Problem 23 - 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. $$-\frac{1}{8}$$

Step-by-Step Solution

Answer

Answer: $$-\frac{1}{8}$$ is approximately located halfway between 0 and $$-\frac{1}{4}$$ on the number line.

Step 1: Draw a number line

Draw a horizontal line representing the number line, and mark its ends with -3 and 3.

Step 2: Mark equal intervals on the number line

Divide the number line into smaller equal intervals, such as 1/4 or 1/2, for better precision. Mark the intervals along the number line. For this exercise, we will use intervals of 1/4.

Step 3: Locate the given number on the number line

Locate the number $$-\frac{1}{8}$$ on the number line by approximating its position relative to the marked intervals. Since $$-\frac{1}{8}$$ is greater than -1 and less than 0, it should be placed in that range. Furthermore, $$-\frac{1}{8}$$ is halfway between 0 and $$-\frac{1}{4}$$, so place a point (closed circle) at the approximate location of $$-\frac{1}{8}$$ between 0 and $$-\frac{1}{4}$$ on the number line. The number $$-\frac{1}{8}$$ is now correctly located on the number line.

Key Concepts

Real Numbers on the Number Line

Understanding real numbers and their representation on a number line is fundamental in math. A number line is a straight, horizontal line that contains points which correspond to the numbers. The position of each number is indicated by a unique point on the line. The real numbers include all the numbers within the number line, so this includes whole numbers, fractions, irrational numbers, and their negative counterparts. When plotting a real number, it's crucial to first identify the scale and interval of the number line. For example, if our number line extends from -3 to 3, we're dealing with a range that includes negatives, zero, and positive numbers. After drawing the number line and marking its endpoints, we need to determine the intervals before locating specific numbers.

Understanding Negative Fractions

Negative fractions are simply fractions with a negative sign, indicating a value less than zero. They behave like any other fraction, except they are positioned to the left of zero on the number line. To plot a negative fraction like \( -\frac{1}{8} \) on the number line, first understand that the number is between 0 and -1. The negative sign signifies its position on the left side of zero, indicating that it is a portion of the whole number -1. If there's a challenge in visualizing where a fraction lies, it is often helpful to think about its decimal equivalent or to compare it with nearby fractions with similar denominators on the number line.

Dividing Intervals on the Number Line

Interval division is the process of splitting the number line into equal parts between the marked numbers. The choice of intervals depends on the precision needed. For instance, dividing the line into quarters (\frac{1}{4}) allows us to accurately place fractions that have a denominator of 4 or multiples of 4. When plotting \( -\frac{1}{8} \) on a number line marked in quarters, we can observe that it falls precisely in the middle between 0 and \( -\frac{1}{4} \). Therefore, by counting the intervals, we are able to place \( -\frac{1}{8} \) correctly. Dividing intervals further into eighths or sixteenths can provide even more precision when plotting fractions with larger denominators.