Irrational Numbers and Properties: Definition, Symbols, Examples

Irrational Numbers and Properties: Irrational numbers are the part of 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.

Irrational Numbers involves Pi, Euler’s Number, and Golden ratio. It also involves several square and cube roots. For instance, \(\sqrt{2}\) is an irrational number. In this article, we will learn about irrational numbers, properties, along with some examples. Scroll down to learn more!

Irrational Numbers: Overview

Definition: An irrational number is defined as the number that cannot be expressed in the form of \(\frac{p}{g}\), where \(p\) and \(q\) are coprime integers and \(q \neq 0\).

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 \(\mathrm{e}=2.718281 \ldots \ldots\), Golden ratio \(\varphi=1.618034 \ldots \ldots\)

Learn the Concepts of Irrational Numbers

How to Identify an Irrational Number?

You know that the irrational numbers are the real numbers only that cannot be expressed in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). For example: \(\sqrt{5}\) and \(\sqrt{3}\), etc. are the irrational numbers. Since, the numbers that can be represented in the form of \(\frac{p}{q}\), such that \(p\) and \(q\) are integers and \(q \neq 0\), are the rational numbers.

Irrational Numbers Symbol

Generally, the symbol used to represent the irrational symbol is “\(P\)”. Since the set of real numbers \((R)\) that are not the rational number \((Q)\) is called an irrational number. The symbol \(P\) is often used because of its association with natural and rational numbers. (i.e.) because of the alphabetic sequence \(P, Q, R\). But mainly, it is represented using the set difference of the real minus rationals, in a way \(R- Q\) or \(R\backslash Q\).

Irrational Numbers List

Irrational numbers consist of Pi, Euler’s Number, and Golden ratio. There are many square roots and cube root numbers that 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 \(4=2 x 2\) and \(\sqrt{4}=\pm 2\), 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 numbers \(1\) and \(2\), there are infinitely many irrational numbers between \(1\) and \(2\). Now, the values of irrational numbers are given below:

Irrational Numbers Properties

The irrational numbers are the subsets of the real numbers, and irrational numbers will also have all the properties of the real number system. The following are the properties of irrational numbers:

Properties of Addition of Two Irrational Numbers

Here you can use this for easy understanding:
Irrational number \(+\) Irrational Number \(=\) may or maynot be an Irrational Number.

For example: \((5+\sqrt{2})+(3+\sqrt{2})=5+\sqrt{2}+3+\sqrt{2}=8+2 \sqrt{2}\) is an irrational number.

But, \((5+\sqrt{2})+(3-\sqrt{2})=5+\sqrt{2}+3-\sqrt{2}=8\) is a rational number.

Properties of Subtraction of Two Irrational Numbers

Irrational Number \(-\) Irrational Number \(=\) may or maynot be an Irrational Number.
Example: Here, we will take the roots as shown below.
\(\sqrt{2}=1.414 \ldots, \sqrt{3}=1.732 \ldots, \sqrt{5}=2.236 \ldots\)
Now, subtract these irrational numbers \(\sqrt{3}-\sqrt{2}\)
\(=1.732 \ldots-1.414 \ldots=0.318 \ldots\)

Here, which we got by subtracting these numbers is non-repeating and non-terminating. So, this means the entire number is irrational.

But, \((5+\sqrt{2})-\sqrt{2}\)
We get \(=5+\sqrt{2}-\sqrt{2}=5\)

\(5\) is a rational number. Here, this example is clear that the result of subtraction of two irrational numbers may or maynot be an irrational number.

Properties of Multiplication of Two Irrational Numbers

Irrational number \(\times\) Irrational number \(=\) May or may not be an Irrational Number. Example: Let the irrational numbers be \(5+\sqrt{2}\) and \(5+\sqrt{3}\).

Their product is \((5+\sqrt{2}) \times(5+\sqrt{3})=25+5 \sqrt{2}+5 \sqrt{3}+\sqrt{6}\). This is an irrational number.

But, for the irrational numbers \(5+\sqrt{2}\) and \(5-\sqrt{2}\), the product is \((5+\sqrt{2}) \times(5-\sqrt{2})=25-(\sqrt{2})^{2}=25-2=23\). This is a rational number.

Hence, the multiplication of two irrational numbers may be rational or irrational.

Properties of Division of Two Irrational Numbers

\(\frac{\text { Irrational Number }}{\text { Irrational Number }}=\) May or may not be an Irrational Number

Example: Let the irrational numbers be \(\sqrt{6}\) and \(\sqrt{3}\).

We have, \(\frac{\sqrt{6}}{\sqrt{3}}=\sqrt{2}\). The quitient \(\sqrt{2}\) is an irrational number.

But, for the irrational numbers \(10+2 \sqrt{3}\) and \(5+\sqrt{3}\), we observe that \(\frac{10+2 \sqrt{3}}{5+\sqrt{3}}=2\). The quotient is a rational number.

Hence, the division of two irrational numbers may be rational or irrational.

Set of Irrational Numbers

The set of irrational numbers can be acquired by writing a few of the irrational numbers within the brackets. The set of irrational numbers are given below:

1. All the square roots that are not perfect square are irrational numbers.
   \({\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{8}}\)
2. Euler’s Number, Golden ratio, and \(\mathrm{Pi}\) are few famous irrational numbers.
   \(\left\{ {e,\emptyset,兀} \right\}\)
3. The square root of any of the prime numbers is the irrational number.

Now, the table explains the list of few irrational numbers:

Solved Examples on Irrational Numbers

Q.1. Show that \(5-\sqrt{3}\) is irrational.
Ans: Let us assume, to the contrary, that \(5-\sqrt{3}\) is rational.
That is, you can find the coprime \(a\) and \(b(b \neq 0)\) such that \(5-\sqrt{3}=\frac{a}{b}\).
Therefore, \(5-\frac{a}{b}=\sqrt{3}\).
Rearranging this equation, we get \(\sqrt{3}=5-\frac{a}{b}=\frac{5 b-a}{b}\).
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 \(b(b \neq 0)\) such that \(3 \sqrt{2}=\frac{a}{b}\).
Rearranging, we get \(\sqrt{2}=\frac{a}{3 b}\).
Thus \(3, a\) and \(b\) are the integers, \(\frac{a}{3 b}\) is the rational, and so \(\sqrt{2}\) is the rational.
But this contradicts the facts that \(\sqrt{2}\) is irrational.
So, we conclude that \(3 \sqrt{2}\) is irrational.

Q.4. 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 \(2\) and \(b\) has \(3\). So, we consider the numbers.
\(c=0.1201001000100001 \ldots \ldots\)
And \(d=0.12101001000100001 \ldots\)
Thus, \(c\) and \(d\) are the irrational numbers such that \(a<c<d<b\).

Q.5. Find one irrational number between the number \(a\) and \(b\) given: \(a=0.1111 \ldots=0 . \overline{1}\) and \(b=0.1101\)
Ans: Clearly, \(a\) and \(b\) are rational numbers, since \(a\) has a repeating decimal and \(b\) has 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 \(b\).

Summary

Irrational numbers can be described as the set of real numbers that cannot be represented in fractions or ratios. Various irrational numbers cannot be represented in a simplified way. The irrational numbers can also be referred to as the subsets of the real numbers. They possess all the properties of the real number system. The irrational numbers are represented by the symbol “P”. Since the set of real numbers (R) that are not the rational number (Q) is called an irrational number. In the above article, we learnt about irrational numbers and how to identify irrational numbers with examples.

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FAQs on Irrational Numbers

Q.1. What is an example of an irrational number?
Ans: 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 \(q \neq 0\).
Example: \(3 \sqrt{2}\) is irrational number.

Q.2. What are \(5\) irrational numbers examples?
Ans: There are many irrational numbers that cannot be written in simplified form. Some of the examples are: \(\sqrt 8 ,\sqrt {11} ,\sqrt {50} \), Euler’s number \(e =2.718281\), Golden ratio, \(\phi=1.618034\).

Q.3. How do you know if a number is irrational?
Ans: You know that the irrational numbers are the real numbers only that cannot be expressed in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). For example: \(\sqrt{5}\) and \(\sqrt{3}\), etc. are the irrational numbers. Since, the numbers that can be represented in the form of \(\frac{p}{a}\), such that \(p\) and \(q\) are integers and \(q \neq 0\), are the rational numbers.

Q.4. Is 4 an irrational number?
Ans: The number \(4\) is the rational number because it can be expressed as the quotient of two integers: \(4 \div 1=\frac{4}{1}\).

Q.5. Is 0 A irrational number?
Ans: Th number \(0\) can be expressed as \(\frac{0}{1}\), which is in the form of a rational number. So, \(0\) is not an irrational number. It is a rational number.

Q.6. Is 81 an irrational number?
Ans: No, the number \(81\) is not irrational because it can be expressed as the quotient of the two integers: \(81 \div 1=\frac{81}{1}\).

Q.7. Is 0.101100101010 an irrational number?
Ans: The number \(0.101100101010\) is a terminating decimal number, and the terminating decimals are considered as rational numbers, so this number is not an irrational number.

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