### Key Concepts

• Fraction

• Rational number

• Decimal number

• Repeating decimal

• Non repeating/terminating decimal

•Non repeating/Non terminating decimal

• Irrational number

• Square roots

• Perfect square

**Irrational Numbers:**

An irrational number is a type of real number that cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. When irrational numbers are expressed in the decimal form, they go on forever, even after the decimal point without repeating numbers. Thus, they are also known as non-terminating, non-repeating numbers.

If N is irrational, then N is not equal to p/q where p and q are integers and q is not equal to 0.

**Let us see an example:**

If we find the value of , √5

the answer will be 2.23606797749979

If we observe the numbers after decimal point, the numbers are non-terminating, non-repeating.

Can we convert this decimal value into a fraction?

It is not possible.

Hence, we can say 2.23606797749979 is an irrational number.

**Irrational numbers in daily life:**

Do you remember the formula to calculate area of a circle?

Area= πr^{2}

The most common irrational number is:

Pi (π) = 22/7= 3.14159265358979…

**1.2.2 Identifying the Square roots as Irrational Numbers**

Perfect squares are numbers which are obtained by squaring a whole number.

If you look at the above picture, you can observe that we can form a square with 4 marbles but not with 6 marbles.

Here, 4 = 2 × 2 = 2^{2}

If *n *is a natural number, then √n is either a natural number or an irrational number.

For any whole number *b *that is not a perfect square, √b is irrational.

**Let us look at some examples:**

The number 4 is a perfect square. So √4 is 2 which is a rational number.

The number 5 is not a perfect square. So √5 is 2.23606797749979 which is an irrational number.

**1.2.3 Classification of Numbers into Rational and Irrational Numbers**

Any integer number that can be expressed in the form of x/y where both the numerator and the denominator are integers is a rational number.

Any integer number that cannot be expressed in the form of x/y where both the numerator and the denominator are integers is an irrational number. In simple words, if the decimal form of a number does not stop and does not repeat, the number is irrational.

Some of the examples of rational numbers.

- Number 4 can be written in the form of 4/1 where 4 and 1 both are integers.
- 0.25 can also be written as 1/4, or 25/100 and all terminating decimals are rational numbers.

- √64 is a rational number, as it can be simplified further to 8, which is also the quotient of 8/1.
- 0.888888 is a rational number because it is recurring in nature.

Some of the examples of irrational numbers.

- 3/0 is an irrational number, with the denominator equals to zero.
- π is an irrational number that has value 3.142, and it is non-recurring and non-terminating in nature.
- √3 is an irrational number, as it cannot be able to simplify further.
- 0.21211211 is an irrational number as it is non-recurring and non-terminating in nature.

## Exercise:

- Identify each of the following as rational or irrational:
- 0.58
- 0.475
- 3.605551275…

- Identify each of the following as rational or irrational:
- √36
- √44
- √81
- √17

- An elementary school has a square playground with an area of 3000 square feet. What is the width of the playground? Is the width a rational or irrational number?
- Which of the following numbers are rational and which are irrational?
- √24
- √25
- √36
- √37

- The area of a square is 50 square feet. What are the lengths of its sides?
- Find √125
- Find √8
- Find √27
- Solve the following:

11√8 + 15√21 - Prove that is 2√3/5 irrational number

### What we have learnt:

• 1.2.1 About fractions, rational numbers, decimal numbers, irrational numbers, application of irrational numbers in everyday life and how to identify irrational numbers.

• 1.2.2 How to identify square roots as irrational numbers

• 1.2.3 How to classify a number as rational or irrational.

**Concept Map**:

**Frequently Asked Questions (FAQ’s):**

**Every real number is an irrational number?**

All numbers are real numbers and all real numbers that are non-terminating are irrational numbers. 2,3,4 etc. Are examples of real numbers that are not irrational numbers.

**Why integers are not irrational numbers?**

Integers that are positive, negative and zero are not irrational because they can be represented in the form of p/q (where Q ≠0)

**What are some commonly used irrational numbers?**

√2, √3, π(pi), e (Euler’s Number) are some common irrational numbers.

**How can you identify an irrational number?**

Every number that is not a rational number is an irrational number. Irrational number can be written in the form of decimals but not in the form of fractions.

**What is the difference between a rational number and an irrational number?**

Any integer number that can be expressed in the form of x/y where both the numerator and the denominator are integers is a rational number.

Any integer number that cannot be expressed in the form of x/y where both the numerator and the denominator are integers is an irrational number.

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