Problem 22 - Elementary and Intermediate Algebra

Question

Graph each rational number on the number line. $$ 3.87 $$

Step-by-Step Solution

Answer

Plot 3.87 between 3.8 and 3.9, closer to 3.9, on the number line.

Step 1: Understand the Rational Number

Identify the given rational number. Here, the rational number is \(3.87\). It is a decimal number between 3 and 4.

Step 2: Divide the Number Line

Sketch a number line and mark the integers 3 and 4. Divide the interval between 3 and 4 into 10 equal parts, since the decimal part goes up to the hundredth place, we are focusing on the first decimal first (i.e., 0.8).

Step 3: Locate 3.8 on the Number Line

Count 8 parts after the number 3 on the number line. This spot represents 3.8.

Step 4: Further Divide Between 3.8 and 3.9

Now to pinpoint 3.87, further divide the segment between 3.8 and 3.9 into 10 smaller parts. Count 7 parts after 3.8.

Step 5: Plot the Number

Mark the point 3.87 on the number line. This is 3.80 plus 0.07 (7 parts).

Key Concepts

Number Line

A number line is a straight horizontal line with numbers placed at equal intervals along its length. This tool helps us visualize numbers, including rational and irrational numbers. To graph a rational number like 3.87, we first need to correctly draw and label our number line.
First, mark the integers (whole numbers) on the line, ensuring equal spacing between them. This setup allows us to easily fit more specific points, like decimal numbers, along the line by further subdividing these intervals.

Decimal Representation

Decimal representation is a method to express fractions and rational numbers using the base-10 system. The number 3.87 can be broken down into its whole part (3) and decimal part (0.87).
This representation is helpful for pinpointing exact positions on a number line. By understanding how to interpret these parts, we can accurately graph our number. For instance, the whole number tells us the basic position, and the decimal part helps us make more granular divisions within that segment.

Plotting Points

Plotting points on a number line involves marking the position of a number accurately. To start, we identify and locate the main interval the number falls between, such as 3 and 4 for 3.87. Next, subdivide this interval into smaller parts to get closer to the precise value.
For 3.87, 3.8 is the first point we plot. Then, subdivide the interval between 3.8 and 3.9 and mark 3.87 within this smaller portion. This process ensures the number is located at the exact spot on the number line.

Subdivision of Intervals

Subdivision of intervals is crucial for accurately graphing numbers, especially decimals. It involves breaking down larger sections of the number line into smaller, equal parts. For the number 3.87, we start by locating the interval between 3 and 4.
Then, we divide it into 10 equal parts to locate 3.8. To find 3.87, we further subdivide the segment between 3.8 and 3.9 into 10 smaller parts and locate the exact seventh part, marking it precisely. This technique ensures that even small decimal differences are accurately represented on the number line.