Irrational numbers are real numbers that cannot be expressed as a simple fraction or ratio of two integers. In other words, they cannot be written in the form a/b where a and b are integers and b ≠ 0. The decimal representation of an irrational number is non-terminating and non-repeating.

In this article, we have covered the definition of an Irrational Number, its examples and others in detail.

What are Irrational Numbers?

A number which is not rational numbers or cannot be written in the form of fraction a/b is defined as Irrational numbers. Here √2 is an irrational number, if the value of √2, it will be √2 = 1.14121356230951..., and the numbers go on into infinity and will not ever repeat, and they don't ever terminate. It can't be written in a /b form where b is not equal to zero. The value getting is non-terminating and there is no pattern in the digits after the decimal.

These types of numbers are called irrational numbers.

Examples of Irrational Numbers

Various examples of Irrational Numbers are:

Square Roots of Non-Perfect Squares:

• √2 ≈ 1.414213...
• √3 ≈ 1.732050...
• √5 ≈ 2.236067...

Mathematical Constants:

• π ≈ 3.141592... (Ratio of Circumference of a Circle to its Diameter)
• e = Euler’s Number, and also an irrational number. The first few digits look like 2.7182818284590452353602874713527654…
• φ = It's an Irrational Number. The first digits look like 1.61803398874989484820…

Properties of Irrational Numbers

Various properties of Irrational Number are:

Addition and Subtraction of Irrational Numbers

According to this, the result of an addition of irrational numbers need not be an irrational number

(4 + √3) + (6 – √3)

= 4 + √3  + 6 – √3 = 10

Here 10 is a rational number.

By this, the result of adding two irrational numbers is not an irrational number.

As per this, the result of Subtraction of irrational number need not be an irrational number

(5+ √2 ) - (3 + √2)

= 5+ √2 - 3 -√2 = 2

So Here 2 is a rational number.

Multiplication and Division of Irrational numbers 

According to this, the product of two irrational numbers can be a rational or irrational number.

√2 × √3 = 6

Here the result is a rational number.

As per this, the result of the division of two irrational numbers can be a rational or irrational number.

√2 ÷ √3 =\sqrt{\frac{2}{3}} 

Here the result is an irrational number.

Is Pi an Irrational Number?

Yes, Pi (π) is an irrational number because it is neither terminating decimals nor repeating decimals. We often take 22/7 as the value of Pi, but it is an approximation.

For more, Is pi Rational or Irrational Number?

Applications of Irrational Number

Various application of Irrational Number are:

• Irrational numbers often appear when calculating the lengths of sides in right triangles.
• Constant π (approximately 3.14159) is irrational and is crucial for calculations involving circles, such as finding the circumference.
• Values of trigonometric functions for certain angles involve irrational numbers, such as sin⁡(45^∘) = cos⁡(45^∘) = √2/2.
• Irrational numbers are essential in the study of limits, infinite series, and the convergence of sequences.
• Certain integrals yield results that are irrational numbers, etc.

Article Related to Irrational Numbers:

• Square Root 2
• Properties of Real Numbers
• Differences between Rational and Irrational Numbers

Irrational Numbers: Examples

Example 1: Which of the numbers are Rational Numbers or Irrational Numbers?

5, -2, -45678..., 6.5, √3, √2, √5

Solution:

Here, 5 , -2 , 6.5 are all rational number as its can be expressed as a fraction and have terminating decimal.

As 5 can be written as 5/1 and -2 can be written as -2/1 , and 6.5 as 65/10.

Whereas, √3 , √2 , √5 , -4.5678.... are all irrational numbers as its cannot be expressed in fraction or having non-terminating, non repeating decimal, here if √5 is equal to non terminating decimal.. √5 = 2.2360679, and same for the √3 = 1.732.. here these represents the Irrational number.

Example 2: Which of these are Irrational numbers?

0.5, π, 1/3, 0.857857

Solution:

Numbers that cannot be expressed as fraction are irrational numbers.

So here 0.5 can be written as 1/2 And 1/3 itself a fraction n 0.857857 can be written as 8578/1000.

So these are rational numbers. π is the only irrational here which can't be expressed as fraction.

Example 3: When you multiply two irrationals, it gives you different results. How? 

Solution:

Multiply √3 × √3, then it will give result as a rational number

√9 = 3 , Here 3 is rational number.

In second case, If we multiply, √3 × √5 , then it will give result as an irrational number.

=√15 , Here √15 Is an irrational number

So multiplication of two irrational numbers can give you both the result as rational or irrational.

Example 4:  Identify whether the following numbers are Rational or Irrational?

√2, 84, 8.432432432…, 3.14159265358979…, √11, 33/3.

Solution:

84, 8.432432432…, and 33/3  are Rational numbers as either they are Integers or their decimal expansions are terminating, repeating.

√2, 3.14159265358979…, and √11 are Irrational numbers as their decimal Expansions are Non-terminating, Non-repeating.

Example 5: Identify is 6.5 a rational or irrational number?

Solution:

Number 4.5 is a rational number.

Since rational numbers can also be expressed as decimals with repeating digits after the decimal point.

Here we can write 6.5 as 65/10 and further write it as 15/2 = 6.5 so its a rational number.