WHOLE NUMBERS - LINKS; CLOSURE, COMMUTATIVITY, ASSOCIATIVITY ETC. PROPERTIES

Please study
Number Systems before Whole Numbers,
if you have not already done so.

There, we covered the base 10 system
called the Hindu-Arabic Number system.

We introduce the digits used and explain the place
value charts for numbers upto five digits and above.

We also explain writing the numbers in words
including in English System.

Studying of it is a part of our study here.

When we begin to count, 1, 2, 3, 4, ...... come naturally to us.
Hence mathematicians call the counting numbers as Natural Numbers.

Number 1 has no predecessor in Natural Numbers.
To the collection of Natural Numbers,
we add zero as the predecessor for 1.

The Natural Numbers along with zero
form the collection of whole numbers.

The set of whole numbers is denoted by W.
Hence, W = { 0, 1, 2, 3, .....}.

Observe that each element in the set W, has a successor
and the smallest element is zero.

The set W is not finite.

For the four basic operations on whole numbers, see

(i) Addition

(ii) Subtraction

(iii) Multiplication  including
Multiplication Tables

(iv) Division  including
Long Division.

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All Natural Numbers are in the set of W.

The set of W has only one extra element '0'
compared to the set of Natural Numbers.

Thus, the following topics are also a part of the study of
W. They include

Elementary Number Theory



Prime Numbers  including
List of PrimeNumbers

Perfect Numbers

Divisibility Rules

Prime Factorization  including
Fundamental Theorem of Arithmetic

Greatest Common Factor

Least Common Multiple.

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Properties of Whole Numbers

Please note the following standard symbols.

a ∈ W means a is an element of the set, W.
a ∉ W means a is not in the set, W.

Closure Property of Addition :

Observe the following Examples :

3 + 7 = 10 ∈ W; 0 + 23 = 23 ∈ W;
1827 + 2345 = 4172 ∈ W; 834 + 16 = 850 ∈ W;

What is seen to be true for in these examples,
is true for any two elements of W. So, we can say :

For any two whole numbers a and b, (a + b) is also a whole number.
This is called the Closure Property of Addition for the set of W.

Commutative Property of Addition :

Look at the following Examples.

12 + 18 = 18 + 12 = 30; 345 + 65 = 65 + 345 = 410

What is seen to be true for in these examples,
is true for any two elements of W. So, we can say :

For any two elements of the set W, a and b, a + b = b + a
This is called the Commutative Property of Addition for the set of W.

Associative Property of Addition :

Look at the following Examples.

(15 + 27) + 18 = 15 + (27 + 18) = 60;
(3 + 6) + 1 = 3 + (6 + 1) = 10

What is seen to be true for in these examples,
is true for any three elements of W. So, we can say :

For any three whole numbers a, b and c, a + (b + c) = (a + b) + c
This is called the Associative Property of Addition for the set of W.

Additive Identity :

Look at the following Examples.

19 + 0 = 0 + 19 = 19 ; 1345 + 0 = 0 + 1345 = 1345

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For every whole number a, a + 0 = 0 + a = a
0 is called the Additive Identity in the set of W.

Observe the following Examples :

30 x 7 = 210 ∈ W; 2 x 23 = 46 ∈ W;

What is seen to be true for in these examples,
is true for any two elements of W. So, we can say :

For any two whole numbers a and b, (a x b) is also a whole number.
This is called the Closure Property of Multiplication for the set of W.

Commutative Property of Multiplication :

Look at the following Examples.

10 x 18 = 18 x 10 = 180; 25 x 4 = 4 x 25 = 100

What is seen to be true for in these examples,
is true for any two elements of W. So, we can say :

For any two whole numbers a and b, a x b = b x a
This is called the Commutative Property of Multiplication for the set of W.

Associative Property of Multiplication :

Look at the following Examples.

(4 x 2) x 3 = 4 x (2 x 3) = 24;
(16 x 9) x 2 = 16 x (9 x 2) = 288

What is seen to be true for in these examples,
is true for any three elements of W. So, we can say :

For any three whole numbers a, b and c, a x (b x c) = (a x b) x c
This is called the Associative Property of Multiplication for the set of W.

Multiplicative Identity :

Look at the following Examples.

14 x 1 = 1 x 14 = 14 ; 9875 x 1 = 1 x 9875 = 9875

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For every whole number a, a x 1 = 1 x a = a
1 is called the Multiplicative Identity in the set of W.

Zero Property of Multiplication :

Look at the following Examples.

234 x 0 = 0 x 234 = 0 ; 5647 x 0 = 0 x 5647 = 0

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For every whole number a, a x 0 = 0 x a = 0
This is called the Zero Property of Multiplication for the set of W.

Also,

For any two whole numbers a and b,
If a x b = 0, then either a = 0 or b = 0 or a = b = 0

Distributive Property of Multiplication over Addition :

Look at the following Examples.

5 x (4 + 3) = 5 x 4 + 5 x 3 = 20 + 15 = 35 same as 5 x 7;
2 x (7 + 8) = 2 x 7 + 2 x 8 = 14 + 16 = 30 same as 2 x 15;

What is seen to be true for in these examples,
is true for any three elements of W. So, we can say :

For any three whole numbers a, b and c, a x (b + c) = (a x b) + a x c
This is called the Distributive Property of Multiplication over Addition.

Closure Property is not true for Subtraction :

Observe the following Examples :

3 - 7 = ? ∉ W; 20 - 23 = ? ∉ W;

Thus For some elements of W,
subtraction does not give elements ∈ W. So, we can say

The Closure Property is not true for Subtraction in the set of W.

Subtraction is not Commutative :

Observe the following Examples :

7 - 4 ≠ 4 - 7; 78 - 67 ≠ 67 - 78

So, we can say Subtraction is not Commutative.

Subtraction is not Commutative in the set of W.

Subtraction is not Associative :

Observe the following Examples :

(8 - 5) - 2 ≠ 8 - (5 - 2); (78 - 67) - 11 ≠ 78 - (67 - 11)

So, we can say Subtraction is not Associative.

Subtraction is not Associative in the set of W.

Closure Property is not true for Division :

Observe the following Examples :

8 ÷ 3 = ? ∉ W; 36 ÷ 5 = ? ∉ W;

Thus For some elements of W,
division does not give elements ∈ W. So, we can say

The Closure Property is not true for Division in the set of W.

Division is not Commutative :

Observe the following Examples :

8 ÷ 4 ≠ 4 ÷ 8; 144 ÷ 9 ≠ 9 ÷ 144

So, we can say Division is not Commutative.

Division is not Commutative in the set of W.

Division is not Associative :

Observe the following Examples :

(8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2);
(54 ÷ 6) ÷ 3 ≠ 54 ÷ (6 ÷ 3)

So, we can say Division is not Associative.

Division is not Associative in the set of W.

Division with 1 of any number gives that number :

Look at the following Examples.

19 ÷ 1 = 19 ; 9776 ÷ 1 = 9776

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For every whole number a, a ÷ 1 = a

Division is inverse operation to Multiplication :

Look at the following Examples.

144 ÷ 18 = 8 and 8 x 18 = 144;
81 ÷ 3 = 27 and 27 x 3 = 81.

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For any three whole numbers a, b(≠0) and c, a ÷ b = c means a = b x c

9 ÷ 0 = ?
Is it possible to assign any whole number to this quotient ?
Suppose, if possible, 9 ÷ 0 = a where a is a whole number.
Since division is the inverse process of multiplication, we have
a x 0 = 9.
But, any whole number x 0 = 0
[see zero property of multiplication above.]
So a x 0 = 0 and not 9.
Thus there is no whole number which when
multiplied by zero gives a non zero whole number.

Therefore 9 ÷ 0 can not be equal to any whole number.

Thus division by zero is not defined.

Division by zero is not defined.

What about 0 ÷ 0 ?

Let 0 ÷ 0 = b
Then 0 = b x 0.
This is satisfied for all values of b.
So 0 ÷ 0 has innumerable Answers.
It can not be any one umber.

So, we can say

0 ÷ 0 is indeterminate.

Division of zero with any number gives zero :

Look at the following Examples.

0 ÷ 34 = 34 ; 0 ÷ 5617 = 0

What is seen to be true for in these examples,
is true for any element of W. So, we can say :

For every whole number a, 0 ÷ a = 0

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