Problem 12 - 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. $$-2$$

Step-by-Step Solution

Answer

A number line has been drawn from -5 to 5 and the number -2 has been correctly plotted on it.

Step 1: Draw a Number Line

Start by drawing a straight horizontal line. This will serve as the number line. Mark multiple equidistant points along the line, each representing an integer value, and label these points from -5 on the left to 5 on the right. Label the center point as 0.

Step 2: Locate and Mark the Number -2

Now, locate the point labelled as -2 on the number line. It is two places to the left of 0.

Step 3: Plot the Number -2

Make a darker or bigger mark on the -2 point you've located in the previous step. This denotes the graphic representation of the number -2 on your number line.

Key Concepts

Integers

Integers are whole numbers that can be positive, negative, or zero. When we talk about integers, we're referring to the set of numbers \(\{...,-3, -2, -1, 0, 1, 2, 3,...\}\). This set does not include fractions or decimals; it is strictly whole numbers. On a number line, integers are represented by equally spaced points.

These numbers help in several basic arithmetic operations, such as addition, subtraction, and multiplication, making them essential in mathematics. The integer 0 is a unique member of this set, acting as a neutral element in addition and a middle mark on a number line.

Understanding integers also involves getting comfortable with terms like 'positive integers' (which are greater than zero), 'negative integers' (which are less than zero), and knowing that each negative integer is the opposite of its positive counterpart.

• Positive integers: 1, 2, 3,...
• Negative integers: ..., -3, -2, -1
• Zero is neither positive nor negative

Real Numbers

Real numbers include all the numbers we come across in everyday life. These not only incorporate integers but also all the fractions and decimals. Essentially, real numbers are all the points on the number line.

Real numbers are classified into two main subsets: rational numbers and irrational numbers. Rational numbers can be expressed as fractions where both the numerator and denominator are integers. This means that any repeating or terminating decimal can also be considered a rational number. On the other hand, irrational numbers cannot be written as simple fractions. These numbers have non-repeating, non-terminating decimals, like \(\pi\) and \(\sqrt{2}\).

• Rational Numbers: Fractions like 1/2, or decimals like 0.333...
• Irrational Numbers: Numbers like \(\pi\) or \(\sqrt{2}\)

Together, they form the entire set of real numbers, providing a complete representation of values that we can plot, calculate, and use for a wide range of applications.

Graphing

Graphing is a visual way to represent numbers and relationships between them on a line or a plane. Let's focus on graphing on a number line, like the task given, which involves plotting integers and real numbers.

Creating a number line begins with drawing a straight horizontal line and marking it with points that correspond to integer values. This helps in visually understanding the relationship and spacing between numbers. However, real numbers, including integers, can be plotted with precision as they can represent any point on the line.

On a number line:

• Each tick mark represents an integer.
• Real numbers are placed depending on their values between these tick marks.
• For instance, \(-2\) would be plotted two spaces to the left of zero.

Graphing serves as a significant tool because it allows math problems to be visualized. It offers students a way to grasp abstract concepts by making them part of a coordinate system.