Rational Number
Integers are made up of whole numbers which are numbers such as 0, 1, 2, 3,… and their opposites which are -1, -2, -3…and so on. A rational number is a number that can be written as \( \frac{x}{y} \) , where x and y are integers and y ≠ 0.
Using a Number Line to find Absolute Value of Rational Number
Absolute value is the distance between 0 to the number on the number line. In other words, it is a number’s magnitude or size which is calculated using a number line. The absolute value (or modulus) a of a real number ‘a’ is its non-negative value, regardless of its sign.
For example: \( \left | ~-~5~ \right |~=~5 \)
\( \left | ~5~ \right |~=~5 \)
In this example, the \( \left | ~-5~ \right | \) and \( \left |~ 5 ~\right |~=~5 \) both are five units away from the ‘0’. So that the absolute value is 5 units.
For example, for an account balance of −30 dollars, write | −30 | = 30 to describe the size of the debt in dollars.
Examples on Finding Absolute Values of Rational Numbers
a) Find the absolute value of -6.
Solution:
Plot -6 on a number line in the graph. Then calculate the distance between -6 and 0. Then the distance = 6.
So, \( \left | ~-6 ~\right |~=~6 \)
b) Find the absolute value of \( 1~\frac{1}{2} \).
Solution:
Plot \( 1~\frac{1}{2} \) on a number line. Then calculate the distance between \( 1~\frac{1}{2} \) and 0 is \( 1\frac{1}{2} \).
So, \( \left | ~1~\frac{1}{2} \right |~=~1\frac{1}{2} \)
c) Find the absolute value of 7.
Solution:
Plot 7 on a number line. Then calculate the distance between 7 and 0. So the distance is 7.
So, \( \left | ~7~ \right |~=~7 \)
Comparing Rational Numbers
On a number line, numbers increase when we move from left to right. Hence if a rational number lies to the right of a given number then that number is greater. Similarly, rational numbers that lie to the left of a given number are lesser than that number.
Greater Than (>), less than (<) and equal to (=) symbols are used to draw comparison between numbers.
a) Compare \( \left | 9 \right | \) and \( \left | ~-9~\right | \).
Solution:
Plot \( \left | 9 \right |~=~9 \) and plot \( \left |~ -9~ \right |~=~9 \) on a number line. Then you can observe that \( \left | 9 \right | \) lies on \( \left | 9 \right | \).
So, \( \left | ~-9 ~\right |~=~9 \)
b) Compare 7 and \( \left | 4.5 \right | \).
Solution:
Plot 7 and plot \( \left | 4.5 \right |~=~4.5 \) on a number line. Then you can observe that \( \left | 4.5 \right | \) is to the left of 7.
So, \( \left | 4.5 \right |< 7 \)
c) Compare \( \left | \frac{-3}{4} \right | \) and \( -\frac{1}{4} \).
Solution:
Plot \( -\frac{1}{4} \) and plot \( \left | \frac{-3}{4} \right |~=~\frac{3}{4} \)on a number line. Then you can observe that \( -\frac{1}{4} \) is to the left of \( ~ -~\frac{3}{4} \).
So, \( ~-~\frac{1}{4}< \left | \frac{-3}{4} \right | \)
Example based on Real Life Situation
1. The table below shows the temperatures of every year at a particular place from 2017 to 2020. Which year had the highest and lowest temperatures?
Year | 2017 | 2018 | 2019 | 2020 |
Average Temperature anomaly (fahrenheit) | -2 | -3 | -3.5 | -1 |
Solution:
To determine which year had the highest and lowest temperatures graph the temperatures on a horizontal scale number line.
The number line shows the temperatures. The values to the left are lower and values to the right are higher in a number line. Hence, the lowest temperatures have to be the leftmost and the highest temperatures will be the rightmost value.
Hence 2020 has the highest temperature and 2019 has the lowest temperature.
2. Mark and 5 friends A, B, C, D and E build tree houses. Mark builds the tallest tree house. The tip of Mark’s tree house is kept as the reference and is taken as the 0. Keeping Mark’s tree house as reference, the elevations of the other tree houses are given below. For example, A’s elevation is -20 m implies that the measure from the top of Mark’s tree house to the top of A’s tree house is \( \left | -20 \right |~=~20 \)m. Which treehouse is the tallest after Mark’s and which treehouse is the shortest? If Mark’s tree house is 120m, find the height of the tallest and shortest treehouses.
House | A | B | C | D | E |
Elevation (in meters) | -20 | -45 | -10 | -25 | -60 |
Solution:
To determine which house is the tallest and shortest, graph the elevations on a number line.
We know that Mark has the tallest tree house so the elevation that is closest to Mark is going to be the tallest tree house and that is farthest from Mark is going to be the shortest. In other words, the elevation closest to zero will be the tallest and the one that is farthest from zero will be the shortest.
Let’s find the absolute values of each,
C: \( \left | ~-10~ \right |~=~10 \) m
A: \( \left | ~-20~ \right |~=~20 \) m
D: \( \left |~ -25~ \right |~=~25 \) m
B: \( \left | ~-45 ~\right |~=~45 \) m
E: \( \left |~ -60~ \right |~=~60 \) m
So, C has the tallest treehouse and E has the shortest treehouse.
Height of C’s treehouse is 120 – 10 = 110 m
Height of E’s treehouse is 120 – 60 = 60 m
Opposites are numbers that are at the same distance from 0. That is, 3 and -3 are opposites and similarly the opposite of -4 is 4. Another term for opposites is additive inverse.
The natural numbers, their additive inverses, and zero make up the set of numbers known as the integers. Whereas numbers that can be expressed as a ratio between two integers (denominator 0) are known as rational numbers. All integers are rational numbers, but all rational numbers are not integers.