Irrational Numbers – Definition, Properties & Examples

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There are real numbers that can be expressed in the form of ratios. These numbers are called rational numbers. But there are real numbers that cannot be expressed in the form of ratios. Such numbers are called irrational numbers. The irrational numbers are the numbers that cannot be expressed in the form of $\frac {p}{q}$, where $p$ and $q$ are integers.

Let’s understand irrational numbers and their properties.

What are Irrational Numbers?

Irrational numbers are real numbers that are not rational numbers. These numbers cannot be expressed as a ratio, i.e., in the form of $\frac {p}{q}$, where $p$ and $q$ are integers and $q \ne 0$. 

Also, the decimal expansion of irrational numbers is non-terminating and non-recurring decimals.

Note: Decimal expansion of rational numbers is either terminating or non-terminating but a recurring decimal number.

Examples of Irrational Numbers

The numbers like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt[3]{4}$, $\sqrt[3]{6}$, $\sqrt[4]{7}$, $\sqrt[5]{8}$ are all irrational numbers. All these numbers have non terminating and non recurring decimal expansions.

Some of the widely used irrational numbers are

• $\pi = 3⋅14159265…$. Since the value of $\pi$ is closer to the fraction $\frac {22}{7}$, we take the value of $\pi$ as $\frac {22}{7}$ or $3.14$
• Euler’s number $e = 2⋅718281⋅⋅⋅⋅$
• Golden ratio, $\phi = 1.61803398874989….$

Locating Irrational Numbers On a Number Line

The irrational numbers along with the rational numbers are real numbers. Hence, a unique point is considered to represent them on the number line. Some irrational numbers in the form of $\sqrt{n}$, where $n$ is a positive integer can be represented on a number line by using the following steps.

Step 1: Split the number inside the square root such that the sum adds up to the number

Step 2: The distance between these two natural numbers should be equal on the number line starting from the origin. One line should be perpendicular to the other

Step 3: Use the Pythagoras Theorem

Step 4: Represent the area as the desired measurement

Let us consider the example of $\sqrt{2}$ to understand this better.

Step 1: Draw a number line with the center as zero, left of zero as $-1$, and right of zero as $1$

Step 2: Keeping the same length as between $0$ and $1$, draw a line perpendicular to point $1$, such that the new line has a length of $1$ unit.

Step 3: Draw a line from $0$ to the end of the perpendicular line constructing a right-angled triangle ABC. With AB as height, BC as the base, and AC as the hypotenuse

Step 4: Find the length of the hypotenuse i.e. AC by applying the Pythagoras theorem

$AC^{2} = AB^{2} + BC^{2} => AC^{2} = 1^{2} + 1^{2} => AC^2 = 2 => AC = \sqrt {2}$

Step 5: Keeping AC as the radius with C as the center, cut an arc on the same number line naming it D. CD will also become the radius of the arc with length $\sqrt{2}$.

Step 6: Therefore, point D represents $\sqrt{2}$ on the number line.

Properties of Irrational Numbers

You can perform any of the following four basic operations on irrational numbers. 

• Addition
• Subtraction
• Multiplication 
• Division

Each of these operations shows one or more of the following properties:

• Commutative Property
• Associative Property

Let’s understand these properties of irrational numbers in detail.

Commutative Property of Irrational Numbers

The commutative property deals with the ordering of numbers in an operation. It states that the result remains the same even if the order of numbers in the operation is changed or swapped.

The commutative property is exhibited by the operations of addition and multiplication in the set of irrational numbers. The operations subtraction and division do not show exhibit commutative property in the set of irrational numbers.

Commutative Property of Addition of Irrational Numbers

It states that for any two irrational numbers their sum remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in \overline{Q}, \text {then  } a + b = b + a$.

For example, $\sqrt{7}$ and $\sqrt{11}$ are two irrational numbers. $\sqrt{7} + \sqrt{11} = \sqrt{11} + \sqrt{7}$ 

Commutative Property of Multiplication of Irrational Numbers

It states that for any two irrational numbers their product remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in \overline{Q}, \text {then  } a \times b = b \times a$.

For example, $\sqrt{3}$ and $\sqrt{5}$ are two irrational numbers. $\sqrt{3} \times \sqrt{5} = \sqrt{15}$ and also $\sqrt{5} \times \sqrt{3} = \sqrt{15}$.

Associative Property of Irrational Numbers

The associative property deals with the grouping of irrational numbers in an operation. It states that the result remains the same even if the grouping of numbers is changed while performing the operation.

The associative property is exhibited by the operations addition and multiplication in the set of irrational numbers. The operations subtraction and division do not show exhibit associative property in the set of irrational numbers.

Associative Property of Addition of Irrational Numbers

It states that the sum of any three irrational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in \overline{Q}, \text {then  } \left (a + b \right) + c = a + \left(b + c \right)$. 

For example, for three irrational numbers $\sqrt{2}$, $\sqrt{5}$ and $\sqrt{7}$, $\left( \sqrt{2} + \sqrt{5} \right) + \sqrt{7}$ and $\sqrt{2} + \left(\sqrt{5} + \sqrt{7} \right)$ are equal.

Associative Property of Multiplication of Irrational Numbers

It states that the product of any three irrational numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in \overline{Q}, \text {then  } \left (a \times b \right) \times c = a \times \left(b \times c \right)$. 

For example, for three irrational numbers $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{11}$, $\left( \sqrt{3} \times \sqrt{5} \right) \times \sqrt{11} = \sqrt{15} \times \sqrt{11} = \sqrt{165}$ and $\sqrt{3} \times \left(\sqrt{5} \times \sqrt{11} \right) = \sqrt{3} \times \sqrt{55} = \sqrt{165}$.

Difference Between Rational Numbers & Irrational Numbers

Following is the differences between rational and irrational numbers.

Parameter	Rational Numbers	Irrational Numbers
Meaning	Rational numbers are the numbers that can be represented as a ratio of two numbers. They exist in the form of $\frac {p}{q}$ where $p$ and $q$ are integers and $q \ne 0$ $7$ can be represented $\frac {7}{1}$	Irrational numbers are numbers that cannot be represented as a ratio of two numbers. The number $\pi = 3.14159265358979…$ cannot be represented in $\frac {p}{q}$ form
Properties	The rational numbers exhibit  a) closure property of addition, subtraction, multiplication, and division b) commutative property of addition and multiplication c) associative property of addition and multiplication d) distributive property of multiplication over addition and subtraction e) existence of additive identity f) existence of multiplicative identity g) existence of the additive inverse h) existence of the multiplicative inverse	The irrational numbers exhibit a) closure property of addition and subtraction (closure property of  multiplication and division do not exist) b) commutative property of addition and multiplication c) associative property of addition and multiplication d) additive identity of irrational numbers does not exist ($0$ is a rational number) e) multiplicative identity of irrational numbers does not exist ($1$ is a rational number) f) existence of the additive inverse g) existence of the multiplicative inverse
Form	Both the numerator and the denominators of the rational numbers are whole numbers, where the denominator of a rational number is not equal to zero	They cannot be represented in fractional form
Decimal form	They are numbers having decimal places that are finite or infinite but recurring in nature. eg: 2.9, 7, 8.66…, 9.1212…	They are non-terminating or non-recurring in nature. e.g: 3.141592…, 1.61803398…

Conclusion

Irrational numbers are numbers that are not rational numbers, i.e., these numbers cannot be expressed in $\frac {p}{q}$ form. When expanded in a decimal form they result in non terminating and non recurring decimal numbers.

Practice Problems

1. Which of the following are irrational numbers
   • $\sqrt{4}$
   • $\sqrt{8}$
   • $\sqrt{6}$
   • $\sqrt[3]{3}$
   • $\sqrt[3]{9}$
   • $\sqrt[3]{18}$
   • $\sqrt[4]{2}$
   • $\sqrt[4]{4}$
   • $\sqrt[4]{6}$
2. State True or False
   • The sum of two irrational numbers is always an irrational number
   • The difference between two irrational numbers is always an irrational number
   • The product of two irrational numbers is always an irrational number
   • The quotient of two irrational numbers is always an irrational number

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties

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