Rational Numbers
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
Hence addition is Commutative.
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) / 7Hence,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.
