Problem 31 - Elementary Algebra

Question

Choose an appropriate scale and graph the following sets of real numbers on a number line. $$ \\{-2,-13,23,53\\} $$

Step-by-Step Solution

Answer

Graph the numbers using a scale of 5 units on a number line from -15 to 55.

Step 1: Identify the Range

First, identify the smallest and largest numbers in the set. The smallest number is \(-13\) and the largest is \(53\). This gives us a range from \(-13\) to \(53\).

Step 2: Choose an Appropriate Scale

Select a scale that covers the full range from \(-13\) to \(53\), such as marking every 5 units on the number line. This way, each mark can represent 5 units, which will comfortably fit all numbers.

Step 3: Draw the Number Line

Draw a horizontal line and mark points with equal spacing. Label the point corresponding to zero and mark every 5 units on either side of zero, from \(-15\) to \(55\) to ensure coverage.

Step 4: Plot the Numbers

Locate and mark the points for each number in the set on the number line. Find \(-13\), \(-2\), \(23\), and \(53\) based on the scale chosen. Ensure each point is accurately positioned according to the scale.

Key Concepts

Real Numbers

Real numbers include all the numbers you can think of, whether they’re whole numbers, fractions, or irrational numbers like the square root of 2. They can be positive, negative, or zero. When dealing with real numbers, it’s essential to know that they fill up the entire number line without any gaps. This means if you pick any two real numbers, there’s always another real number in between them. Real numbers are important in mathematics because they provide a complete and continuous way to represent quantities and measurements.

Choosing a Scale

Selecting an appropriate scale on a number line is crucial for accurately representing the numbers in your data set. The scale determines the spacing of numbers and impacts how easy it is to read and interpret the graph.

• First, identify the smallest and largest numbers in your set to establish the range you need to cover.
• Then, choose intervals that will allow you to fit all numbers properly. For example, if dealing with a range from (-13) to 53, marking every 5 units is a common choice.

This choice can help avoid crowding numbers too close together or spreading them too far apart, both of which can make understanding the graph difficult.

Plotting Numbers

Plotting numbers on a number line involves placing each number at the correct position based on the chosen scale. Start by drawing the line and marking consistent intervals.

• Label significant points, such as zero, based on your scale.
• Next, locate where each number in your set should be. For example, (-13) would be plotted between (-15) and (-10) if using a scale of 5 units.

This visual representation helps in quickly understanding the distribution of numbers and comparing them at a glance. Accurately positioning numbers ensures an easy-to-read graph that reflects correct relationships between values.

Range of Numbers

The range of a set of numbers is the difference between the largest and smallest numbers. It plays a critical role in determining both the size of your number line and the scale you’ll use.

• To find the range, simply subtract the smallest number from the largest number in your set.
• In our case, the range is (53 - (-13)) = 66.

Recognizing the range gives you the necessary context to strategize how your number line should look, ensuring all numbers fit within it comfortably.