In mathematics, numbers are the key building blocks. there are a number of rational numbers in a pair of rational numbers. The number system is categorized into real and imaginary numbers. The imaginary numbers are the numbers that cannot be represented on the number line. On the contrary, real numbers are the number that can be represented on the number line. The numbers that can be written in the form of \(\frac{a}{b},\,b \ne 0,\) where \(a,\,b\) are any integers are known as rational numbers.

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Rational Numbers: Details

The word “rational number” has originated from the concept of “ratio”. The numbers that can be written in the form of \(\frac{p}{q},\,q \ne 0,\) where \(p,\,q\) are integers are known as rational numbers.

Example:
\( – \frac{1}{7},\,\frac{3}{5}\) are few examples of rational numbers.
In another way, we can say that two integers that can be written in the fractional form are called rational numbers.
All composite numbers, even numbers, odd numbers, prime numbers are examples of rational numbers. All the integers can be written in rational numbers by dividing the given integers by one.
Example: \(\frac{2}{1},\, – \frac{5}{1}\)
Note: The integer zero is also a rational number that can be written in the form of \(\frac{p}{q}\) By dividing the zero by one.
\(0 = \frac{0}{1}.\)
All counting numbers are known as natural numbers. All the natural numbers plus zero are called whole numbers. Whole numbers, along with the negative of natural numbers, is called integers. All integers are known as rational numbers. All rational numbers are represented on the number line.
Numbers that can not be written in the form of \(\frac{p}{q}\) are known as irrational numbers. Irrational numbers are real numbers that have non-terminating and non-recurring decimal expansions.
Example: \(\sqrt 2 ,\,\sqrt 7 \) etc.

Rational Numbers Between Two Rational Numbers

The numbers that can be written in the form of \(\frac{p}{q},\,q \ne 0,\) where \(p,\,q\) are integers, are known as rational numbers. Every integer on the number line is also a rational number. In between two rational numbers, there is countless number of rational numbers.
As we know that every fraction and every decimal (terminating, non-terminating, repeating) are known as rational numbers. Thus, in between two rational numbers, we have infinite rational numbers.
Example:
Some of the rational numbers between \(2\) and \(3\) are \(2.1 = \frac{{21}}{{10}},\,2.3 = \frac{{23}}{{10}},\,2.7 = \frac{{27}}{{10}},\,2.9 = \frac{{29}}{{10}}.\)

Irrational Numbers Between Two Rational Numbers

The numbers that can be written in the form of \(\frac{p}{q},\,q \ne 0,\) where \(p,\,q\) are integers are known as rational numbers. Numbers that can not be written in the form of \(\frac{p}{q}\) are known as irrational numbers.

In the number line, in between any two numbers, we get countless of a number of decimals that are non-terminating and non-recurring in nature. We know that non-terminating non-recurring decimals are known as irrational numbers.
Therefore, in between any two rational numbers, we have infinite irrational numbers.
Example:
Some of the irrational numbers between two rational numbers \(2\) and \(3\) are
\(2.11233……,\,2.3445666,\,2.999965467.\)

Rational Numbers between Two Rational Numbers with The Same Denominator

The first step in finding the rational numbers between two rational numbers is identifying the denominators of the rational numbers.
1. If the denominators of the given rational numbers are the same, then proceed with the further steps.
2. Now, check for the numerators of the rational numbers, whose denominators are equal.
3. Suppose the numerators of the rational numbers differ by the larger values. The rational numbers between two rational numbers can be written by increasing the one for the numerator and writing the rational numbers by keeping the same denominator.
Example: The five rational numbers between \(\frac{1}{9}\) and \(\frac{8}{9}\) are \(\frac{2}{9},\,\frac{3}{9},\,\frac{4}{9},\,\frac{5}{9},\,\frac{6}{9}\) etc.
1. If the numerators of the rational number differ by smaller values, multiply both numerators and denominators of given rational numbers by \(10.\) And, write the rational numbers between two rational numbers by increasing the one for the numerator.
Example: The ten rational numbers between \(\frac{2}{7}\) and \(\frac{4}{7}\) are found by multiplying and dividing by \(10.\)
\( \Rightarrow \frac{2}{7} \times \frac{{10}}{{10}} = \frac{{20}}{{70}}\) and \(\frac{4}{7} \times \frac{{10}}{{10}} = \frac{{40}}{{70}}\) are \(\frac{{41}}{{70}},\,\frac{{42}}{{70}},\,\frac{{43}}{{70}},\,\frac{{44}}{{70}},\,\frac{{45}}{{70}},\,\frac{{46}}{{70}},\,\frac{{47}}{{70}},\,\frac{{48}}{{70}},\,\frac{{51}}{{70}},\,\frac{{53}}{{70}},\) etc.

The mean of the two numbers is obtained by half of the sum of the numbers. The mean of two numbers a,b is \(\frac{{a + b}}{2}.\)

Rational Numbers Between Two Rational Numbers with Different Denominators

1. To find the rational numbers between two rational numbers with different denominators, equate their denominators and proceed as mentioned above method.
2. By taking the LCM of denominators of the denominators, equate their real numbers to LCM.
3. We can also equate their denominators by multiplying and dividing the rational numbers with some constant number.
4. Once the denominators are equalled, proceed with the rational numbers between two rational numbers with the same denominator.
Example: The rational numbers between \(\frac{1}{4}\) and \(\frac{1}{2}\) are found by making their denominators equal to \(8.\)
\(\frac{2}{8}\) and \(\frac{4}{8}\) and multiply and divide with \(10.\)
\(\frac{{20}}{{80}}\) and \(\frac{{40}}{{80}}\) are \(\frac{{21}}{{80}},\,\frac{{22}}{{80}},\,\frac{{23}}{{80}},\,\frac{{25}}{{80}},\,\frac{{31}}{{80}}\) etc.

Mean Method of Finding Rational Numbers Between Two Rational Numbers

Let us consider two rational numbers are \(a\) and \(b\)
1. Find the mean of the given numbers
a. \(\frac{{a + b}}{2}.\)
b. That gives the rational middle number between the given rational numbers.
2. Next, find the mean of \(a\) and \(\frac{{a + b}}{2}.\)
a. \(\frac{{\left( {a + \frac{{a + b}}{2}} \right)}}{2} = \frac{{3a + b}}{4}\)
3. Similarly, find the mean of \(b\) and \(\frac{{a + b}}{2}.\)
a. \(\frac{{\left( {b + \frac{{a + b}}{2}} \right)}}{2} = \frac{{a + 3b}}{4}\)
4. Again, find the mean of \(a\) and \(\frac{{3a + b}}{4}\) also, the mean of the \(b\) and \(\frac{{a + 3b}}{4}\) proceed in the same way.
Example:
The five rational numbers between \(2\) and \(3\) are found as follows:
1. The mean of \(2\) and \(3\) is \(\frac{{2 + 3}}{2} = \frac{5}{2}\)
2. Next, the mean of \(2\) and \(\frac{5}{2}\) is \(\frac{{2 + \frac{5}{2}}}{2} = \frac{9}{4}\)
3. Mean of \({\frac{5}{2}}\) and \(3\) is \(\frac{{\frac{5}{2} + 3}}{2} = \frac{{11}}{4}\)
4. Mean of \(2\) and \(\frac{9}{4}\) is \(\frac{{2 + \frac{9}{2}}}{2} = \frac{{17}}{8}\)
5. Mean of \(3\) and \({\frac{{11}}{4}}\) is \(\frac{{3 + \frac{{11}}{4}}}{2} = \frac{{23}}{8}\)
Therefore, the five rational numbers between \(2\) and \(3\) are \(\frac{{17}}{8},\,\frac{{11}}{4},\,\frac{5}{2},\,\frac{9}{4},\,\frac{{23}}{8}.\)

Important Tricks

We know that in between two rational numbers, we have infinite rational numbers as well as infinite irrational numbers. We can find the middlemost rational and irrational number between given two rational numbers as follows:
Let \(a\) and \(b\) are two rational numbers, then
1. The middlemost rational number lies between the two rational numbers \(a\) and \(b\) is found by calculating the arithmetic mean
\(\frac{{a + b}}{2}.\)
Example: The rational middle number between \(2\) and \(3\) is \(\frac{{2 + 3}}{2} = \frac{5}{2}.\)
1. The middlemost irrational number lies between the two rational numbers \(a\) and \(b\) is found by calculating the geometric mean
\(\sqrt {a \times b} .\)
Example: The irrational number between \(2\) and \(3\) is \(\sqrt {2 \times 3} = \sqrt 6 .\)

Solved Examples – Rational Number Between Two Rational Numbers

Q.1. Find the rational number lying in the middle of the rational numbers \(\frac{2}{7}\) and \(\frac{3}{4}.\)
Ans: Given rational numbers are \(\frac{2}{7}\) and \(\frac{3}{4}.\)
The rational middle number between two rational numbers can be found by calculating the arithmetic mean of the given numbers, and that equals half of the sum of the given numbers.
So, the required rational number is \(\frac{{\frac{2}{7} + \frac{3}{4}}}{2}\)
\( = \frac{{8 + 56}}{{56}}\)
\( = \frac{{64}}{{56}}\)
\( = \frac{8}{7}\)
Therefore, the rational number lying the middle of the rational numbers \(\frac{2}{7}\) and \(\frac{3}{4}\) is \(\frac{8}{7}.\)

Q.2. Find ten five rational numbers between 2 and 3.
Ans: The five rational numbers between \(2\) and \(3\) are found as follows:
1. The mean of \(2\) and \(3\) is \(\frac{{2 + 3}}{2} = \frac{5}{2}\)
2. Next, the mean of \(2\) and \(\frac{5}{2}\) is \(\frac{{2 + \frac{5}{2}}}{2} = \frac{9}{4}\)
3. Mean of \({\frac{5}{2}}\) and \(3\) is \(\frac{{\frac{5}{2} + 3}}{2} = \frac{{11}}{4}\)
4. Mean of \(2\) and \(\frac{9}{4}\) is \(\frac{{2 + \frac{9}{2}}}{2} = \frac{{17}}{8}\)
5. Mean of \(3\) and \({\frac{{11}}{4}}\) is \(\frac{{3 + \frac{{11}}{4}}}{2} = \frac{{23}}{8}\)
Therefore, the five rational numbers between \(2\) and \(3\) are \(\frac{{17}}{8},\,\frac{{11}}{4},\,\frac{5}{2},\,\frac{9}{4},\,\frac{{23}}{8}.\)

Q.3. Find the ten rational numbers between \( – \frac{3}{{11}}\) and \(\frac{8}{{11}}.\)
Ans: Given rational numbers \( – \frac{3}{{11}}\) and \(\frac{8}{{11}}\) has the same denominator.
So, the rational numbers between these are obtained by increasing the numerator by one and so on.
The ten rational numbers between \( – \frac{3}{{11}}\) and \(\frac{8}{{11}}\) are \( – \frac{2}{{11}},\, – \frac{1}{{11}},\,\frac{0}{{11}},\,\frac{1}{{11}},\,\frac{2}{{11}},\,\frac{3}{{11}},\,\frac{4}{{11}},\,\frac{5}{{11}},\,\frac{6}{{11}},\,\frac{7}{{11}}.\)

Q.4. Find out three rational numbers between 3 and 4.
Ans: The three rational numbers between \(3\) and \(4\) are found as follows:
1. The mean of \(3\) and \(4\) is \(\frac{{3 + 4}}{2} = \frac{7}{2}.\)
2. Next, the mean of \(3\) and \(\frac{7}{2}\) is \(\frac{{3 + \frac{7}{2}}}{2} = \frac{{13}}{4}.\)
3. Mean of \(\frac{7}{2}\) and \(4\) is \(\frac{{\frac{7}{2} + 4}}{2} = \frac{{15}}{4}.\)
Therefore, the three rational numbers between \(3\) and \(4\) are \(\frac{{13}}{4},\,\frac{7}{2},\,\frac{{15}}{4}.\)

Q.5. Find the irrational number between 3 and 5.
Ans: The irrational number between two rational numbers is the geometric mean of the given rational numbers.
So, the irrational number between \(3\) and \(5\) is \(\sqrt {3 \times 5} = \sqrt {15}. \)
Therefore, the irrational number between \(3\) and \(5\) is \(\sqrt {15} .\)

Summary

In this article, we have studied the definition of rational numbers and irrational numbers. This article gives rational numbers between two rational numbers and irrational numbers between two rational numbers.
We also studied finding the rational numbers between two rational numbers with the same denominator and finding the rational numbers between two rational numbers with different denominators and the mean method and the solved examples.
This article also gives finding the rational and irrational numbers between two rational numbers.

Frequently Asked Questions (FAQs)- Rational Number Between Two Rational Numbers

Frequently asked questions related to rational number between two rational numbers is listed as follows:

Q. How to find the rational numbers between two rational numbers.
Ans: Finding the LCM of the denominators and making their denominators equal, increasing the numerators, we will find the rational numbers.

Q. How many irrational numbers lie in between two rational numbers.
Ans: There are infinite irrational numbers that lie in between two rational numbers.

Q. How many rational numbers lie in between two rational numbers.
Ans: There are infinite rational numbers that lie in between two rational numbers.

Q. How to find the irrational number between two rational numbers?
Ans: The irrational number between two rational numbers can be found by calculating the geometric mean of the given numbers.

Q. How to find the rational number between two rational numbers?
Ans: The rational number between two rational numbers can be found by calculating the arithmetic mean of the given numbers.

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