# RATIONAL AND IRRATIONAL NUMBERS

## About "Rational and irrational numbers"

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.

## Rational numbers

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.

## Irrational 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.

## 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.

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

## How to convert non terminating repeating decimal to fraction?

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

## Some more problems

To have better understanding on conversion of non terminating repeating decimals to fraction, 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 "Rational and 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 "Rational and 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 "Rational and 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 "Rational and 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 "Rational and 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, "Rational and irrational numbers"

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