -5 to +5.

Step 3: Locate the Fraction on the Number Line

Since \(\frac{7}{3}\) equals to approximately 2.33, it should be positioned about one third of the way between 2 and 3 on the number line. Make a dot or a small mark on this spot and label it as \(\frac{7}{3}\).

Key Concepts

Understanding Integers

Integers are whole numbers that can be positive, negative, or zero. Think of them like the steps on a ladder, where each step represents a whole number.
On a number line, these are the numbers you usually see marked at equal intervals, like

• ...,-3,-2,-1,0,+1,+2,+3,...

They are the building blocks for other number types, like fractions and decimals. Knowing where integers are on the number line is essential to placing other numbers in context. For the exercise, you mark integers from -5 to +5, setting up your line with evenly spaced intervals.

Exploring Real Numbers

Real numbers include all the numbers on the number line. This means both the integers and numbers that aren't whole, like fractions and decimals.
Real numbers can be rational (like 1/2 or 0.75) or irrational (like \(\pi\) or \(\sqrt{2}\)). They fill in all the gaps between the whole numbers on the number line. When you're graphing, remember that not only integers are real numbers; every point on the line represents a real number.

Fractions and Their Places

Fractions take pieces of a whole number. For instance, \(\frac{7}{3}\) means seven parts divided by three, which cannot be represented by an integer alone.
When plotting a fraction on a number line, like \(\frac{7}{3}\), you convert it to a decimal for easier placement. This fraction becomes approximately 2.33, so it's found slightly right of the number 2. Fractions can be plotted by either converting them to decimals or estimating their places based on the denominator.

Decimals as Part of the Line

Decimals express numbers that aren't whole. They are written in the form of a base number with a point, like 2.33. Converting fractions to decimals helps pinpoint their exact spot on the number line.
For example, when you convert \(\frac{7}{3}\), you get 2.33. This helps determine its precise position between the integers 2 and 3. Decimals make it easier to graph fractions, especially when they extend beyond whole numbers. They're just another way to show parts of a whole and work seamlessly with other types of numbers on the number line.