6th whole numbers

 

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2.Whole Numbers

The numbers 1,2, 3, are called natural numbers or counting numbers.
 
Let us add one more number i.e., zero (0), to the collection of natural numbers. Now the numbers are 0,1,2, … These numbers are called whole numbers

We can say that whole nos. consist of zero and the natural numbers. Therefore, except zero all the whole nos. are natural numbers.

Facts of Whole numbers

1) The smallest natural number is 1.

2) The number 0 is the first and the smallest whole nos.

3) There are infinitely many or uncountable number of whole-numbers.

4) All natural numbers are whole-numbers.

5) All whole-numbers are not natural numbers. For example, 0 is a whole-number but it is not a natural number.

The first 50 whole nos. are

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,14, 15, 16, 17, 18, 19, 20, 21,

22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,

41, 42, 43, 44, 45, 46, 47, 48, 49, 50
Some other Important terms to remember

SUCCESSOR	The successor of a whole number is the number obtained by adding 1 to it. Clearly, the successor of 1 is 2; successor of 2 is 3; successor of 3 is 4 and so on.
PREDECESSOR	The predecessor of a whole number is one less than the given number. Clearly, the predecessor of 1 is 0; predecessor of 2 is 1; predecessor of 3 is 2 and so on. The whole number 0 does not have any predecessor.

Example
1) Write the successor of
(a) $10701$ (b) $100499$         (c) $5099999$         (d) $5670$

2) Write the predecessor of
(a) $14$                 (b) $100000$ (c) $8090$                 (d) $4321$
Solution
1)
a) $10702$
b) $100500$
c) $5100000$
d) $5671$
2)
a) $12$
b) $99999$
c) $8089$

Properties of Whole Numbers

Closure Property

Closure property on Addition for Whole Number

$0+2=2$
$1+3=4$
$5+6=11$

So Whole number are closed on Addition
Closure property on Multiplication for Whole Number

$0\times 2=0$
$1\times 4=4$
$5\times 1=5$

So Whole number are closed on Multiplication
Closure property on subtraction of Whole number

$5-0=5$
$0-5=?$
$1-3=?$
$3-1=2$

So Whole number are not closed on Subtraction
Closure property on Division of Whole number

$\frac{2}{1}=2$
$\frac{1}{2}=?$
$\frac{0}{2}=0$
$\frac{2}{0}=?$ ( Division by Zero is undefined)

So Whole Number are not closed on Division
In short

Closure Property

If a and b are any two whole numbers, then a+b, axb are also whole numbers.

Commutative property

Commutativity property on Addition for Whole Number

$0+2=2+0=2$

So Whole number are Commutative on Addition
Commutativity property on Multiplication for Whole Number

$0\times 2=0or2\times 0=0$

So Whole number are Commutative on Multiplication
Commutativity property on subtraction of Whole number

$5-0=5$ but $0-5=?$

So Whole number are not Commutative on Subtraction
Commutativity property on Division of Whole number

$\frac{2}{1}=2$   but $\frac{1}{2}=?$

So Whole Number are not Commutative on Division
In short
You can add two whole numbers in any order. You can multiply two whole numbers in
any order.

Commutative property

If a and b are any two whole numbers, then $a+b=b+a$ and $a\times b=b\times a$

Associative property

Associativity property on Addition for Whole Number

$0+(2+3)=(0+2)+3=5$
$1+(2+3)=6=(1+2)+3$

So Whole number are Associative on Addition
Associativity property on Multiplication for Whole Number

$0\times (2\times 3)=0$ or $(0\times 2)\times 3=0$

So Whole number are Associative on Multiplication
Associativity property on subtraction of Whole number

$10-(2-1)=9$ but $(10-2)-1=7$

So Whole number are not Associative on Subtraction
Associativity property on Division of Whole number

$16÷(4÷2)=8$ but $(16÷4)÷2=2$

So Whole Number are not Associative on Division
So in Short
If a, b and c are any two whole numbers, then (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c).

Distributive property

If a, b and c are any two whole numbers, then $a(b+c)=a\times b+a\times c$

Additive Identity

If a is any whole number, then $a+0=a=0+a$.

Example
$2+0=2$
$0+3=3$
$5+0=5$

Multiplicative Identity

If a is any whole number, then $a\times 1=a=1\times a$

Example
$1\times 1=1$
$5\times 1=5$
$6\times 1=6$

 

Multiplication by zero

If a is any whole number, then $a\times 0=0=0\times a$.

Example
$1\times 0=0$
$5\times 0=0$
$0\times 0=0$