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## Definition of Rational Number

The word **‘Rational’** comes from the word **‘Ratio’**. From the name itself, it can be assumed that the **Rational Number** has something related to the **Ratio**. Certainty, the **Rational Numbers** are the numbers that can be expressed in terms of the ratio of **Integers**.

The **Rational Numbers** can be defined as the numbers that can be expressed in the form \frac{p}{q}, where, p and q are the Integers and q\ne 0. The **Rational Numbers** are denoted by the letter Q.

Therefore, for a number to be a **Rational Number**, two conditions must be satisfied:

- The number must be expressible in \frac{p}{q} form where, p and q are integers and
- The denominator, q, must not be equal to zero (0) or q\ne 0.

## Rational Numbers Examples

All the Natural Numbers, Whole Numbers and Integers are **Rational Numbers** because all of those can be expressed in \frac {p}{q} form, where q\ne 0.

For example, let us consider some numbers such as 2,\, 5,\,15,\, -5,\, -3

The number 2 in the above example can be expressed as \frac{2}{1}, here, p=2, q=1 and q\ne 0.

Similarly,

5=\frac{5}{1}, p=5, q=1 and q\ne 0

15=\frac{15}{1}, p=15, q=1 and q\ne 0

-5=\frac{-5}{1}, p=-5, q=1 and q\ne 0

-3=\frac{-3}{1}, p=-3, q=1 and q\ne 0

The number, zero (0), is also a rational number because zero can be expressed as,

\frac{0}{1}, \frac{0}{2}, \frac{0}{5}, \frac{0}{120}, etc.

In the above examples p and q are integers and q\ne 0, which satisfies both the conditions of a **Rational Number**. Therefore, it can be said that 0 is also a **Rational Number**.

From the above discussion, it can be concluded that all Integers and fractions can be considered as **Rational Numbers**. However, \pi is not a **Rational Number** even if the value of pi is \frac {22}{7}. \pi is considered as an **Irrational Number**.

## Equivalent Rational Number

To understand the concept of **Equivalent Rational Number** let us consider a number as \frac {2}{3}.

Now, the number \frac{2}{3} can also be represented as,

\frac{2}{3}=\frac{4}{6}=\frac{6}{9}=\frac{8}{12}=……… and so on.

Therefore, it can be said that **Rational Numbers** do not have a unique representation and they can be represented in many forms. These different forms of a **Rational Number** are called **Equivalent Rational Numbers** or Fractions.

However, while representing such a number on the Number Line we have to consider the number in which p and q does not have any common factor or p and q are co-prime. For the above example, 2 and 3 do not have any common factor in the fraction \frac{2}{3}. Therefore, \frac {2}{3} will be used for representing all of the **Equivalent Rational Numbers** on the **Number Line**.

## Rational Numbers Properties

### Closure Property of Rational Numbers

The **Closure Property of Rational Numbers** states that for any two **Rational Numbers** say a and b, the addition, subtraction and multiplication between them results in a **Rational Number** only. However, the division is not considered under closure property because we cannot define the division of a **Rational Number** by zero (0).

**Addition of two Rational Numbers is a Rational Number**

a+b=c

Example: 1+1=2,\,5+9=14,\, \frac{1}{2}+\frac{1}{2}=1 etc.

**Subtraction of two Rational Numbers is a Rational Number**

a-b=c

Example: 1-1=0,\,4-2=2,\,15-6=9,\,\frac{3}{2}-\frac{1}{2}=1 etc.

**Multiplication of two Rational Numbers is a Rational Number**

a\times b=c

Example: 1\times 1=1,\,3\times 2=6, \,12\times 5=60,\, \frac{1}{2} \times \frac{1}{3}=\frac{1}{6} etc.

### Commutative Property of Rational Numbers

The **Commutative Property of Rational Numbers** states that for any two Rational Numbers say a and b, the addition and multiplication between them are **commutative**. However, the **Commutative Property** is not applicable for Subtraction and Division between two or more **Rational Numbers**.

**Commutative Property for Addition of two Rational Numbers**

If a and b are two Rational Numbers, then,

a+b=b+a

**Commutative Property for Multiplication of two Rational Numbers**

If a and b are two Rational Numbers, then,

a\times b=b\times a

#### Why Commutative Property is not applicable for Subtraction between two Rational Numbers?

To understand why **Commutative Property** is not applicable for Subtraction between two or more Rational Numbers let us consider two Rational Numbers as

a=4 and b=2

Now, a-b=4-2=2 and b-a=2-4=-2

Here, a-b\ne=b-a

Hence, **Commutative Property** is not applicable for Subtraction between two Rational Numbers.

#### Why Commutative Property is not applicable for Division between two Rational Numbers?

Let us consider two Rational Numbers a and b such that,

a=2 and b=5

Now, \frac {a}{b}=\frac{2}{5}=0.4

and, \frac{b}{a}=\frac{5}{2}=2.5

Here, \frac{a}{b}\ne\frac{b}{a}

Therefore, it can be concluded that the **Commutative Property** is not applicable for Division between two Rational Numbers.

### Associative Property of Rational Numbers

The **Associative Property of Rational Numbers** states that for any three **Rational Numbers** say a, b and c, the addition and multiplication between them are **Associative**. However, the **Associative Property** is not applicable for Subtraction and Division between two **Rational Numbers**.

**Associative Property for Addition of two Rational Numbers**

If a, b and c are three **Rational Numbers**, then,

a+(b+c)=(a+b)+c

**Associative Property for Multiplication of two Rational Numbers**

If a, b and c are three **Rational Numbers**, then,

a\times (b\times c)=(a\times b)\times c

#### Why Associative Property is not applicable for Subtraction of Rational Numbers?

To understand why **Associative Property** is not applicable for Subtraction of Rational Numbers let us consider three Rational Numbers as

a=4, b=3 and c=2

Now, a-(b-c)=4-(3-2)=4-1=3 and (a-b)-c=(4-3)-2=1-2=-1

Here, a-(b-c)\ne=(a-b)-c

Hence, A**ssociative Property** is not applicable for Subtraction between Rational Numbers.

#### Why Associative Property is not applicable for Division of Rational Numbers?

The Associative Property is not applicable for Division of Rational Numbers because the division of a Rational Number by zero (0) is not defined.

### Distributive Property of Rational Numbers

To understand the **Distributive Property of Rational Numbers**, let us consider three Rational Numbers a, b and c. Now, as per **Distributive Property**,

a\times (b+c)=(a\times b)+(a\times c)

#### Proof of Distributive Property of Rational Numbers

Let, a=1, b=2, and c=3

LHS=a\times (b+c)

LHS=1\times (2+3)

LHS=1\times 5

LHS=5

And

RHS=(a\times b)+(a\times c)

RHS=(1\times 2)+(1\times 3)

RHS=2+3

RHS=5

From above, LHS=RHS. Hence Proved.

### Identity Property of Rational Numbers

#### Additive Identity of Rational Numbers

For any Rational Number, the Number zero (0) is called the **Additive identity** because any Rational Number added to zero gives the same Rational Number.

For Example: 5+0=5,\, 11+0=11,\, \frac{1}{5}+0=\frac{1}{5} etc.

#### Multiplicative Identity of Rational Numbers

The Number 1 is called **Multiplicative Identity** of any Rational Number because any Rational Number Multiplied with 1 gives the same number.

For Example: 15\times 1=15,\, \frac{3}{8}\times 1=\frac{3}{8} etc.

### Inverse Property of Rational Numbers

#### Additive Inverse of Rational Numbers

For any Rational Number say \frac{p}{q},\, -\frac{p}{q} is called the **Additive Inverse** of the Rational Number.

#### Multiplicative Inverse of Rational Numbers

For any Rational Number say \frac{p}{q},\, \frac{q}{p} is called the **Multiplicative Inverse** of the Rational Number.

## Frequently Asked Questions (FAQ)

### What is Rational Number?

The Rational Numbers can be defined as the numbers that can be expressed in the form p/q, where, p and q are the integers and q not equal to 0. The Rational Numbers are denoted by the letter Q.

### Is 0 a rational number?

Yes, zero (0) is a Rational Number because it can be expressed in p/q form where p and q are integers and q not equal to 0. For example:

0=0/1=0/3=0/9 etc.

In all of the above examples, both p and q are integers and q not equal to 0. Therefore, 0 is a Rational Number.

### Is the reciprocal of a positive rational number positive?

Yes, the reciprocal of a Positive Rational Number is always positive. For example, 2, 5, 8, 10 etc. are the positive Rational Numbers.

The reciprocal of the numbers is,

1/2=0.5

2/5=0.2

1/8=0.125

1/10=0.1

### Is every rational number an integer?

No, every Rational Number is not an Integer. For example, the Rational Numbers such as 1/2, 2/5, 1/-7, -2/9 etc. are not Integers as they are fractions.

However, all the Integers are Rational Numbers.

### What is the Rational Number that is equal to its negative?

The Rational Number zero (0) is equal to its negative.

0=-0

### The denominator of the rational number 0 is?

The denominator of the Rational Number 0 is a Rational Number.

For example:

0= 0/1=0/3=0/-2=0/-9

### What is rational number example?

The Rational Numbers can be defined as the numbers that can be expressed in the form p/q, where, p and q are the integers and q not equal to 0.

Examples of Rational Numbers are: 0, 1, 2, 3, 4, 1/2, 2/3, 4/5 etc.

### Is 3.14 a rational number?

Yes, 3.14 is a Rational Number because it can be expressed in term of p/q where p and q are integers and q not equal to 0.

However, π whose value is 3.14159…… is not a Rational Number.

### Is 2/5 an irrational number?

Yes, 2/5 is a Rational Number as it is in the form p/q. Both p and q are integers and q not equal to 0.

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