Problem 18 - 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. $$-3.4$$

Step-by-Step Solution

Answer

The point representing -3.4 is plotted on the number line between -4 and -3, but it's located closer to -3.

Step 1: Draw a Number Line

Start by drawing a straight horizontal line. Mark this line evenly to represent integers from -5 to 5. Make sure the distances between integers are equal.

Step 2: Identify the Position of -3.4

The number -3.4 is located between -4 and -3 on the number line. It is closer to -3, as the decimal part .4 is on the higher end of the range from 0 to 1.

Step 3: Plot the Number -3.4

Plot the point on the number line that accurately represents -3.4. This point should be between -4 and -3, but closer to -3.

Key Concepts

Integers and Their Place on the Number Line

Integers are whole numbers that can be positive, negative, or zero. They are represented without fractions or decimals and include numbers like \(-3, 0, 1,\) and many more.
When we talk about integers on a number line, we focus on evenly spaced points. Each point represents an integer value. For instance, if you draw a line from \(-5\) to \(5\), every point between these two numbers marks a specific integer.

• Integers are crucial for basic arithmetic operations such as addition and subtraction.
• They form the foundation of more complex number systems.
• Understanding integers helps us to grasp other number types like real numbers.

A clear understanding of integers allows you to see how other numbers, such as fractions and decimals, fit into the number system.

Understanding Real Numbers

Real numbers encompass a broad category that includes not only integers but also fractions, decimals, and irrational numbers like \(\sqrt{2}\) or \pi\.
Real numbers fill the gaps between integers on the number line. This means any point on the number line represents a real number.

• All integers are real numbers, but not all real numbers are integers.
• Real numbers can be positive, negative, or zero.
• They are used in various fields such as engineering, sciences, and everyday life calculations.

The concept of real numbers extends our understanding from whole and "countable" numbers to more precise values that reflect the continuous nature of the world.

Graphing Numbers on a Number Line

Graphing numbers involves placing them in their correct positions on the number line. This helps in visualizing both integers and real numbers' relationships and sizes.
Here's how you can graph a number like \-3.4\ on a number line:

• First, identify the two integers between which the number falls; in this case, \(-3.4\) lies between \(-4\) and \(-3\).
• Determine the position relative to these integers. Since \(-3.4\) is closer to \(-3\), it should be plotted nearer to \(-3\) than \(-4\).
• Make a mark to represent this position accurately.

Graphing numbers allows you to:

• Accurately understand the size and order of diverse numbers.
• Visualize numerical relationships, aiding in tasks like comparison and arithmetic operations.
• Develop an intuition for where numbers "live" on the number line.

Creating visual representations of numbers enhances comprehension and aids in solving mathematical problems efficiently.