Problem 9 - Introductory and Intermediate Algebra for College Students 4th

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

We start by drawing a number line representing integers from -5 to 5. Then, beginning from zero, we locate and mark the number 2 to the right on the line.

Step 1: Drawing a Number Line

First, a number line is needed. Start by drawing a straight horizontal line. Mark the center of the line as 0. Keep the same distance while marking the numbers to the left as negative (until -5) and to the right as positive (until 5). This will represent the integers from \(-5\) to \(5\).

Step 2: Plotting the Number 2

The next step requires the number 2 to be plotted. Starting from zero, count two units to the right and make a visible mark or dot. That mark represents the real number 2 on the number line.

Key Concepts

Integers

Integers are a type of number which include all the whole numbers and their negatives, like

• -3, -2, -1, 0, 1, 2, 3.

These numbers are often represented on a number line, which is a simple visual tool to understand the sequence and spacing of these numbers.
You can imagine a number line as an endless path extending both left and right,

• where left implies negative
• and right represents positive values.

Zero sits right at the center, serving as a neutral ground between the positive and negative sides. Each integer is evenly spaced from its neighbor. This equal spacing is crucial because it helps maintain accurate relationships between these numbers. For example, by drawing a line starting at

• -5 on the left
• and extending to 5 on the right,

you create a simple integer number line. In any counting or calculation involving integers, keeping these consistent intervals is very important.

Plotting Points

Plotting points on a number line is a fundamental skill in understanding how numbers are related in a sequence.
When you plot a point, you are essentially placing a mark at a specific spot on the line that corresponds to a number. To successfully plot a point:

• First locate zero on the number line to orient yourself.
• If the number is positive, move right from zero;
• if negative, move left.

For instance, if you need to plot the number 2:
You would find zero, then move two equal spaces to the right until you reach the spot corresponding to 2. Mark this spot clearly with a dot or any small mark,

• making it obvious where the number lives.

This process not only helps visualize numbers but also lays the groundwork for more complex graphing tasks in mathematics.

Real Numbers

Real numbers include all the possible numbers you can find on a number line. This includes:

• integers like -2, 0, and 3,
• rational numbers such as fractions and repeating decimals, and
• irrational numbers like \( \pi \) or the square root of non-perfect squares.

Every point on a number line represents a real number.
Real numbers are comprehensive and form the basis of most everyday math operations.To illustrate this concept with our exercise, when we plot the number 2 on the number line, we’re identifying it as a real number.

• This means it’s not just an integer but also part of a larger system including decimals and fractions.

Understanding real numbers helps to see how arithmetic, algebra, and calculus operate since all these fields rely on manipulating and understanding these numbers.Each specific point on a number line is important because it provides precise value whether as a distance (like 2) or more abstract representations (like \( \pi \)).
Recognizing and plotting real numbers is therefore an essential skill in mathematics.