Irrational Numbers between Two Rational Numbers: Irrational numbers are those real numbers that cannot be represented in a ratio. They are expressed usually in the form of \(R\backslash Q,\) in which the backward slash symbol denotes the ‘set minus’. This can also be expressed as \(R – Q,\) which states the difference between a set of real numbers and rational numbers. In this article, we will learn about irrational numbers, sets, properties, along with some examples.

What is an Irrational Number?

Definition: An irrational number is defined as the number that cannot be expressed in the form of \(\frac{p}{q},\) where \(p\) and \(q\) are coprime integers and \(q \ne 0.\) Irrational numbers are the set of real numbers that cannot be expressed in fractions or ratios. There are plenty of irrational numbers which cannot be written in a simplified way.
Example: \(\sqrt 8 ,\sqrt {11} ,\sqrt {50} \) and Euler’s number \(e = 2.718281….,\) Golden ratio \(\varphi = 1.618034…..\)

Irrational Number Between Two Rational Numbers

Assume that we have two rational numbers \(a\) and \(b,\) then the irrational numbers between the two will be \(\sqrt {ab} .\) Now see the example given below for a better understanding:
Example-1: Find the irrational number between the two rational numbers \(2\) and \(5.\)
Then the irrational number between \(2\) and \(5\) is \(\sqrt {2 \times 5} = \sqrt {10} .\)
Example-2: Find the irrational number between the two rational numbers \(11\) and \(13.\)
Then the irrational number between \(11\) and \(13\) is \(\sqrt {11 \times 13} = \sqrt {143} .\)

Irrational Number Between Two Irrational Numbers

Let us consider an example
\(\sqrt 2 \) and \(\sqrt 3 \) are irrational numbers
\(\sqrt 2 = 1.4142\) (nearly)
\(\sqrt 3 = 1.7321\) (nealry)
Now we have to find an irrational number which should lie between \(1.4142\) and \(1.7321\)
The irrational number is \(1.50500500050000….\)

Rational Number Between Two Rational Numbers

The number of integers between two integers is limited (finite). Will the same happen in the case of rational numbers too, we will see that below.
Rahul took two rational numbers \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)
He converted them into rational numbers with the same denominators.
So, \(\frac{{ – 3}}{5} = \frac{{ – 9}}{{15}}\) and \(\frac{{ – 1}}{3} = \frac{{ – 5}}{{15}}\)
We have \(\frac{{ – 9}}{{15}} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}} < \frac{{ – 5}}{{15}}\) or \(\frac{{ – 3}}{5} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 2}}{5} < \frac{{ – 1}}{3}\)
He could find rational numbers \(\frac{{ – 8}}{5} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}}\) between \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)
Are the numbers \(\frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}}\) the only rational numbers between \( – \frac{3}{5}\) and \( – \frac{1}{3}\)?
We have \(\frac{{ – 3}}{5} = \frac{{ – 18}}{{30}}\) and \(\frac{{ – 8}}{{15}} = \frac{{ – 16}}{{30}}\)
And \(\frac{{ – 18}}{{30}} < \frac{{ – 17}}{{30}} < \frac{{ – 16}}{{30}}.i.e.,\frac{{ – 3}}{5} < \frac{{ – 17}}{{30}} < \frac{{ – 8}}{{15}}\)
Hence, \(\frac{{ – 3}}{5} < \frac{{ – 17}}{{30}} < \frac{{ – 8}}{{15}} < \frac{{ – 7}}{{15}} < \frac{{ – 6}}{{15}} < \frac{{ – 1}}{3}\)
So, he could find one more rational number between \(\frac{{ – 3}}{5}\) and \(\frac{{ – 1}}{3}.\)
We can insert as many rational numbers as possible between two different rational numbers using the above method.
We can find an unlimited number of rational numbers between any two rational numbers.

Between two irrational numbers, is there a rational number?

Yes, between any two distinct irrational numbers, we have a rational number, in fact, an infinite number of them. We know that we have an irrational number between the two rational numbers, and between two irrational numbers, there is a rational number.

How to Find Irrational Numbers Between Decimals?

The Irrational numbers have non-recurring and non-terminating decimals. It is like the square root of \(2 = 1.41421356237305904……\) It doesn’t have the specific pattern of repeating the numbers after the decimal point, and these numbers are never-ending.
So we have to remember that we can write the numbers which are not repeating in any manner (not like \(2.17171717……\)), whatever we wish after the decimal point and indicate it is never-ending.

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List of Irrational Numbers

The irrational numbers consist of Pi, Euler’s Number, and Golden ratio etc. Many square roots and cube root numbers are also irrational; however, not all of them are. For example, \(\sqrt 5 \) is an irrational number, but \(\sqrt 4 \) is a rational number, as \(4\) is a perfect square, such as \(4 = 2×2\) and \(\sqrt 4 = 2,\) which is a rational number. It is noted that there are infinite irrational numbers between any two real numbers.
For example: Let us take the examples of the numbers \(1\) and \(2.\) There are infinitely many irrational numbers between \(1\) and \(2.\) Now, the values of irrational numbers are given below:

Pi \(\pi \)	\(3.14159265358979 \ldots \)
Euler’s number \(e\)	\(2.71828182845904 \ldots \)
Golden \(\varphi \)	\(1.61803398874989 \ldots \)

Solved Examples – Irrational Numbers Between Two Rational Numbers

Q.1. Show that \(5 – \sqrt 3 \) is irrational.
Ans: Let us assume, on the contrary, that \(5 – \sqrt 3 \) is rational.
That is, you can find the coprime \(a\) and \(b\left({b \ne 0} \right)\) such that \(5 – \sqrt 3 = \frac{a}{b}.\)
Therefore, \(5 – \frac{a}{b} = \sqrt 3 .\)
Rearranging this equation, we get \(\sqrt 3 = 5 – \frac{a}{b} = \frac{{5b – a}}{b}.\)
Since \(a\) and \(b\) are integers, we get \(5 – \frac{a}{b}\) is rational, and so \(\sqrt 3 \) is rational.
But this contradicts the facts that \(\sqrt 3 \) is irrational.
This contradiction has come up because of our incorrect assumption that \(5 – \sqrt 3 \) is rational.
So, we conclude that \(5 – \sqrt 3 \) is irrational.

Q.2. Show that \(3\sqrt 2 \) is irrational.
Ans: Let us assume, to the contrary, that \(3\sqrt 2 \) is rational.
That is, you can find the coprime \(a\) and \(b\left({b \ne 0} \right)\) such that \(3\sqrt 2 = \frac{a}{b}.\)
Rearranging, we get \(\sqrt 2 = \frac{a}{{3b}}.\)
Thus \(3,\) \(a\) and \(b\) are the integers, \(\frac{a}{{3b}}\) is the rational, and so \(\sqrt 2 \) is a rational number.
But this contradicts the fact that \(\sqrt 2 \) is irrational.
So, we conclude that \(3\sqrt 2 \) is irrational.

Q.3. Write the two irrational numbers between the given numbers 0.12 and 0.13.
Ans: Let \(a = 0.12\) and \(b = 0.13.\)
Here, \(a\) and \(b\) are the rational numbers such that \(a < b.\)
We observe that the numbers \(a\) and \(b\) have a \(1\) in the first place of decimal. But in the second place of decimal \(a\) has a \(2\) and \(b\) has \(3.\) So, we consider the numbers.
\(c = 0.1201001000100001 \ldots ..\)
And \(d = 0.12101001000100001 \ldots \)
Thus, \(c\) and \(d\) are irrational numbers such that \(a < c < d < b.\)

Q.4. Find one irrational number between the number a and b given: \(a = 0.1111 \ldots = 0.\underline{1} \) and \(b = 0.1101\)
Ans: Clearly, \(a\) and \(b\) are rational numbers since \(a\) has a repeating decimal and \(b\) has a terminating decimal. We observe that in the third place of decimal \(a\) has a \(1,\) while \(b\) has a zero.
\(\therefore \,\,\,\,a > b\)
Consider the number \(c\) given by: \(c = 0.111101001000100001 \ldots \)
Clearly, \(c\) is an irrational number as it has non-repeating and non-terminating decimal representation.
We observe that in the first two places of their decimal representations, \(b\) and \(c\) have the same digits. But in the third place, \(b\) has a zero, whereas \(c\) has a \(1.\)
\(\therefore \,\,\,\,b < c\)
Also, \(c\) and \(a\) have the same digits in the first four places of their decimal representations, but in the fifth place, \(c\) has a zero, and \(a\) has a \(1.\)
\(\therefore \,\,\,\,c < a\)
Hence, \(b < c < a\)
Thus, \(c\) is the required irrational number between \(a\) and \(b.\)

Q.5. Write each of the following rational numbers with positive denominators:
\(\frac{3}{{ – 8}},\frac{7}{{ – 12}},\frac{{ – 5}}{{ – 2}},\frac{{ – 13}}{{ – 8}}\)
Ans: We have \(\frac{3}{{ – 8}} = \frac{{3 \times \left({ – 1} \right)}}{{\left({ – 8} \right) \times \left({ – 1} \right)}} = \frac{{ – 3}}{8};\)
\(\frac{7}{{ – 12}} = \frac{{7 \times \left({ – 1} \right)}}{{\left({ – 12} \right) \times \left({ – 1} \right)}} = \frac{{ – 7}}{{12}};\)
\(\frac{{ – 5}}{{ – 2}} = \frac{{\left({ – 5} \right) \times \left({ – 1} \right)}}{{\left({ – 2} \right) \times \left({ – 1} \right)}} = \frac{5}{2};\)
\(\frac{{ – 13}}{{ – 8}} = \frac{{\left({ – 13} \right) \times \left({ – 1} \right)}}{{\left({ – 8} \right) \times \left({ – 1} \right)}} = \frac{{13}}{8}\)

Summary

In the given article, we have discussed irrational numbers, and then we talked about the irrational numbers between the rational numbers. Then we have provided information about irrational numbers between two irrational numbers, followed by rational numbers between two rational numbers.

We also glanced at the number between two irrational numbers, followed by finding the irrational number between decimals. Finally, we have given solved examples along with a few FAQs.

Learn About Irrational Numbers and Properties

Frequently Asked Questions (FAQs)

Q.1. How do you find irrational numbers between rational numbers?
Ans: Assume that we have two rational numbers \(a\) and \(b,\) then the irrational numbers between the two will be \(\sqrt {ab} .\) Now see the example given below for a better understanding:
Example: Find the irrational number between the two rational numbers \(3\) and \(7.\)
Then the irrational number between \(3\) and \(7\) is \(\sqrt {3 \times 7} = \sqrt {21} .\)

Q.2. How do you find two irrational numbers between two irrational numbers?
Ans: Let us consider an example
\(\sqrt 2 \) and \(\sqrt 3 \) are irrational numbers
\(\sqrt 2 = 1.4142\) (nearly)
\(\sqrt 3 = 1.7321\) (nealry)
Now we have to find an irrational number which should lie between \(1.4142\) and \(1.7321\) The irrational number is \(1.50500500050000 \ldots \)
We shall find the second irrational number between \(1.4141\) and \(1.50500500050000 \ldots \) or between \(1.50500500050000 \ldots \) and \(1.7321.\) Let us choose \(1.6010010001 \ldots \) between \(1.50500500050000 \ldots \) and \(1.7321.\)
Hence, the two irrational numbers between irrational numbers \(\sqrt 2 \) and \(\sqrt 3 \) are \(1.50500500050000 \ldots \) and \(1.6010010001 \ldots \)

Q.3. Is there a rational number between two rational numbers?
Ans: The number of integers between two integers is limited (finite). Will the same happen in the case of rational numbers too. We can insert as many rational numbers as possible between two different rational numbers using the above method. We can find an unlimited number of rational numbers between any two rational numbers.

Q.4. How do you find the irrational number between 2 and 3?
Ans: To find the irrational numbers between the numbers \(2\) and \(3.\)
As we know that the square root of \(4\) is \(2,\sqrt 4 = 2\)
And the square of the number \(9\) s \(3,\sqrt 9 = 3\)
Hence, the irrational numbers between the numbers \(2\) and \(3\) are \(\sqrt 5 ,\sqrt 6 ,\sqrt 7 ,\) and \(\sqrt 8 ,\) as they are not perfect squares and cannot be simplified further.

Q.5. How do you know a number is irrational?
Ans: The irrational number does not have a perfect square root, and it cannot be written as the ratio of two integers. The decimal form of an irrational number does not stop and does not repeat.

We hope this detailed article on irrational numbers between two rational numbers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!