Rational Number

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:

  1. The number must be expressible in \frac{p}{q} form where, p and q are integers and
  2. 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, Associative 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)

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.

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.

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

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.

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

0=-0

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

For example:

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

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.

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.

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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