Problem 14 - 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. $$2 \frac{1}{4}$$

Step-by-Step Solution

Answer

The required number line representing integers from -5 to 5 with the real number 2.25 accurately graphed, lies between 2 and 3 and is closer to 2.

Step 1: Drawing the Number Line

Draw a straight line and mark it with equal intervals. Label these intervals from -5 to 5. Each division represents 1 unit to depict integers.

Step 2: Converting the Fraction to Decimal

Now, convert the mixed fraction 2 1/4 to decimal. This is achieved by dividing the numerator by the denominator for the fractional part and adding the whole number to the result. It gives the decimal equivalent as 2.25.

Step 3: Graphing on the Number Line

On the number line, the decimal 2.25 will be between 2 and 3, closer to 2. Mark this point accurately to represent 2.25 on the number line.

Key Concepts

Integers

When we talk about integers, we are referring to a fundamental concept in mathematics. Integers are whole numbers that include all the positive numbers, negative numbers, and zero. Here's a simple way to think about it:

• Positive integers: 1, 2, 3, ...
• Negative integers: -1, -2, -3, ...
• Zero: 0

Integers do not have fractional or decimal parts. This is why on a number line, integers are represented as distinct and evenly spaced points. For example, in the exercise you would draw a number line from -5 to 5. Each step between the numbers is exactly one unit. The number line helps to visually understand the sequence and order of integers. You can compare sizes, see the effects of addition and subtraction, and easily spot where each integer lies in relation to others.

Real Numbers

Real numbers include all the numbers that can be found on the number line. This means every number that falls into the categories of rational and irrational numbers. But don’t worry—it’s simpler than it sounds!Here's what makes up real numbers:

• Integers (as we just discussed)
• Fractions
• Decimals, including both finite and repeating (like 0.25 or 0.333...)
• Irrational numbers (like \( \sqrt{2} \) and \( \pi \), which cannot be written as exact fractions or repeating decimals)

In the exercise, you take a mixed fraction \(2 \frac{1}{4}\) and place it on the number line as a real number. After converting the mixed fraction to a decimal (2.25), you can find its exact position between 2 and 3. Real numbers can be precisely plotted on a number line, which lays the foundation for understanding more complex mathematical concepts.

Mixed Fractions

Mixed fractions are a combination of whole numbers and fractions. These are often seen when you have a number that includes both an integer and a fractional component. Understanding mixed fractions is key because they show up frequently in everyday situations. For instance, consider the mixed fraction \(2 \frac{1}{4}\). It means you have 2 whole units and an additional \(\frac{1}{4}\) unit. Converting mixed fractions to decimals helps in many calculations—especially when plotting on a number line or working with digital tools.Here's how to convert a mixed fraction like \(2 \frac{1}{4}\) into a decimal:

• Keep the whole number part as it is: 2
• Convert the fractional part \(\frac{1}{4}\) to a decimal by performing the division (1 divided by 4), which is 0.25
• Add the two results together: 2 plus 0.25 equals 2.25

This conversion shows how the number fits precisely within the sequence of other real numbers, lending clarity when visualizing numbers on a line or performing various types of arithmetic.