40.
"The sum of any two whole numbers
is a whole number."
This property of whole numbers is
referred to as
commutative property
(2
associative property
о
(3)
distributive property
(4)
closure property
Answers
Answer with full explanation:
The whole numbers are the part of the number system in which it includes all the positive integers from 0 to infinity. These numbers exist in the number line. Hence, they are all real numbers. We can say, all the whole numbers are real numbers, but not all the real numbers are whole numbers.
The complete set of natural numbers along with ‘0’ are called whole numbers. The examples are: 0, 11, 25, 36, 999, 1200, etc.
Whole Numbers Definition
The whole numbers are the numbers without fractions and it is a collection of positive integers and zero. It is represented by the symbol “W” and the set of numbers are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,……………}. Zero as a whole represents nothing or a null value.
Whole Numbers: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10……}
Natural Numbers: N = {1, 2, 3, 4, 5, 6, 7, 8, 9,…}
Integers: Z = {….-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…}
Counting Numbers: {1, 2, 3, 4, 5, 6, 7,….}
These numbers are positive integers including zero and do not include fractional or decimal parts (3/4, 2.2 and 5.3 are not whole numbers). Addition, Subtraction, Multiplication and Division operations are possible on whole numbers.
Symbol
The symbol to represent whole numbers is the alphabet ‘W’ in capital letter.
W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…
Facts:
All the natural numbers are whole numbers
All counting numbers are whole numbers
All positive integers including zero are whole numbers
All whole numbers are real numbers
Whole Numbers Properties
The properties of whole numbers are based on arithmetic operations such as addition, subtraction, division and multiplication. Two whole numbers if added or multiplied will give a whole number itself. Subtraction of two whole numbers may not result in whole numbers, i.e. it can be an integer too. Also, division of two whole numbers results in getting a fraction in some cases. Now let us see some more properties here;
Closure Property
They can be closed under addition and multiplication, i.e., if x and y are two whole numbers then x. y or x + y is also a whole number.
Commutative Property of Addition and Multiplication
The sum and product of two whole numbers will be the same whatever the order they are added or multiplied in, i.e., if x and y are two whole numbers, then x + y = y + x and x . y = y . x
Additive identity
When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x + 0 = 0 + x = x
Multiplicative identity
When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.1 = x = 1.x
Associative Property
When whole numbers are being added or multiplied as a set, they can be grouped in any order, and the result will be the same, i.e. if x, y and z are whole numbers then x + (y + z) = (x + y) + z and x. (y.z)=(x.y).z
Distributive Property
If x, y and z are three whole numbers, the distributive property of multiplication over addition is x. (y + z) = (x.y) + (x.z), similarly, the distributive property of multiplication over subtraction is x. (y – z) = (x.y) – (x.z)
Multiplication by zero
When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0
Division by zero
Division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.
Can Whole Numbers be negative?
The whole number can’t be negative!
As per definition: {0, 1, 2, 3, 4, 5, 6, 7,……till positive infinity} are whole numbers. There is no place for negative numbers.
Is 0 a whole number?
Whole numbers are the set of all the natural numbers including zero. So yes, 0 (zero) is not only a whole number but the first whole number.
Solved Examples
Example 1: Are 100, 227, 198, 4321 whole numbers?
Solution: Yes. 100, 227, 198, 4321 are all whole numbers.
Example 2: Solve 10 × (5 + 10) using the distributive property.
Solution: Distributive property of multiplication over the addition of whole numbers is:
x × (y + z) = (x × y) + (x × z)
10 × (5 + 10) = (10 × 5) + (10 × 10)
= 50 + 100
= 150
Therefore, 10 × (5 + 10) = 150