Problem 13 - 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. $$3 \frac{1}{2}$$

Step-by-Step Solution

Answer

The number \(3 \frac{1}{2}\) or 3.5 should be plotted halfway between the whole numbers 3 and 4 on the number line.

Step 1: Draw The Number Line

Start by drawing a straight horizontal line and marking it at equal intervals. Each of these intervals will represent one unit. Mark it from -5 to 5, with 0 in the middle.

Step 2: Identify The Fractional Number

The real number that needs to be plotted is \(3 \frac{1}{2}\), which is a fraction more than 3, and also known as 3.5 in decimal form.

Step 3: Plot The Real Number

Locate the number 3 on the number line and plot a point halfway between 3 and 4, that will be \(3 \frac{1}{2}\) or 3.5.

Key Concepts

Understanding Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Here are some important aspects of integers:

• Positive integers are the numbers greater than zero (e.g., 1, 2, 3).
• Negative integers are the numbers less than zero (e.g., -1, -2, -3).
• Zero is considered an integer, neither positive nor negative.

Integers are used in mathematics to describe whole quantities, and you frequently encounter them in situations like counting objects or measuring temperatures. Unlike decimals or fractions, integers represent complete values. They are fundamental in constructing a number line, serving as the basic points or markers that make up its structure.

Plotting Numbers on a Number Line

A number line is a straight horizontal line that helps visualize numbers, showing their order and relative distance from each other. It's a useful tool for plotting and understanding different number types, including integers and real numbers. To effectively plot numbers, follow these simple steps:

• Label the Line: Begin by marking equal intervals along the line. Each mark represents a unit, allowing you to plot integers clearly, like the interval from egative five to five.


• Locate the Numbers: Identify the specific number you want to plot, such as integers like -3 or 2.


• Plot Real Numbers: For numbers that include fractions or decimals, such as the example of 3.5, place them appropriately between the integer marks. When plotting 3.5, position it halfway between 3 and 4.

By using a number line, you can easily visualize the location of numbers, see how they relate to each other, and grasp concepts like ordering and magnitude.

The Concept of Real Numbers

Real numbers encompass both rational and irrational numbers, forming a broad category that includes integers, fractions, and decimals. Here's a deeper dive into real numbers:

• Rational Numbers: These are numbers that can be expressed as fractions, such as 1/2 or 3/4. Every integer is also a rational number since it can be written as a fraction with a denominator of one (e.g., 5 = 5/1).
• Irrational Numbers: These numbers cannot be expressed as simple fractions. Examples include the square root of two ( a non-terminating and non-repeating decimal).

Real numbers are significant as they represent every possible distance on a number line. They allow us to measure more than just whole numbers, accounting for precise values like 3.5, which needs to be accurately positioned between integers. Understanding real numbers is crucial for comprehending both the vastness and continuity of the number line.