**Introduction to irrational numbers :**

We all know that a number that is expressed in the form a/b is called as rational number.

Here both "a" and "b" are integers and also b ≠ 0.

Let us consider the decimal number that is given below.

0.21757575................

Here, after the decimal, we have the repeated pattern 75. This is the decimal number which is non terminating with repeated pattern.

So, this decimal number can be converted in to fraction. That is in the form a/b.

So, 0.21757575................ is rational.

Let us consider the decimal number which is non terminating and also it has no repeated pattern.

2.17526325890748......................

**A non terminating decimal which does not have repeated pattern is called as irrational number.**

To have better understanding of irrational numbers, let us know the difference between rational and irrational numbers.

**Example : **

**Rational : 1.2626262626.............(Repeated pattern is 26)**

**Irrational : 1.4142135623............(No repeated pattern)**

**More clearly, **

**A non terminating decimal which has repeated pattern is called as rational number.**

**Because, the non terminating decimal which has repeated pattern can be converted into fraction. **

**A non terminating decimal which does not have repeated pattern is called as irrational number.**

**Because, the non terminating decimal which does not have repeated pattern can not be converted into fraction. **

To have better understanding on "Difference between rational and irrational numbers", let us come to know about rational numbers and irrational numbers more clearly.

**What are rational and irrational numbers ? **

First let us come to know, what is rational number. Because, once we understand rational number, we can easily understand irrational number.

A rational number has to be in the form as given below.

So, any number in the form of fraction can be treated as rational number.

**Examples of rational number : **

5, 2.3, 0.02, 5/6

Because all these numbers can be written as fractions.

5 = **5/1 **

2.3 =** 23/10**

0.02 = 2/100 = **1/50**

**5/6** (This is already a fraction)

Apart from the above examples, sometimes we will have recurring decimals like 1.262626..........

1.262626........ is a non terminating recurring decimal.

All these recurring decimals can be converted into fractions and they are also rational numbers.

A number which can not be converted into fraction is called as irrational numbers.

**Examples of irrational number : **

All the above non terminating numbers can not be converted into fractions.

Because, they do not have repeated patterns.

When we are trying to find square of a number which is not a perfect square, we get this non repeating non terminating decimal.

**And these non recurring decimals can never be converted in to fractions and they are called as irrational numbers. **

Already we know the stuff that recurring decimal can be converted into fraction and it is rational.

Now our question is, how a non terminating recurring decimal can be converted into fraction. That we are going to see in the next section.

**Step 1 : **

**Let x = Given decimal number **

**For example, **

**If the given decimal number is 2.0343434......... **

**then, let x = 2.0343434...........**

**Step 2 : **

**Identify the repeated pattern**

**For example,**

**In 2.0343434..........., the repeated pattern is 34**

**(Because 34 is being repeated)**

**Step 3 :**

**Identify the first repeated pattern and second repeated pattern as as explained in the example given below. **

**Step 4 :**

**Count the number of digits between the decimal point and first repeated pattern as given in the picture below. **

**Step 5 :**

**Since there is 1 digit between the decimal point and the first repeated pattern, we have to multiply the given decimal by 10 as given in the picture below. **

**(If there are two digits -----------> multiply by 100, **

**three digits -----------> multiply by 1000 and so on )**

**Note : In (1), we have only repeated patterns after the decimal.**

**Step 6 : **

**Count the number of digits between the decimal point and second repeated pattern as given in the picture below.**

**Step 7 :**

**Since there are 3 digits between the decimal point and the second repeated pattern, we have to multiply the given decimal by 1000 as given in the picture below. **

**Note : In (2), we have only repeated patterns after the decimal.**

**Step 8 :**

**Now, we have to subtract the result of step 5 from step 7 as given in the picture below. **

**Now we got the fraction which is equal to the given decimal**

To have better understanding on "Introduction to irrational numbers" let us look at some problems.

**Problem 1 :**

Covert the given non terminating decimal into fraction

**32.03256256256..........**

**Solution : **

Let X = 32.03256256256.............

Here, the repeated pattern is 256

No. of digits between the 1st repeated pattern and decimal = 2

So, multiply the given decimal by 100. Then, we have

**100X = 3203.256256256...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 5

So, multiply the given decimal by 100000. Then, we have

**100000X = 3203256.256256256...............----------(2)**

(2) - (1) --------> 99900X = 3200053

X = 3200053 / 99900

**Hence, 32.03256256256.......... = 3200053 / 99900**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

Let us look at the next problem on "Introduction to irrational numbers"

**Problem 2 :**

Covert the given non terminating decimal into fraction

**0.01232222........**

**Solution : **

Let X = 0.01232222.............

Here, the repeated pattern is 2

No. of digits between the 1st repeated pattern and decimal = 4

(Here, the first repeated pattern starts after four digits of the decimal)

So, multiply the given decimal by 10000. Then, we have

**10000X = 123.2222...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 5

So, multiply the given decimal by 100000. Then, we have

**100000X = 1232.2222...............----------(2)**

(2) - (1) --------> 90000X = 1109

X = 1109 / 90000

**Hence, 0.01232222........... = 1109 / 90000**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

Let us look at the next problem on "Introduction to irrational numbers"

**Problem 3 :**

Covert the given non terminating decimal into fraction

**2.03323232..........**

**Solution : **

Let X = 2.03323232.............

Here, the repeated pattern is 32

No. of digits between the 1st repeated pattern and decimal = 2

(Here, the first repeated pattern starts after two digits of the decimal)

So, multiply the given decimal by 100. Then, we have

**100X = 203.323232...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 4

So, multiply the given decimal by 10000. Then, we have

**10000X = 20332.323232...............----------(2)**

(2) - (1) --------> 9900X = 20129

X = 9900 / 20129

**Hence, 2.03323232.......... = 9900 / 20129**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

Let us look at the next problem on "Introduction to irrational numbers"

**Problem 4 :**

Covert the given non terminating decimal into fraction

**0.252525..........**

**Solution : **

Let X = 0.252525.............

Here, the repeated pattern is 25

No. of digits between the 1st repeated pattern and decimal = 0

So, multiply the given decimal by 1. Then, we have

**X = 0.252525...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 2

So, multiply the given decimal by 100. Then, we have

**100X = 25.252525...............----------(2)**

(2) - (1) --------> 99X = 25

X = 25 / 99

**Hence, 0.252525.......... = 25 / 99**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

Let us look at the next problem on "Introduction to irrational numbers"

**Problem 5 :**

Covert the given non terminating decimal into fraction

**3.3333..........**

**Solution : **

Let X = 3.3333.............

Here, the repeated pattern is 3

No. of digits between the 1st repeated pattern and decimal = 0

(Here, the first repeated pattern is "3" which comes right after the decimal point)

So, multiply the given decimal by 1. Then, we have

**X = 3.3333...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 1

(Here, the second repeated pattern is "3" which comes one digit after the decimal point)

So, multiply the given decimal by 10. Then, we have

**10X = 33.3333...............----------(2)**

(2) - (1) --------> 9X = 30

X = 30 / 9 = 10 / 3

**Hence, 3.3333.............. = 10 / 9**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

Let us look at the next problem on "Introduction to irrational numbers"

**Problem 6 :**

Covert the given non terminating decimal into fraction

**1.023562562562..........**

**Solution : **

Let X = 1.023562562562.............

Here, the repeated pattern is 562

No. of digits between the 1st repeated pattern and decimal = 3

So, multiply the given decimal by 1000. Then, we have

**1000X = 1023.562562562...............----------(1) **

No. of digits between the 2nd repeated pattern and decimal = 6

So, multiply the given decimal by 1000000. Then, we have

**1000000X = 1023562.562562562...............----------(2)**

(2) - (1) --------> 999000X = 1022538

X = 1022539 / 999000

**Hence, 1.023562562562.......... = ****1022539 / 999000**

Since the given non terminating recurring decimal can be converted into fraction, it is rational.

After having gone through the stuff and examples, we hope that the students would have understood "Introduction to irrational numbers"

If you want to know more about "Introduction to irrational numbers", please click here.

Apart from "Introduction to irrational numbers", you can also visit the following pages.

__Converting percent into fractions__

__Converting improper fractions into mixed fractions__

__Converting mixed fractions into improper fractions__

**Converting decimals into fractions**

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