Rational Numbers Properties
Rational Numbers show several properties under the different operations (two of such common operations are addition and multiplication), which are as follows:
- Closure Property
- Commutative Property
- Associative Property
- Identity Property
- Inverse Property
- Distributive Property
Closure Property
- Closure Property for Addition: Rational numbers are closed under addition, i.e., for any two rational numbers a and b, the sum a + b is also a rational number.
- Closure Property for Multiplication: Rational numbers are closed under multiplication, i.e., for any two rational numbers a and b, their product ab is also a rational number
Example: Verify Closure Property for the sum and product of 3/4 and -1/2
Solution:
For a = 3 / 4 and b = (-1) / 2
Now, a + b = 3 / 4 + (-1) / 2
β a + b = (3- 2)/ 4
β a + b = 1/4, is Rational Number.
Also, a Γ b = 3/4 Γ (-1)/2 = -3/8, which is also Rational number.
Commutative Property
- Commutative Property for Addition: Rational numbers hold commutative property under addition operation, i.e., for any two rational numbers a and b, a + b = b + a.
- Commutative Property for Multiplication: Rational numbers hold commutative property under multiplication operation as well, i.e., for any two rational numbers a and b, ab = ba.
Example: For a = (-7) / 8 and b = 3 / 5, verify commutative property.
Solution:
Now, a + b = -7/8 + 3/5
β a + b = (-7 x 5 + 3 x 8)/40 = (-35 + 24) / 40
β a + b = (-11) / 40
And, b + a = 3/5 + (-7)/8
β b + a = (3 x 8 + (-7) x 5)/ 40 = (24 β 35)/40
β b + a = -11/40 = a + b
Now, ab = (-7)/8 x 3/5 = (-7 x 3)/(8 x 5)
β ab = -21/40
And, ba = 3/5 x (-7)/8 = (3 x 7 )/(5 x 8)
β ba =(-21)/40 = ab
Associative Property
- Associative Property for Addition: Rational Numbers are associative under addition operation, i.e., for any three Rational Numbers a, b, and c, a + (b + c) = (a + b) + c
- Associative Property for Multiplication: Rational Numbers are associative under multiplication operation as well, i.e., for any three Rational numbers a, b, and c, a(bc) = (ab)c
Example: For three Rational numbers a, b, c where a = -1/2, b = 3/5, c = -7/10, verify associative property.
Solution:
Now, a + b = -1/2 + 3/5 = (-5 + 6)/10 = 1/10
and (a + b) + c = 1/10 + (-7)/10
β (a + b) + c = (1 β 7)/10 = -6/10 = -3/5
Also, b + c = 3/5 + (-7)/10
β b + c = (6 β 7)/10 = -1/10
And, a + (b + c) = -1/2 + (-1)/10
β a + (b + c) = (-5 β 1)/10= -6/10 = -3/5
Thus, (a + b) + c = a + (b + c) is true for Rational Numbers.
Similarly, for multiplication
a Γ b = -1/2 Γ 3/5 = -3/10
And, (a Γ b)Γ c = -3/10 Γ -7/10= -3Γ (-7)/100
β (a Γ b)Γ c = 21/100
Also, bΓ c = 3/5 Γ (-7)/10 = -21/50
and, a Γ ( b Γ c ) = -1/2 Γ (-21)/50
β a Γ ( b Γ c ) = 21/100
Thus, (aΓ b)Γ c = a Γ ( b Γ c ) is true for Rational Numbers.
Identity Property
- Identity Property for Addition: For any rational number a, there exists a unique rational number 0 such that 0 + a = a = a + 0, where 0 is called the identity of the rational number under the addition operation.
- Identity Property for Multiplication: For any rational number a, there exists a unique rational number 1 such that a Γ 1 = a = a Γ 1, where 1 is called the identity of the rational number under the multiplication operation.
Inverse Property
- Additive Inverse property: For any rational number a, there exists a unique rational number -a such that a + (-a) = (-a) + a = 0, and -a is called the inverse of element a under the operation of addition. Also, 0 is the additive identity.
- Multiplicative Inverse property: For any rational number b, there exists a unique rational number 1/b such that b Γ 1 / b = 1 / b Γ b = 1, and 1/b is called the inverse of the element b under the multiplication operation. Here, 1 is the multiplicative identity.
Example: Find the Additive and Multiplicative Inverse of -11/23
Solution:
For a = -11/23
a + (-a) = -11/23 β (-11)/23
a + (-a) = -11/23 + 11/23 = (-11 + 11)/23 = 0
Similarly, (-a) + a = 0
Thus, 11/23 is the additive inverse of -11/23.
Now, for b = -17/29
b Γ 1/b = -17/29 Γ -29/17 = 1
Similarly, 1 / b Γ b = 29/17 Γ -17/29 = 1
Thus, -29 / 17 is the multiplicative inverse of -17/23.
Distributive Property
Distributive property for any two operations holds if one distributes over the other. For example, multiplicative is distributive over addition for the collection of rational numbers, for any three rational numbers a, b, and c the distributive law of multiplication of addition is
a Γ (b +c) = (aΓ b) + (a Γ c), and it is true for all the rational numbers.
Example: For rational number a, b, c i.e., a = -7 / 9, b = 11 / 18 and c = -14 / 27, verify distributive.
Solution:
Now, b + c = 11/18 + (-14)/27
β b + c = 33/54 + (-28)/54 = (33 β 28)/54 = 5/54
and, a Γ ( b + c ) = -7/9 Γ 5/54
β a Γ ( b + c ) = (-7 Γ 5)/(9 Γ 54) = -35/486 . . .(1)
Also, a Γ b = -7/9 Γ 11/18
β a Γ b = (-7 Γ 11)/9 Γ 18 = -77/9 Γ 9 Γ 2
and a Γ c = (-7)/ 9 Γ(-14)/27
β a Γ c = (7 Γ 14)/9 Γ 9 Γ 3 = 98/9 Γ 9 Γ 3
Now, (a Γ b) + (a Γ c) = (-77/9 Γ 9 Γ 2 ) + ( 98/9 Γ 9 Γ 3)
β (a Γ b) + (a Γ c) = (-77 Γ 3 + 98 Γ 2)/9 Γ 9 Γ 2 Γ 3
β (a Γ b) + (a Γ c) = (-231 + 196)/486 = -35/486 . . .(2)
(1) and(2) shows that a Γ ( b + c ) = ( a Γ b ) + ( a Γ c ).
Hence, multiplication is distributive over addition for the collection Q of rational numbers.
Rational Numbers: Definition, Examples, Worksheet
Rational Numbers are numbers written in terms of the ratio of two integers, where the denominator is not zero. In maths, Rational numbers are a type of real numbers that can be written in the form of p/q, where q β 0. Any fraction is a rational number provided its denominator should not be zero.
Examples of Rational Numbers include 12/21β, 34/2, -22 etc. In other words, a rational number is any number that can be written in the form of a/bβ, where a and b are integers and b is not equal to zero.
In this article, we have provided everything related to Rational numbers including, definitions, examples, types, a list of rational numbers, and how to identify rational numbers.
Table of Content
- What is a Rational Numbers?
- Examples of Rational Numbers
- Representation of Rational Numbers
- Types of Rational Numbers
- How to Identify Rational Numbers?
- List of Rational Numbers in Number System
- Arithmetic Operations on Rational Numbers
- Addition of Rational Numbers
- Subtraction of Rational Numbers
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Equivalent Rational Numbers
- Decimal Expansion of Rational Numbers
- Multiplicative Inverse of a Rational Number
- Rational Numbers Properties
- Find Rational Numbers between Two Rational Numbers?
- Representing Rational Numbers on Real Line
- Rational and Irrational Numbers