# Irrational number

A number which we cannot express in the fractional form i.e. p/q where p & q are some integers & q≠0.

Example:- √(15), √(3), √(2), π, 0.202200222....., etc.

Symbol of an irrational number is P.

## How to Identify an irrational number?

It is very simple to identify any number if it is irrational. It should be

**non-terminating & non-recurring decimal expansion.**Let's discuss-### Non-terminating

Non-terminating means when we divide numerator from denominator & we do not get zero as a reminder so it will be

**non-terminating decimal expansion.***Such as (22÷7) = 3.14159265358....*

You noticed when we divide numerator 22 from denominator 7 so we do not get zero as remainder.

We continue solving but not get remainder zero so it will be non-terminating. Remember if we get zero as remainder so it will be terminating.

### Non-recurring(non-repeating)

Non-recurring means when we divide numerator from the denominator & we do not get repeating numbers in quotient so it will be

**non-recurring decimal expansion.**

*Such as (22÷7) = 3.14159265358....*

You noticed when we divide numerator 22 from denominator 7 so we get non-repeating numbers in quotient 3.14159265358.....

We continue solving but we do not get a repeating number in quotient so it will be non- recurring. Remember if we get repeating numbers in quotient so it will be recurring.

## Questions of an irrational number

### 1. Why √2 is an irrational number?

Because when we solve √(2) as decimal expansion so we get 1.4142135624.....

And here we get non-terminating

*(not getting zero in the remainder).*And Non-recurring

*(continue getting the different number in quotient)*decimal expansion so √(2) will be an irrational number.### 2. What is the product of rational & irrational number?

Suppose, we have a rational number (1/2) & irrational number √(3).

After the product of (1/2) & √(3), we get √(3)/2.

We get it's a decimal expansion (1.73205080...)/2 & after solving we get 0.8660254038....

We may notice here we are getting non-terminating & non-recurring decimal expansion.

*So the product of rational & irrational number will be irrational.*### 3. Write irrational number without root.

We may write many irrational numbers without root such as-

1.23463100112...(non-terminating & non-recurring decimal expansion).

5.31267340012...(non-terminating & non-recurring decimal expansion).

In this way, we may write different irrational numbers.

### 4. Why π is an irrational number?

Because when we find it's decimal expansion then we get

*3.14159265358....*

And here we get non-terminating & non-recurring decimal expansion so π is an irrational number.

### 5. Why we write π=(22/7) while an irrational number is not in fractional form.

Yes, π is an irrational number & also irrational number not is in fractional form.

Remember, we use the approximate value of π as (22/7) or 3.14. Because we cannot show it's overall value so we do it.

### 6. What is the sum of two irrational number?

Suppose we have two irrational number √2 & √3.

After adding both numbers we get (√2+√3). If we add it's decimal expansion so we get the irrational number.

Now we can say that the sum of two irrational numbers is irrational.

### 6. Write the list of irrational numbers.

There is a different irrational number-

√2, √5, π, √6, 1.101100011011...., etc. ( All numbers have non-terminating & non-recurring decimal expansion.

### 7. Difference between rational & irrational number.

We write a rational number in fractional form or in the form of p/q where q ≠ 0.

While the irrational number is not in fractional form or not in form of p/q.

While the irrational number is not in fractional form or not in form of p/q.

All-natural numbers are a rational number. Because we can show natural numbers as p/q.

While a natural number is not an irrational number.

### 8. Find irrational between 3.5 & 4.5.

We can write an uncountable irrational number between 3.5 & 4.5.

3.6912789543....

*(non-terminating & non-recurring decimal expansion).*

4.01278953472....

*(non-terminating & non-recurring decimal expansion).*

3.901276541299...

*(non-terminating & non-recurring decimal expansion).*

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