**Classifying rational numbers :**

Students who are studying math in primary level must be aware of classification of rational numbers.

We can use Venn diagram to classify rational numbers.

A Venn diagram is a visual representation used to show the relationships between groups.

The Venn diagram below shows how rational numbers, integers, and whole numbers are related.

The following are the classifications of rational numbers.

**(i) Natural numbers :**

N = { 1, 2, 3, 4, ...... }

**(ii) Whole numbers :**

W = { 0, 1, 2, 3, ....... }

**(iii) Integers :**

Z = {.........-2, -1, 0, 1, 2, 3, .........}

**(iv) Fractions :**

Examples : -1/2, 7/3, 3/4, 2 1/7

**(v) Decimals :**

Examples : 0.2, -1.23, 0.0085

**(vi) Recurring decimal : **

Examples : 0.333333......, 0.257257..........

**Caution : **

Non terminating recurring decimal is a rational number.

Examples : 0.333333......, 0.257257..........

But, non terminating non recurring decimal is not a rational number and it is irrational number.

Examples:

√2 = 1.41421356......................

√3 = 1.73205080....................

**Question 1 : **

Place the following numbers in the Venn diagram. Then classify each number by indicating in which set or sets each number belongs.

0.35, -3, 75, 3/4

**Solution :**

**75 : **

The number 75 belongs in the sets of whole numbers, integers, and rational numbers.

**-3 :**

The number -3 belongs in the sets of integers and rational numbers.

**3/4 :**

The number -3/4 belongs in the set of rational numbers.

**0.35 :**

The number 0.35 belongs in the set of rational numbers.

**Question 2 :**

Classify the following two numbers as rational and irrational and also explain your answer.

5.312312312.......................

5.385164807.......................

**Solution : **

5.312312312....................... ---> Rational number

5.385164807....................... ---> Irrational number

**Explanation : **

Even though 5.312312312.......... is a non terminating decimal, there is a repeated pattern 312 in it.

So, 5.312312312....... is non terminating recurring decimal.

Hence, 5.312312312....... is a rational number.

5.385164807............ is a non terminating decimal and also there is no repeated pattern in it.

So, 5.385164807............ is non terminating non recurring decimal.

Hence, 5.385164807............ is an irrational number.

**Question 3 : **

Why is the non terminating recurring decimal

2.0343434 .........

considered to be a rational number ? Explain.

**Solution : **

Rational number is usually expressed in the form a/b.

So, if we can express any number in the form "a/b", the number can be considered as rational number.

Now, let us see, how to express the number 2.0343434...... in the form a/b, say fraction.

**Step 1 : **

Let x = 2.0343434...........

**Step 2 : **

Identify the repeated pattern

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 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.

Since the given non terminating recurring decimal can be written as a fraction, it is considered to be a rational number.

After having gone through the stuff given above, we hope that the students would have understood "Classifying rational numbers".

Apart from the stuff given above, if you want to know more about "Classifying rational numbers", please click here

Apart from "Classifying rational numbers", if you need any other stuff in math, please use our google custom search here.

- Rational numbers
- Write an improper fraction as a mixed number
- Write a mixed number as an improper fraction
- Comparing an ordering decimals
- Classifying rational numbers
- Representing division as a fraction
- Identifying opposites and absolute value of rational numbers
- Positive and negative rational numbers
- Rational numbers and opposites on a number line
- Absolute value of rational numbers
- Comparing and ordering rational numbers
- Equivalent fractions and decimals
- Ordering fractions and decimals

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**