Representation of rational number on a number line:
Given a rational number \(\frac{p}{q}\), then the number is represented on the number line using the following procedure.
Step -1: Draw a number line and mark the integers with the positive integers to the right of \(0\) and the negative integers to the left of \(0\).
Step - 2: Divide the interval between the two consecutive integers into \(q\) equal parts.
Step - 3: Starting from \(0\), move \(p\) parts to the right of \(0\) if the number is positive or to the left of \(0\) if the number is negative.
Step - 4: Now, locate the number \(\frac{p}{q}\) on the number line.
Important!
If the denominator is lesser than the numerator ((i.e.)improper fraction), then convert it into mixed fraction. Then, locate accordingly on the number line.
Example: If the number is \(\frac{11}{7}\), then \(\frac{11}{7} = 1 \frac{4}{7}\).
Locate \(\frac{4}{7}\) between the interval \(1\) and \(2\) on the number line.
Absolute value of a rational number:
The absolute value of a rational number \(a\), written as \(|a|\), represents the distance of \(a\) from \(0\) on the number line.
The absolute value of a positive number is the number itself and the absolute value of a negative number is its positive value.
The absolute value of any rational number \(x\) is always positive. (i.e.) \(\left|x\right|>0\)
The Density of Rational Numbers:
Between any two distinct rational numbers, there exist infinitely many other rational numbers.
Between any two rational numbers \(a\) and \(b\), there is always a rational number \(\frac{1}{2}(a+b)\), such that \(a < \frac{1}{2}(a+b) < b\).
Method to find the rational number between any two rational numbers:
There are many ways to find a rational number between the given two rational numbers. Some of them are mentioned below:
Method 1: Average method
1. Let the rational number be \(a\) and \(b\).
2. Add \(a\) and \(b\) and divide the sum by \(2\). That is, \(c = \frac{a+b}{2}\) which might lie in between those two number.
3. To get another rational number, find the average of \(c\) and \(a\), for one more rational number, find the average of \(c\) and \(b\). Proceeding in this way, you can find the infinite number of irrational between two rational numbers.
Example:
Let the rational numbers be \(6\) and \(7\). Now follow the steps to find the rational numbers.
Add the rational numbers \(6\) and \(7\) and divide the sum by \(2\).
That is .
Method 2: Same denominator method
This method gives all the required number of rational numbers between \(a\) and \(b\) in one step.
If we want to find \(n\) rational numbers between the numbers \(a\) and \(b\), we write \(a\) and \(b\) as rational numbers with denominator \(n + 1\).
That is, make it as .
Now you can verify that the numbers between and are all rational numbers between \(a\) and \(b\).
Example:
Find the set of \(4\) rational numbers between the numbers \(6\) and \(7\).
Here \(a = 6, b = 7\) and \(n = 4\).
Substituting the known values, we will have
and .
Thus the number between \(30/5\) and \(35/5\) are \(31/5, 32/5, 33/5\) and \(34/5\).
Therefore, the four rational numbers are \(31/5, 32/5, 33/5\) and \(34/5\).
Method 3: Increasing the decimal space
Given any two decimal numbers, rewrite them with more decimal places to create more space so as to obtain the decimal numbers lying between them.
Example:
To find the three rational numbers between \(3.15\) and \(3.16\), first we will write them with more decimal places as follows:
\(3.15 = 3.150\) and \(3.16 = 3.160\)
Now, we directly write some of the decimals lying between them as \(3.151\), \(3.152\), \(3.153\), \(3.154\), \(3.155\), \(3.156\), \(3.157\), \(3.158\) and \(3.159\).
Important!
In the same way, we can find as many rational numbers between two rational numbers. Thus, there are infinitely many rational numbers between any two given rational numbers.
Rational number \(Q\) does not have a unique representation in the form of \(p/q\).
. These are called an equivalent rational number.