Problem 22 - Elementary Algebra

Question

For the following problems, draw a number line that extends from -3 to 3\. Locate each real number on the number line by placing a point (closed circle) at its approximate location. $$-2$$

Step-by-Step Solution

Answer

Answer: The real number -2 is located at the second tick mark from the left end of the number line, between -3 and -1.

Step 1: Draw the number line

Begin by drawing a straight horizontal line. Place two arrows at each end, pointing outward, to indicate that the line represents the number line.

Step 2: Mark the range of the number line

Now, label the left end of the number line with -3 and the right end with 3. This will represent the range of the given number line.

Step 3: Divide the number line into equal intervals

Divide the number line into 6 equal intervals. To do this, you can place 6 equally spaced dashes (tick marks) between -3 and 3, representing the integers from -2 to 2.

Step 4: Label the intervals

Next, label each interval with the respective integer. Starting from the left end, you should have the following sequence of numbers: -3, -2, -1, 0, 1, 2, and 3.

Step 5: Place the point on -2

Locate the number -2 on your number line and place a closed circle (point) above the tick mark for -2. This represents the approximate position of the real number -2 on the given number line.

Key Concepts

Real Numbers

Understanding real numbers begins with grasping the comprehensive set of values that describe quantities along a continuous line - hence the name 'real'. This broad category includes rational numbers like whole numbers (0, 1, 2, ...), integers (-1, -2, -3, ...), and fractions (1/2, 2/3, 5/4, ...), as well as irrational numbers such as \( \sqrt{2} \) or \(\pi\).

In simple terms, if you can place a number on an endless line stretching from negative to positive infinity, it's a real number. This is why on the number line in our exercise, not only is -2 a real number, but so are all the possible values between the tick marks, even if they are not marked or labeled. Some real numbers are easier to represent on this line than others, like integers, while others, like irrational numbers, can only be approximated because they have non-terminating decimal expansions.

Plotting Numbers

Learning how to accurately plot numbers is a fundamental skill in mathematics that enables students to visualize the relationships between different numbers.

To plot a number, you first need a number line, which is essentially a horizontal line with numbers placed at intervals corresponding to their value. In the exercise, the number line was drawn to range from -3 to 3. When plotting an integer such as -2, you identify where on the line this number falls—in this case, two units to the left of zero. Marking it with a closed circle indicates that -2 is exactly at this point. For numbers that are not whole numbers, such as fractions or decimals, you would find the appropriate space between integers that best represents their value. The precision of your plot depends on how finely you divide your number line.

Integer Intervals

An interval in mathematics refers to all the numbers lying between two distinct points on the number line. When those points are integers, we refer to this as an 'integer interval.' These intervals can be open or closed and can be bounded or unbounded, depending on the context.

In this problem, we looked at the integer interval [-3, 3], which means all the integers from -3 up to and including 3. This is known as a 'closed interval' because we include the end points (signified by the square brackets). In visual terms, this interval would be represented by a segment of the number line with closed circles on both -3 and 3. By dividing this interval into equal parts, as in the exercise, we can easily locate and plot individual integers or other numbers within the interval.