Problem 32 - Elementary Algebra

Question

Choose an appropriate scale and graph the following sets of real numbers on a number line. $$ \\{-52,-12,0,12,2\\} $$

Step-by-Step Solution

Answer

Plot the numbers \(-52, -12, 0, 12, 2\) on a number line scaled by 10.

Step 1: Identify the Range

Determine the smallest and largest numbers in the set to establish the range for the number line. In this set, the smallest number is \(-52\) and the largest is \(12\).

Step 2: Choose an Appropriate Scale

Decide on a scale that will allow all numbers to fit on the number line. A reasonable choice is a scale where each unit represents 10 units. This allows all numbers from \(-52\) to \(12\) to fit comfortably.

Step 3: Draw the Number Line

Draw a horizontal line and label it with evenly spaced marks according to the chosen scale. Start from \(-60\) and go to \(20\), marking every 10 units (\(-60, -50, -40, ..., 0, ..., 20\)).

Step 4: Plot the Numbers

Place each number from the set \{-52, -12, 0, 12, 2\} on the number line. \(-52\) is slightly to the left of \(-50\), \(-12\) is slightly to the left of \(-10\), \(0\) is right on the zero mark, \(2\) is just to the right of \(0\), and \(12\) is slightly to the right of \(10\).

Key Concepts

Real Numbers

Real numbers include all the numbers you can think of along a continuous line. They combine both rational and irrational numbers. Here's a breakdown:

• Rational Numbers: These can be expressed as fractions, such as \( \frac{1}{2}, -3, 0.75 \).
• Irrational Numbers: Unlike rational numbers, these cannot be expressed as simple fractions. Examples include \( \pi \) and \( \sqrt{2} \).

In the exercise, the numbers \(-52, -12, 0, 12, 2\) are real because they can all be placed on a number line and have precise locations. Understanding that all these numbers fit into one larger category—real numbers—helps us see that while they look different, they share the same space on a graph.

Graphing

Graphing numbers on a number line is a simple way to visualize their positions relative to each other. Each number corresponds to a specific point.

• Start with a Line: Imagine or draw a straight, horizontal line.
• Mark Points: You evenly space out the points according to your scale.
• Plot the Numbers: Place each number at its appropriate position on the line.

This visual representation makes it easier to compare sizes and distances between the numbers. For example, you clearly see that \(-52\) is much further from zero than \(2\). Graphing can help you immediately understand the relative magnitude and order of numbers.

Scale

Choosing the right scale is crucial when graphing numbers, especially when dealing with wide ranges. The scale defines the spacing between marks on the line, helping to represent data clearly.

• Determine Your Range: Find the smallest and largest number to understand the necessary scale. Here, \(-52\) to \(12\).
• Select a Unit: A logical choice is needed to fit all numbers; here, each mark represents 10 units. This makes spacing even and consistent.
• Draw and Mark: Start from a little before your smallest number and extend slightly beyond your largest number, marking every 10 units from \(-60\) to \(20\).

The right scale makes your number line easy to read and ensures every number fits without crowding.

Number Sets

Understanding number sets helps organize real numbers based on certain rules. In this context, we can categorize numbers in different ways:

• Integers: Whole numbers that can be positive, negative, or zero (e.g., \(-52, 0, 12\)).
• Subsets: Within the broader category of real numbers, there are subsets like even numbers, odd numbers, or negative numbers.

Sets are useful because they help categorize numbers based on shared properties. This organization can aid in mathematical analysis and problem-solving. When you know which set a number belongs to, you understand more about its properties and potential uses in calculations.