# Rational Numbers

• Difficulty Level : Medium
• Last Updated : 24 Nov, 2020

Even after coining integers, one could not relax! 10 ÷ 5 is no doubt fine, giving the answer 2 but is 8 ÷ 5 comfortable? Numbers between numbers are needed. 8 ÷ 5 seen as 1.6, is a number between 1 and 2. But, where does (-3) ÷ 4 lies? Between 0 and -1. Thus, a ratio made by dividing an integer by another integer is called a rational number. The collection of all rational numbers is denoted by Q.

A Rational number is a number of the fractional form a / b, where a and b are integers and b ≠ 0.

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Examples: 1 / 4 , 3 / 7 , (-11) / (-6)

• All-natural numbers, whole numbers, integers, and fractions are rational numbers.
• Every rational number can be represented on a number line.
• 0 is neither a positive nor a negative rational number.

## Properties of Rational Numbers

### Closure property for the collection Q of rational numbers

• Closure property for Addition: For any two rational numbers a and b, the sum a + b is also a rational number.
• Closure property for Multiplication:  For any two rational numbers a and b, the sum ab is also a rational number

Example:

Take a = 3 / 4 and b = (-1) / 2

Now,

a + b = 3 / 4 + (-1) / 2

= 3 / 4 + (-2) / 4

= (3 – 2) / 4

= 1 / 4 is in Q
Also,
a * b = 3 / 4 * (-1) / 2

= (-3) / 8 is in Q

### The commutative property for the collection Q of rational numbers

• The commutative property for Addition: For any two rational numbers a and b, a + b = b + a.
• The commutative property for Multiplication:  For any two rational numbers a and b, ab = ba.

Example:

Take a = (-7) / 8 and b = 3 / 5
Now,

a + b = (-7) / 8 + 3 / 5

= -7 x 5 + 3 x 8 / 40

= (-35 + 24) / 40

= (-11) / 40
Also
b + a = 3 / 5 + (-7) / 8

= 3 x 8 + (-7) x 5 / 40

= (24 – 35) / 40

=  (-11) / 40

Further,

ab = (-7) / 8 x 3 / 5

= (-7 x 3) / (8 x 5)

= (-21) / 40
Also,

ba = 3 / 5 x (-7) / 8

= (3 x 7 ) / (5 x 8)

= (-21) / 40

Hence multiplication is Commutative.

### Associative property for the collection Q of rational numbers

• Associative property for Addition: For any three rational numbers a, b, and c, a + ( b + c ) = ( a + b ) + c
• Associative property for Multiplication: For any three rational numbers a, b, and c, a (b c) = (a b)c

Example:

Take rational numbers a,b,c as a = -1 / 2, b = 3 / 5, c = -7 / 10
Now,

a + b = -1 / 2 + 3 / 5

= -5 / 10 + 6 / 10
= -5 + 6 / 10

= 1 / 10

(a + b) + c = 1 / 10 + (-7) / 10

= 1 – 7 / 10

= -6 / 10

= -3 / 5………………………………………………….( 1 )

Also,

b + c = 3 / 5 + (-7) / 10

= 6 / 10 + (-7) / 10

= 6 – 7 / 10

= -1 / 10

a + (b + c) = -1 / 2 + (-1) / 10

= -5 / 10 + (-1) / 10

= -5 – 1 / 10

= -6 / 10

= -3 / 5 ………………………………………………..( 2 )

(1) and (2) shows that (a + b) + c = a + (b + c) is true for rational numbers.

Similarly,

a ∗ b = -1 / 2 ∗ 3 / 5

= -3 / 10

(a ∗ b)∗ c = -3 / 10 ∗ -7 / 10

= -3 ∗ (-7) / 100

= 21 / 100  …………………………………………( 3 )

Also,

b∗ c = 3 / 5 ∗ (-7) / 10

= -21 / 50
a ∗ ( b ∗ c ) = -1 / 2 ∗ (-21) / 50

= 21 / 100     …………………………………….( 4 )

( 3 ) and ( 4 ) shows that (a∗ b)∗ c = a ∗ ( b ∗ c ) is true for rational numbers. Thus,the associative property is true for addition and multiplication of rational numbers.

### The identity property for the collection Q of rational numbers

• The identity property for Addition: For any rational number a, there exists a unique rational numbers 0 such that 0 + a = a = a + 0.
• The identity property for Multiplication: For any rational number a, there exists a unique rational number 1 such that a ∗ 1 = a = a ∗ 1.

Example:

Take a = 3 / (-7) that is a = -3 / 7

Now,
-3 / 7 + 0 = -3 / 7 = 0 + (-3) / 7

Hence,0 is the additive identity for -3 / 7

Also,

-3 / 7 ∗ 1 = -3 / 7 = 1 ∗ 3 / 7

Hence,1 is the multiplicative identity for -3 / 7.

### Inverse property for the collection Q of the rational numbers

• Additive Inverse property: For any rational number a, there exist a unique rational number -a such that a + (-a) = (-a) +  a = 0.Here, 0 is the additive identity.
• Multiplicative Inverse property: For any rational number b, there exist a unique rational number 1/b such that b ∗ 1 / b = 1 / b ∗ b = 1.Here, 1 is the multiplicative identity.

Example:

Take a = -11 / 23

Now, -a = -(-11) / 23

= 11 / 23
So,

a + (-a) = -11 / 23 + 11 / 23

= -11 + 11 / 23

= 0 / 23

=0
Also,
(-a) + a  = 11 / 23 + (-11) / 23

= 11-11 / 23

= 0 / 23

= 0

Hence a + (-a) = (-a) + a = 0 is true.
Also,

Take b = -17/29

Now,

1 / b = 29 / (-17) = -29 / 17
b ∗ 1 / b = -17 / 29 ∗ -29 / 17 = 1
Also,
1 / b ∗ b = 29 / 17 ∗ -17 / 29 = 1
Hence, b ∗ 1 / b = 1 / b ∗ b = 1 is true.

### The distributive property for the collection Q of rational numbers

Multiplicative is distributive over addition for the collection of rational numbers. For any three rational numbers a, b, and c the distributive law is  a ∗ ( b +c ) = ( a ∗ b ) + ( a ∗ c )

Example:

Take rational number a, b, c as a = -7 / 9, b = 11 / 18 and c = -14 / 27

Now,
b + c = 11 / 18 + (-14) / 27

= 33 / 54 + (-28) / 54

= 33 – 28 /54

= 5 / 54

a ∗ ( b + c ) = -7 / 9 ∗ 5 / 54

= (-7) ∗ 5 / 9 ∗ 54

= -35 / 486………………………………………………………………….(1)

Also,
a ∗ b = -7 / 9 ∗ 11 / 18

= (-7) ∗ 11 / 9 ∗ 18

= -77 / 9 ∗ 9 ∗ 2

a ∗ c = (-7) /  9 ∗ (-14) / 27

= 7 ∗ 14 / 9 ∗ 9∗ 3

= 98 / 9 ∗ 9 ∗ 3

(a ∗ b) + (a ∗ c) = (-77 / 9 ∗ 9 ∗ 2 ) + ( 98 / 9 ∗ 9 ∗ 3)

= (-77) ∗ 3 + 98 ∗ 2 / 9 ∗ 9 ∗ 2 ∗ 3

= -231 + 196 / 486

= (-35) / 486…………………………………………………………….(2)

( 1 ) and( 2 ) shows that  a ∗ ( b + c ) = ( a ∗ b ) + ( a ∗ c ). Hence,multiplication is distributive over addition for the collection Q of rational numbers.

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