What is a Rational Number?
Integers, fractions, and decimals make up the set of rational numbers. A rational number is a number that can be written as \( \frac{a}{b} \), where “a” and “b” are integers, and b ≠ 0.
Graphing Rational Numbers
We can use a horizontal or vertical number line to plot or graph rational numbers. Here we have graphed rational numbers on a horizontal number line.
We start by drawing a number line with numbers at equal intervals along the length of the line. Then we mark the numbers on the number line. For example, we can use these number lines to graph rational numbers.
For example, let’s graph \( \frac{1}{4} \) and the negative form of \( \frac{1}{4} \).
First, draw a horizontal line. Mark the number 0 in the middle. Mark +1 on the right and -1 on the left side, at an equal distance from the number 0. Now check for the denominator of the given fraction, which is 4.
So, in the next step we divide the distance from 0 to 1 and -1 to 0 into 4 equal parts.
As one \( \frac{1}{4} \) is \( \frac{1}{4} \), two \( \frac{1}{4} \) is \( \frac{2}{4} \), and so on, we mark the first equal part on the right of the number 0 as \( \frac{1}{4} \). Then mark the part at the same distance on the left of the number 0 as \( -\frac{1}{4} \).
Therefore, the numbers \( \frac{1}{4} \) and \( -\frac{1}{4} \) can be plotted on the number line as,
Similarly, let’s graph 1.2 and the negative of 1.2.
First, draw a horizontal line. Mark the number 0 in the middle.
We see that 1.2 and -1.2 are equivalent to \( \frac{12}{10} \) and \( -\frac{12}{10} \), respectively. So, 1.2 is located between 0 and 2, and -1.2 is located between -2 and 0.
Therefore, mark +2 on the right and -2 on the left, at an equal distance from the number 0. Since the denominator of the equivalent fractions is 10, divide the distance between -2 and 2 into 10 equal parts, that is, 5 equal parts on each side of the number 0.
Hence, we mark -1.6, -1.2, -0.8, -0.4, 0.4, 0.8, 1.2 and 1.6 on the number line. Now we can easily locate the number 1.2 and its negative form, -1.2, on the number line as shown in the image.
Comparing Fractions
The comparison of fractions is the same as the comparison of numbers on the number line. On a horizontal number line, the number on the right is always greater than the number on the left. Similarly, on a vertical number line, the number above is always greater than the number below.
Hence, to compare two or more fractions, we need to graph them on a number line, and then observe their position with respect to each other.
For example, let’s compare \( -\frac{1}{4} \) and \( -\frac{3}{4} \) .
First, draw a number line. Then, divide the line into suitable intervals, and mark the fractions of the numbers on the line such that the two given fractions lie on the number line.
Locate and mark the given fractions, and observe their position with respect to each other. The number on the right side of the line will be greater than the other number.
\( -\frac{1}{4} \) is on the right of \( -\frac{3}{4} \). So, \( -\frac{3}{4}< -\frac{1}{4} \).
The Absolute Value of a Number
The absolute value of a number is the distance between the number and 0 on a number line. The absolute value of a number a is denoted as |a|.
For a positive number, the absolute value is the number itself. For a negative number, the opposite of the number represents the absolute value. Otherwise, we can also omit the negative sign of a negative number to know its absolute value.
Absolute value formula:
\( \left | a \right |=a \) and \( \left | a \right |=a \)
Where “a” is a positive number.
For example, the absolute value of the number -3 is obtained from the following observation:
The distance from 0 to -3, is the same as the distance from 0 to 3 on a number line that is 3. Therefore, the absolute value of -3 will be,
\( \left | -3 \right |=3 \)
Solved Examples
Example 1: Find the absolute value of \( -3\frac{1}{4} \).
Solution:
First, plot the number \( -3\frac{1}{4} \) on the number line. Then, look at the number line. The distance between \( -3\frac{1}{4} \) and 0, is \( 3\frac{1}{4} \) units.
Therefore, the absolute value of \( -3\frac{1}{4} \) is
\( \left | -3\frac{1}{4} \right |=3\frac{1}{4} \)
Example 2: The given line shows the swimming distance (in feet) of two animals from the sea level.
Which animal is closer to the sea level?
Solution:
The sea level is at 0 feet. So, use the absolute values to find the distances and then compare.
Distance from the sea level that the sea lion swims \( =\left | 5 \right |=5\) units [Take absolute value]
Distance from the sea level that the shark swims \( =\left | -4 \right |=4\) units [Take absolute value]
Now, 4 < 5
Therefore, the shark is closer to sea level in this situation.
Example 3: Compare 1 and |-4|.
Solution:
First, draw a number line. Plot 1 on the number line.
Now, \( \left | -4 \right |=4 \) , so plot 4 on the number line.
Number 1 is on the left of \( \left | -4 \right | \), that is, 4.
Therefore, \( 1< \left | -4 \right | \)
Example 4: Compare \( \left | -5 \right |,~6,~\left | 3 \right |,~0 \).
Solution:
To compare the given numbers, we will plot them on a number line, and then observe their position with respect to each other.
First, draw a number line,
Take the absolute value of \( \left | -5 \right |=5 \). Mark 5 on the number line.
Take the absolute value of \( \left | 3 \right |=3 \). Mark 3 on the number line.
Then mark the numbers 6 and 0 on the line.
Now observe the numbers from left to right and make the comparison,
\( 0< \left | 3 \right |< \left | -5 \right |< 6 \)
Example 5: A fraction temperature indicator indicated two temperatures as \( -3\frac{5}{6}~F \) and \( -3\frac{1}{6}~F \) . Compare the temperatures.
Solution:
To compare the two temperatures, plot each on a number line and observe their position with respect to each other.
\( -3\frac{5}{6} \) is to the left of \( -3\frac{1}{6} \).
Therefore, \( -3\frac{5}{6}~F< -3\frac{1}{6}~F \).
The absolute value of a negative number is called the modulus of the number. For a negative number, the absolute value is the opposite of the number. For example, the modulus of -5 is \( \left | -5 \right |=5 \).
The absolute value of a positive number is the number itself. For example, the absolute value of 9 is \( \left | 9 \right |=9 \).
Yes, whole numbers can be written in the form \( \frac{a}{b} \) where [b≠0]. Therefore, whole numbers are fractions as well.
For example, 5 can be written as \( \frac{5}{1} \). Therefore, the number 5 is a fraction as well.