Irrational numbers are real numbers that cannot be written as a simple fraction. It can be represented in the form of a ratio and, it is a contradiction of rational numbers. Examples √8, √5, and so on. The general form of irrational numbers are a/b, where a, b are two integers and value of b is not equal to zero. Obtain the definition, properties and example questions on irrational numbers on this page.

## What are Irrational Numbers?

Irrational numbers are the set of real numbers that cannot be represented in the form of a fraction, a/b where a and b are integers. The denominator b is not equal to zero. Also, the decimal expansion of irrational numbers is neither repeating nor terminating.

Some of the most used examples of irrational numbers are ㄫ, √3, Euler’s number e and so on. To know more useful details about the concept of irrational numbers, tap on the below links.

- Definition of Irrational Numbers
- Decimal Representation of Irrational Number
- Representation of Irrational Numbers on The Number Line
- Comparison between Two Irrational Numbers
- Comparison between Rational and Irrational Numbers
- Real number between Two Unequal Real Numbers
- Rationalization
- Problems on Irrational Numbers
- Problems on Rationalizing the Denominator
- Worksheet on Irrational Numbers

### How to Identify an Irrational Number?

We already know that irrational numbers can’t be represented in the form of a fraction. So, the numbers in the form of √2, √3, √5, √7, π, etc are examples of irrational numbers. It can also be stated as a number that cannot be expressed in the form of a/b, where a, b are integers and b ≠ 0 are the irrational numbers.

The symbol is used to represent irrational numbers is “P”. The set difference of the real numbers minus rational numbers, like R – Q or R/Q is P.

### Irrational Number Proof

The below-provided theorem is used to prove how a non-perfect square is an irrational number statement.

**To Prove:** √2 is an irrational number

Consider, √2 is a rational number.

By the definition of rational number, it can write as √2 = a/b —- (i)

Where a, b are co-prime integers and b ≠ 0.

On squaring both sides of the equation (i), we ger

2 = \(\frac { a² }{ b² } \)

a² = 2 x b² —- (ii)

The theorem states that “Given a is a prime number and p² is divisible by a, (where p is any positive integer), then it can be concluded that a also divides p”. If 2 is a prime factor of a², then 2 is also a prime factor of a.

So, a = 2 x c where c is an integer

Substituting this value of a in equation (ii), we get

(2c)² = 2b²

4c² = 2b²

b²= 2c²

So, 2 is a prime factor of b² also. Again from the theorem, it can be said that 2 is also a prime factor of q.

According to the initial assumption, a and b are co-primes but the result obtained above contradicts this assumption as a and b have

2 as a common prime factor other than 1. This contradiction arises due to the incorrect assumption that √2 is rational.

So, √2 is irrational.

### Properties of Irrational Numbers

The given properties of irrational numbers are useful to identify irrational numbers from a group of real numbers.

- Irrational numbers are real numbers only.
- These numbers consist of non-terminating and non-recurring decimals.
- For any two irrational numbers, their LCM may or may not exist.
- The addition of rational and irrational numbers is an irrational number.
- When an irrational number is multiplied by nonzero rational numbers, then the product is an irrational number.
- The addition or the multiplication of two irrational numbers may be rational numbers
- The arithmetic operations of two irrational numbers may or may not be rational numbers.

### List of Irrational Numbers

The famous irrational numbers are π, Euler’s number, the golden ratio. Many square roots, cube roots numbers are irrational. The set of irrational numbers can be obtained by some properties.

- All square roots which are not perfect squares are irrational numbers.
- Euler’s number, golden ratio, and Pi are famous irrational numbers.
- The square root of any prime number is also an irrational number.

### Sum and Product of Two Rational Numbers

Here, we will discuss the sum and product of two irrational numbers.

**Product of Two Irrational Numbers**

**Statement:** The product of two irrational numbers is rational or irrational.

For example, √2 is an irrational number, when √2 is multiplied by √2, the product is 2, which is a rational number.

√2 x √2 = 2

In the same way, π is an irrational number, if π is multiplied by π, the result is π², it is an irrational number.

π x π = π²

Therefore, when two irrational numbers are multiplied, then the product may or may or may not be a rational number.

**Sum of Two Irrational Numbers**

**Statement:** The sum of two irrational numbers is rational or irrational.

For example, add two irrational numbers 3√2+ 4√3, a sum is an irrational number.

In another example, (3+4√2) + (-4√2), the sum is 3, which is a rational number.

So, the addition of two irrational numbers might result in an irrational or rational number.

### Difference Between Rational & Irrational Number

- A rational number can be expressed in the form of a ratio or fraction. But an irrational number can’t be expressed in the form of a fraction or ratio.
- The examples of rational numbers are 0.555, 0.6868, 1.75. The examples of irrational numbers are π, √2, √8.
- The decimal expansion of rational numbers is terminating or non-terminating recurring. The decimal expansion of irrational numbers is non-terminating and non-recurring.

### Questions on Irrational Number

**Question 1:**

Check if the below numbers are rational or irrational.

5, 6/11, -6.15, 0.25

**Solution:**

Since the decimal expansion of a rational number either repeats or terminates.

So, 5, 6/11, -6.15, 0.25 are rational numbers.

**Question 2:**

John has a box with five irrational numbers. He wants only one irrational number which is closest to 3 and should not exceed 5. Help John to find the correct one. The irrational numbers in the box are √2, √3, √5, √10.

**Solution:**

Find the values of irrational numbers.

√2 = 1.41421

√3 = 1.7320

√5 = 2.2360

√10 = 3.16

Therefore, √10 is the closest number to 3.

### FAQ’s on Irrational Number

**1. Is Pi an irrational number?**

π is an irrational number because it is non-terminating. The approximate value of π is 22/7 or 3.14159.

**2. What are 5 irrational numbers examples?**

The five examples of irrational numbers are √2, √3, Pi, Euler’s Number e = 2.718281, golden ratio φ= 1.618034.

**3. Are integers irrational numbers?**

Integers are rational numbers but not irrational. All integers can be written in the form of a/b.