Addition and Subtraction of Irrational Numbers(With Examples)

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The set of real numbers consists of two broad categories of numbers – rational numbers and irrational numbers. A rational number can be written as a ratio or as a fraction, where a numerator and a denominator are integers. On the other hand, the numbers that cannot be expressed as a ratio of two integers are irrational numbers, i.e., the numbers that are not rational are irrational. The rational and irrational numbers together form the real numbers. 

In this article, we will discuss the addition and subtraction of irrational numbers.

Addition and Subtraction of Irrational Numbers

You can perform addition and subtraction between rational and irrational numbers and also between two irrational numbers. The following pointers are to be kept in mind when you deal with real numbers and mathematical operations on them:

• When the addition or subtraction operation is done on a rational and irrational number, the result is an irrational number.
• When the addition or subtraction operation is done on an irrational and irrational number, the result can be an irrational or rational number.

Addition of Irrational Numbers

You can add an irrational number with any real number, i.e., an irrational number can be added to a natural number, whole number, integer, or a rational number. 

Note: 

• All natural numbers, whole numbers and integers are rational numbers
• The rational numbers and irrational numbers together form a set of real numbers

Whenever an irrational number is added to a rational number, the sum is always an irrational number and when two or more irrational numbers are added the sum can be an irrational number or a rational number.

Addition of Rational Number With Irrational Number

If $a$ is a rational number and $\sqrt{b}$ is an irrational number, then the sum of these numbers can be written as either

• $a + \sqrt{b}$
• $\sqrt{b} + a$

You cannot simplify $a + \sqrt{b}$ or $\sqrt{b} + a$ further.

Examples

Let’s consider the following examples to understand the addition of an irrational number with a rational number.

Ex 1: Add $3$ and $\sqrt {2}$

Sum of $3$ and $\sqrt {2}$ will be $3 + \sqrt {2}$ or $\sqrt {2} + 3$. You cannot simplify it further.

Note: The addition of numbers is commutative.

You can also represent the answer in decimal form.

$3 + \sqrt {2}$ = $3 + \sqrt {2} = 3 + 1.4142135624… = 4.4142135624…$

You cannot represent the answer in $\frac {p}{q}$ form because the number $3 + \sqrt{2}$ or $\sqrt{2} + 3$ is not a rational number.

Ex 2: Add $-5$ and $\sqrt {3}$

Sum of $-5$ and $\sqrt {3}$ will be $-5 + \sqrt {3}$ or $\sqrt {3} + \left(-5 \right) = \sqrt {3} – 5 $. 

You cannot simplify it further.

Note: The addition of numbers is commutative.

You can also represent the answer in decimal form.

$-5 + \sqrt {3}$ = $-5 + \sqrt {3} = -5 + 1.7320508076… = -3.2679491924…$.

You cannot represent the answer in $\frac {p}{q}$ form because the number $-5 + \sqrt{3}$ or $\sqrt{3} – 5$ is not a rational number.

Addition of Irrational Number With Irrational Number

The sum of an irrational number with another irrational number can be an irrational number or it can also be a rational number.

Examples

Let’s consider some examples to understand the sum of two or more irrational numbers.

Ex 1: Add $\sqrt {7}$ and $\sqrt{7}$

$\sqrt {7} + \sqrt {7} = \sqrt {7} \times 1 + \sqrt {7} \times 1 = \sqrt {7}\left(1 + 1 \right) = \sqrt{7} \times 2 = 2\sqrt{7}$.

Note: A sum is an irrational number.

Ex 2: Add $\sqrt{3}$ and $\sqrt{5}$

Sum of $\sqrt{3}$ and $\sqrt{5}$ will be $\sqrt{3} + \sqrt{5}$ or $\sqrt{5} + \sqrt{3}$.

You cannot simplify $\sqrt{3} + \sqrt{5}$ further although you can represent the sum in decimal form but cannot be represented in $\frac {p}{q}$ form.

$\sqrt{3} + \sqrt{5} = 1.7320508076… + 2.2360679775… = 3.9681187851…$.

Note: A sum is an irrational number.

Ex 3: Add $\sqrt{2}$ and $\sqrt{8}$

$\sqrt{8}$ can be written as $\sqrt{2 \times 2 \times 2} = \sqrt{2^{2} \times 2} = \sqrt{2^{2}} \times \sqrt{2} = 2\sqrt{2}$.

Therefore, $\sqrt{2} + \sqrt{8} = \sqrt{2} + 2\sqrt{2} = 1 \times \sqrt{2} + 2 \times \sqrt{2} =\left(1 + 2 \right)\sqrt{2} = 3\sqrt{2}$.

Note: A sum is an irrational number.

Ex 4: Add $\sqrt{5}$ and $-\sqrt{5}$

$\sqrt{5} + \left(-\sqrt{5} \right) = \sqrt{5} – \sqrt{5} = 0$.

A sum is a rational number.

Note: $0$ is a rational number.

Subtraction of Irrational Numbers

You can subtract an irrational number with any real number, i.e., an irrational number can be subtracted from/to a natural number, whole number, integer, or a rational number. 

Whenever an irrational number is subtracted to/from a rational number, the difference is always an irrational number and when two irrational numbers are subtracted the difference can be an irrational number or a rational number.

Subtraction of Rational Number From Irrational Number

If $a$ is a rational number and $\sqrt{b}$ is an irrational number, then the difference between these numbers can be written as either

• $a – \sqrt{b}$
• $\sqrt{b} – a$

You cannot simplify $a – \sqrt{b}$ or $\sqrt{b} – a$ further.

Examples

Let’s consider the following examples to understand the addition of an irrational number with a rational number.

Ex 1: Subtract $\sqrt {3}$ fom $2$  

Difference of $2$ and $\sqrt {3}$ will be $2 – \sqrt {3}$. You cannot simplify it further.

You can also represent the answer in decimal form.

$2 – \sqrt {3} = 2 – 1.7320508076… = 0.2679491924…$

Similarly,  $\sqrt {3} – 2 = 1.7320508076… – 2 = -0.2679491924…

Ex 2: Subtract $-3$ from $\sqrt {5}$

Difference of $-3$ from $\sqrt {5}$ = $\sqrt {5} – \left(-3 \right) = \sqrt {5} + 3$ or $3 + \sqrt {5}$.

You cannot simplify it further.

You can also represent the answer in decimal form.

$3 + \sqrt {5} = 3 + 2.2360679775… = 5.2360679775…$

Subtraction of Irrational Number From Irrational Number

The difference between an irrational number with another irrational number can be an irrational number or it can also be a rational number.

Examples

Let’s consider some examples to understand the difference between two irrational numbers.

Ex 1: Subtract $\sqrt {2}$ from $\sqrt{7}$

Difference of $\sqrt {2}$ from $\sqrt {7}$ is $\sqrt {7} – \sqrt {2}$.

You cannot simplify it further.

You can also represent the answer in decimal form.

$\sqrt{7} – \sqrt {2} = 2.64575131106… – 1.41421356237… = 1.23153774869…$ which is an irrational number.

Ex 2: Subtract $\sqrt{3}$ from $\sqrt{3}$

Difference of $\sqrt{3}$ from $\sqrt{3}$ is $\sqrt{3} – \sqrt{3} = 0$ 

A difference is a rational number.

Conclusion

You can perform the operations of addition and subtraction on irrational numbers. Whenever you add or subtract an irrational number and a rational number, the result is always an irrational number. But when two irrational numbers are added or subtracted, the result can be an irrational number or a rational number.

Practice Problems

Perform the following operations

• $2 + \sqrt{5}$
• $-3 – \sqrt{2}$
• $\sqrt{7} + \sqrt{5}$
• $\sqrt{8} + \sqrt{2}$
• $2\sqrt{5} – \sqrt{3}$
• $7\sqrt{3} – \sqrt{3}$
• $7 + 7\sqrt{2}$
• $5\sqrt{2} -3\sqrt{2}$
• $\sqrt{8} – \sqrt{2}$

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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