KAPDEC® | Elite STEM Learning Platform | https://kapdec.com
Unit: Number System
Chapter: Properties of Whole Numbers
Reference: – Closure Property, Commutative Property, Associative Property, Distributive Property, Identity Elements (Additive Identity and Multiplicative Identity), Zero Property, Properties of Subtraction and Division
After studying this chapter, you should be able to understand:
- Closure Property of Whole Numbers
- Commutative Property of Addition and Multiplication
- Associative Property of Addition and Multiplication
- Distributive Property of Multiplication over Addition and Subtraction
- Additive Identity (0) and Multiplicative Identity (1)
- Properties That Do Not Hold for Subtraction and Division
Properties of Whole Numbers
- Closure Property
Whole numbers are closed under addition and multiplication. Whole numbers are not closed under subtraction and division. By closure we mean-
- For addition – If a and b are any two whole numbers, then (a + b) is also a whole number.
Example- 2 and 3 are two whole numbers. So, 2 + 3 i.e. 5 is also a whole number.
- For multiplication – If a and b are any two whole numbers, then (a x b) is also a whole number.
Example- 2 and 3 are two whole numbers. So, 2 × 3 i.e. 6 is also a whole number.
- For subtraction – If a and b are any two whole numbers, then (a – b) is not necessarily a whole number.
Example- 2 and 3 are two whole numbers. So, 3 – 2 i.e. 1 is a whole number but 2 – 3 i.e. -1 is an integer but not a whole number.
- For division – If a and b are any two whole numbers, then (a – b) is not necessarily a whole number.
Example- 2 and 4 are two whole numbers. So, 4 ÷ 2 i.e. 2 is a whole number but 2 ÷ 4 i.e. 12
Source: Kapdec.com
is a fraction and not a whole number.
- Commutative Property
- Commutative Property of Addition- The sum of two whole numbers is the same, no matter in which order they are added.
Example: 19 + 22 = 41
22+ 19 = 41
So, 19 + 22 = 22+ 19
- Commutative Property of Multiplication- The product of two whole numbers is the same, no matter in which order they are multiplied.
Example: 18 × 9 = 162
9 × 18 = 162
So, 18 × 9 = 9 × 18
- Subtraction and Division are not commutative in whole numbers.
Examples:
- 19 – 12 = 7
12- 19 = -7
So, 19 – 12≠12- 19
- 18 ÷ 3 = 6
3÷ 18 = 16
Source: Kapdec.com
So, 18 ÷ 3 ≠ 3 ÷ 18
- Additive Identity
A whole number when added to ‘0’ remains the same. Thus, ‘0’ is called the additive identity in whole numbers. If a is any whole number, then a + 0 = a = 0 + a.
Example-3 + 0 = 3 = 0 + 3
- Multiplicative Identity
A whole number when multiplied by 1 remains the same. Thus, 1 is called the multiplicative identity in whole numbers. If a is any whole number, then a × 1 = a = 1 × a.
Example-3 × 1 = 3 = 1 × 3
- Associative property
Associative Property of Addition- While adding whole numbers, we can group the numbers in any order, the sum remains the same.
If a, b and c are any two whole numbers, then (a + b) + c = a + (b + c).
Example- 22+ 8 +16 = (22+ 8) + 16 = 30+ 16 = 46
22+ 8 +16 = 22+(8 + 16) = 22+24 = 46
So, (22 + 8) + 16 = 22 + (8 + 16) = 46
Associative property of multiplication- While multiplying whole numbers, we can group them in any order, the product remains the same.
If a, b and c are any two whole numbers, then (a × b) × c = a × (b × c).
Example-
60 × 8 × 100 = (60 × 8) × 100 = 480 × 100 = 48000
60 × 8 × 100 = 60 × (8 × 100) = 600 × 800 = 48000
60 × 8 × 100 = (60 × 100) × 8 = 6000 × 8 = 48000
- Distributive Property
Distributive Property of Multiplication over Addition- The product of a whole number with the sum of the two other whole numbers is equal to the sum of the products of the whole number with other two whole numbers.
If a, b and c are any two whole numbers, then a (b + c) = a × b + a × c.
Example- 24 X 125 = 24 X (100 + 20 + 5)
= 24 X 100 + 24 X 20 + 24 X 5
= 2400 + 480 + 120
= 3000
Distributive Property of Multiplication over Subtraction- The product of a whole number with the difference of the two other whole numbers is equal to the difference of the products of the whole number with other two whole number.
If a, b and c are any two whole numbers, then a(b–c) = a×b – a×c.
Example- 18 X 98 = 18 X (100 – 2)
= (18 X 100) – (18 X 2)
= 1800-36
=1764
- Multiplication by Zero
Product of a whole number by zero is equal to zero.
If a is any whole number, then a × 0 = 0 = 0 × a.
Example-5 × 0 = 0
- Properties of Division
- Division by 0: Division by 0 is not possible.
If a is any whole number, then a ÷ 0 is not defined.
Example: 51,841 ÷ 0 = Not Possible
- Division by 1: If a number is divided by 1, the result is the number itself.
If a is any whole number, then a ÷ 1 = a.
Example: 8,59,283 ÷ 1 = 8,59,283
- Division by number itself: If a number is divided by itself, the result is always 1.
If a is any non-zero whole number, then a ÷ a = 1.
Example: 4,55,993 ÷ 4,55,993 = 1
- Dividing 0: If 0 is divided by any number (except 0), the result is always 0.
If a is any non-zero whole number, then 0 ÷ a = 0.
Example: 0 ÷ 4,93,661 = 0
- Division Algorithm-
Dividend = Divisor X Quotient + Remainder
This equation is used to check division.
Example- 725 ÷ 16
16) 725 (45
-64
085
– 80
05
Divisor = 16
Quotient = 45
Remainder = 5
So,
Divisor X Quotient + Remainder
= 16 X 45 + 5 = 720 + 5 = 725 = Dividend
