Rational Numbers
Introduction to Rational
Numbers
A number which can be written in the
form of p/q, where p and q are integers and q ≠0 is called a rational
number. Numbers written in fraction are rational numbers.
Ø -2/3
Ø 6/7
Ø -3/1
Ø 4/0 is an undefined
number and hence not a rational number.
For example, if there are 10 chocolates
to be divided into 4 children, it is not possible to give 3 complete chocolates
to each one of them. But, if 2 chocolates are halved, then there will be 4half
pieces and 8 full pieces of the chocolates. So each child would get 2 full
pieces and 1 half piece
Types of numbers
To solve different types of equations,
following types of numbers are usually used. The scope of these numbers may
overlap. Like,
o
All natural numbers are whole
numbers too. Whole numbers have a single extra number zero which is not a
natural number.
o
For example, 5, 50, 23643 etc all are natural numbers as well as whole numbers.
o
All positive integers are natural
as well as whole numbers.
o
For example, integers like 5, 50,
23643 etc all are natural numbers as well as whole
numbers.
|
Type of Number |
Range of numbers |
Equation |
Value of x to solve the
equation |
|
Natural Numbers |
1 to ∞ |
x + 2 = 13 |
11 |
|
Whole Numbers |
0 to ∞ |
x + 5 = 5 |
0 |
|
Integer (Positive) |
1 to ∞ |
x + 18 = 22 |
4 |
|
Integer (Negative) |
-∞ to -1 |
x + 18 = 5 |
-13 |
|
Rational Numbers (Positive) |
p/q; q ≠0 |
2x = 3 |
3/2 |
|
Rational Numbers (Negative) |
-p/q; q ≠0 |
5x + 7 = 0 |
-7/5 |
Properties of the types of
numbers - Closure
A set of numbers is said to be closed for
a specific mathematical operation if the result obtained when an operation is
performed on any two numbers in the set, is itself a member of the set. If a
set of numbers is closed for a particular operation then it is said to possess
the closure property for that operation.
1. Whole Numbers
o
Addition - Adding two
whole numbers results in another whole number. Hence, whole numbers under
addition are closed.
- 2+3 = 5
- 0 + 6 = 6
o
Subtraction - Subtracting
two whole numbers may result in a negative number which is not a whole number.
Hence, whole numbers under subtraction are not closed.
- 5-3 = 2 (whole)
- 0 - 6 = -6 (not whole)
o
Multiplication - Multiplying
two whole numbers results in another whole number. Hence, whole numbers under
multiplication are closed.
- 5 x 3 = 15
- 2 x 0 = 0
o
Division -
Dividing two whole numbers may result in a fraction or a number with decimal
point which is not a whole number. Hence, whole numbers under subtraction are not
closed.
- 4 ÷ 2 = 2 (whole)
- 10 ÷ 4 = 10/4 = 2.5 (not whole)
2. Integers
o
Addition - Adding two integers results in another integer. Hence, integers under
addition are closed.
- (-2) +3 = 1
- (-7) + (-5) = -12
o
Subtraction -
Subtracting two integers results in another integer.
Hence, integers under subtraction are closed.
- 5-(-3) = 8
- (-3) - 6 = -9
o
Multiplication – Multiplying
two integers results in another integer. Hence,
integers under multiplication are closed.
- 5 x (-3) = -15
- (-2) x (-5) = 10
o
Division -
Dividing two integers may result in a fraction or a number with decimal point
which is not an integer. Hence, integers under division are not closed.
- 4 ÷ (-2) = -2 (Integer)
- (-10) ÷ 4 = -10/4 = -2.5 (not an
integer)
3.
Rational Numbers
o
Addition - Adding two
rational numbers results in another rational number. Hence, rational numbers
under addition are closed.
o 8/5 + (-2)/5
= 6/5
o 3/8 + (-5)/7
= (21 + (-40))/56 = -19/56
o
Subtraction - Subtracting
two rational numbers results in another rational number. Hence, rational
numbers under subtraction are closed.
o 8/5 - (-2)/5
= 10/5 = 2 or 2/1
o 3/8 -5/7 =
(21 - 40)/56 = -19/56
o
Multiplication - Multiplying
two rational numbers results in another rational number. Hence, rational numbersunder multiplication are closed.
o 8/5 x (-2)/5
= -16/25
o (-3)/8
x(-5)/7 = 15/56
o
Division -
Dividing two rational numbers may result in an undefined number with which is
not a rational number. Hence, rational numbers under division are not
closed.
o 4/3 ÷ (-2)/7
= -28/6 = -14/3 (rational number)
o (-10)/3 ÷
0/1 = -10/0 = undefined (not a rational number)
Properties of the types of
numbers - Commutativity
A set of numbers is said to be commutative for
a specific mathematical operation if the result obtained when changing order of
the operands does not change the result.
1.
Whole Numbers
o
Addition – Changing the
order of operands in addition of whole numbers does not change the result.
Hence, whole numbers under addition are commutative.
- 2+3 = 3 + 2
- 0 + 6 = 6 + 0
o
Subtraction -
Changing the order of operands in subtraction of whole numbers changes the
result. Hence, whole numbers under subtraction are not commutative.
- 5-3 ≠3 - 5
- 0 - 6 ≠ 6 - 0
o
Multiplication - Changing the
order of operands in multiplication of whole numbers does not change the
result. Hence, whole numbers under multiplication are commutative.
- 5 x 3 = 3 x 5
- 2 x 0 = 0 x 2
o
Division -
Changing the order of operands in division of whole numbers changes the result.
Hence, whole numbers under division are not commutative.
- 4 ÷ 2 ≠ 2÷ 4
- 10 ÷ 4 ≠4 ÷ 10
2. Integers
o
Addition – Changing the
order of operands in addition of integers does not change the result. Hence,
integers under addition are commutative.
- 2 + (-3) = (-3) + 2
- (-1) + 6 = 6 + (-1)
o
Subtraction -
Changing the order of operands in subtraction of integers changes the result.
Hence, integers under subtraction are not commutative.
- 5 -(-3) ≠ (-3) - 5
- (-1) - 6 ≠ 6 - (-1)
o
Multiplication - Changing the
order of operands in multiplication of integers does not change the result.
Hence, integers under multiplication are commutative.
- 5 x (-3) = (-3) x 5
- (-2) x 0 = 0 x (-2)
o
Division -
Changing the order of operands in division of integers changes the result. Hence,
integers under division are not commutative.
- 4 ÷ (-2)≠(-2)÷ 4
- (-10) ÷ 4 ≠4 ÷ (-10)
3. Rational Numbers
o
Addition - Changing the
order of operands in addition of rational numbers does not change the result.
Hence, rational numbers under addition are commutative.
§ 8/5 + (-2)/5
= (-2)/5 + 8/5
§ 3/8 + (-5)/7
= (-5)/7 + 3/8
o
Subtraction - Changing the
order of operands in subtractionof rational numbers
changes the result. Hence, rational numbers under subtraction are not
commutative.
§ 8/5 - (-2)/5
≠(-2)/5 - 8/5
§ 3/8 -5/7
≠5/7 - 3/8
o
Multiplication - Changing the
order of operands in multiplication of rational numbers does not change the
result. Hence, rational numbers under multiplication are commutative.
§ 8/5 x (-2)/5
= (-2)/5 x 8/5
§ (-3)/8
x(-5)/7 = (-5)/7 x (-3)/8
o
Division -
Changing the order of operands in divisionof rational
numbers changes the result. Hence, rational numbers under division are not
commutative.
- 4/3 ÷ (-2)/7≠-(-2)/7 ÷ 4/3
- (-10)/3 ÷ 4/1≠4/1 ÷(-10)/3
Properties of the types of
numbers - Associativity
A set of numbers is said to be associative for
a specific mathematical operation if the result obtained when changing grouping
(parenthesizing) of the operands does not change the result.
1. Whole Numbers
o
Addition – Changing the
grouping of operands in addition of whole numbers does not change the result.
Hence, whole numbers under addition are associative.
§ 2 + (3 + 6)
= (2 + 3) + 6
§ (0 + 6)+ 8 =
0 + (6 + 8)
o
Subtraction -
Changing the grouping of operands in subtraction of whole numbers changes the
result. Hence, whole numbers under subtraction are not associative.
§ 5 - (3 - 4) ≠
(5 - 3) - 4
§ (2 - 0) - 6 ≠
2 - (0 - 6)
o
Multiplication - Changing the
grouping of operands in multiplication of whole numbers does not change the
result. Hence, whole numbers under multiplication are associative.
§ 5 x (3 x 6)
= (5 x 3) x 6
§ (2 x 0)x 9 =
2 x (0 x 9)
o
Division -
Changing the grouping of operands in division of whole numbers changes the
result. Hence, whole numbers under division are not associative.
§ 4 ÷ (2 ÷ 6) ≠(4
÷ 2) ÷ 6
§ (10 ÷ 4)÷ 7 ≠10
÷ (4 ÷ 7)
2. Integers
o
Addition – Changing the
grouping of operands in addition of integers does not change the result. Hence,
integers under addition are associative.
- 2 + (3 + (-6)) = (2 + 3) + (-6)
- (0 + (-6)) + (-8) = 0 + ((-6) + (-8))
o
Subtraction -
Changing the grouping of operands in subtraction of integers changes the
result. Hence, integers under subtraction are not associative.
- 5 - (3 - (-4)) ≠ (5 - 3) - (-4)
- ((-2) - 0) - 6 ≠ (-2) - (0 - 6)
o
Multiplication - Changing the
grouping of operands in multiplication of integers does not change the result.
Hence, integers under multiplication are associative.
- 5 x ((-3) x 6) = (5 x (-3)) x 6
- ((-2) x 0)x (-9)= (-2) x (0 x (-9))
o
Division -
Changing the grouping of operands in division of integers changes the result.
Hence, integers under division are not associative.
- 4 ÷ (2 ÷(-6)) ≠(4 ÷ 2) ÷(-6)
- ((-10) ÷ 4) ÷ 7 ≠(-10) ÷ (4 ÷ 7)
3. Rational Numbers
o
Addition - Changing the
grouping of operands in addition of rational numbers does not change the
result. Hence, rational numbers under addition are associative.
§ 2/3 + (3/2 +
(-6)/7) = (2/3 + 3/2) + (-6)/7
§ (0/1 +
(-6)/5) + (-8)/3 = 0/1 + ((-6)/5 + (-8)/3)
o
Subtraction - Changing the
grouping of operands in subtraction of rational numbers changes the result.
Hence, rational numbers under subtraction are not associative.
§ 5/4 - (3/11
- (-4)/7) ≠ (5/4 - 3/11) - (-4)/7
§ ((-2)/7 -
0/3) - 6/5 ≠ (-2)/7 - (0/3 - 6/5)
o
Multiplication - Changing the
grouping of operands in multiplication of rational numbers does not change the
result. Hence, rational numbers under multiplication are associative.
§ 5/3 x
((-3)/4 x 6/11) = (5/3 x (-3)/4) x 6/11
§ ((-2)/5 x
0/1)x (-9)/2 = (-2)/5 x (0/1 x (-9)/2)
o
Division -
Changing the grouping of operands in division of rational numbers changes the
result. Hence, rational numbers under division are not associative.
- 4/3 ÷ (2/3 ÷ (-6)/7) ≠(4/3 ÷
2/3) ÷ (-6)/7
- ((-10)/3 ÷ 4/7) ÷ 7/6 ≠(-10)/3 ÷
(4/7 ÷ 7/6)
Additive and Multiplicative
identity of Rational numbers
o
Additive identity: Zero is the
additive identity for Rational, natural, whole numbers and integers, since
adding it to them does not change the result.
- 3 + 0 = 3
- -4/5 + 0 = -4/5
Hence, 0 + a = a + 0 = a, where a can be rational number or natural
number or whole number of integer.
o
Multiplicative identity: One is the
additive identity for Rational, natural, whole numbers and integers, since
multiplying it to them does not change the result.
- 3 x 1 = 3
- -4/5 x 1 = -4/5
Hence, 1x a = ax1 = a, where a can be rational number or natural number
or whole number of integer.
Additive inverse and
Multiplicative inverse of Rational numbers
o
Negative or Additive inverse: -
or minus is the additive inverse for
Rational, natural, whole numbers and integers, since adding its additive
inverse to a number results in zero.
- 3 + (-3) = 0
- (-4/5) + (-(-4/5)) = (-4/5) + 4/5 = 0
Hence, a + (-a) = (-a) + a = 0, where a can be rational number or
natural number or whole number of integer.
o
Reciprocal or Multiplicative
inverse: Dividing a number by 1 is the multiplicative
inverse for Rational, natural, whole numbers and integers, since multiplying it
to the original number always results in 1.
- 3 x 1/3 = 1
- (-4/5) x (1/(-4/5) = (-4/5) x (-5/4) =
1
Hence, ax 1/a = 1/a x
a = 1, where a can be rational number or natural number or integer.
Distributive property of
multiplication over addition and subtraction for rational numbers
The distributive property of multiplication is:
o
Over addition: a(b + c) =
ab + ac
- -3/4 {2/3 + (-5/6)} = -3/4
{(4+(-5))/6} = (-3/4) x (-1/6) = 3/24 = 1/8
- -3/4 {2/3 + (-5/6)} = -3/4 x 2/3 +
(-3/4) x (-5/6) = -1/2 + 5/8 = (-4+5)/8 = 1/8
o
Over subtraction: a(b - c) =
ab - ac
- -3/4 {2/3 - 5/6} = -3/4 {(4-5)/6} =
(-3/4) x (-1/6) = 3/24 = 1/8
- -3/4 {2/3 - 5/6} = -3/4 x 2/3 - (-3/4)
x 5/6 = -1/2 + 5/8 = (-4+5)/8 = 1/8
Representation of number types
on number line
While representing rational numbers on number line,
o
Between two integers, the line
should be divided into number of equal parts which is same as the denominator
of the rational number.
o
For eg,
to represent 1/3, three equal parts are made from 0 to 1 representing 1/3, 2/3
and 3/3 (that is 1) respectively.
Rational numbers between two
rational numbers
There are always definite amount of numbers between two natural/whole
numbers or integers. But, there can be indefinite amount of
numbers between two rational numbers.
o
Natural numbers between 4 and 22
are 17 (5, 6, …, 20, 21).
o
Whole numbers between 0 and 5 are
4 (1,2,3 and 4).
o
Integers between -4 and 9 are 12
(-3, -2, …., 7, 8)
o
Rational numbers between 3/10 and
7/10 can be:
o
4/10, 5/10, 6/10
o
Or, 31/100, 32/100, …., 68/100, 69/100 since 3/10 can also be written as
30/100 and 7/10 can be written as 70/100.
o
Since 3/10 can also be written as
300/1000 and 7/10 can be written as 700/1000, there can be numbers 301/1000 to
699/1000.
o
And so on.
Between any 2 numbers, it is not necessary that there will be an integer
or a whole number but there is always a rational number.
Example, there are no integer or whole or natural numbers between 1 and
2, but there are rational numbers like, 1/2, 1/3 and 2/3, 1/4, 2/4, 3/4 etc.