In our daily routine we meet such situations that we count objects like:
How many people are there in crowd?
How many students are there in this school?
How many books are there in the library?
How many cakes are in the box?
How many eggs are there in the crate?
To answer these questions we associate different objects with numbers for example single object with number 1 two objects with numbers 2 and so on.It means we denote them by different symbols as below.
1,2,3,4,5,6,7,8,9,......
These symbols are called numerals.
Natural numbers: قدرتی اعداد
1,2,3,4,5,6,.... These are numbers that we use for counting the different objects so these are called counting numbers or natural numbers. We note that the natural numbers are starting form 1 so we said that the smallest natural number is 1. If we think which is the largest natural number then we have no answer because the largest natural number does not exist.
Example : مثالیں
If a person think that the largest natural number is 100 ( suppose )
And if we add 1 in 100 then it will be the next natural number. If we suppose the largest natural number is 1000000000000......... and we add 1 in it then it will be the next natural number. So no one tell or judge the largest natural number.
Whole Numbers: مکمل اعداد
In the starting of natural numbers if we add 0 before 1 then these numbers are known as whole numbers. The set of whole Numbers is denoted by W.
0,1,2,3,4,5,6,7,8,9,........
The smallest number or we say that the whole numbers are starting with 0 and we cannot tell the largest number of whole Numbers like natural numbers.
Representing Whole Numbers on Number Line:
نمبر لائین پر مکمل اعداد کا اظہار
We draw a line and Mark a point with 1st whole number which is 0.
On this line we Mark points at equal distance and each point we write 1,2,3,4,5,6,7,8,9,... respectively.
Properties of Whole Numbers:
مکمل اعداد کی خصوصیات
1- There is no whole number on left side of zero (0).
2- zero (0) is the smallest whole number.
3- Each number on number line is successor of its previous number, mean each number is 1 more than its previous number or we say that each number is predecessor of its next number.
4- Each whole number on number line is greater than the each whole number on its left,
e.g 6 is greater than 2, 5 is greater than 3, and 8 is greater than 0.
Addition of Whole Numbers:
مکمل اعداد کی جمع
When we add whole numbers we write them in vertical column and each digit is place according to its place value.
Example
Subtraction of Whole Numbers:
مکمل اعداد کی تفریق
When we subtract whole numbers we write them in vertical column and each digit is place according to its place value.
Examples:
Laws of Addition of Whole numbers:
Commutative law: قانون مبادلہ
Suppose two whole numbers x and y if we add them like x+y or y+x the result remains unchanged. Means the result x+y = y+x
Example: 7+5 = 12 if we change their order like 5+7 = 12 so the result is not changed by changing their order.
Associative Law : قانون تلازم
If we add three whole numbers 1st of all we add two numbers and the result is added to 3rd whole number. For example there are three numbers 4,5 and 7
(4+5)+7 = 4+(5+7)
9+7 = 4+12
16 = 16
So (4+5)+7 = 4+(5+7) , This law is called associative law of Addition.
Note:
Commutative and associative laws do not hold in subtraction and division.
These laws only hold in addition and multiplicative.
Additive Identity: جمعی ذاتی عنصر
If we observe the following examples
1+0 = 1 and 9+0 = 9 and 7+0 = 7
We note that the sum of zero and a whole number is always the whole number itself so zero (0) is known as the additive Identity.
Laws of Multiplication
Commutative law :
Suppose two whole numbers x,y then there multiplication is x×y = y×x this is called Commutative law w.r t multiplication.
Example:
If there are two whole numbers 4 ,5 then
4×5 =20 if we change their order then 5×4 = 20 , Hence the result is same after changing their order. So 4×5= 5×4
Associative law:
Suppose three whole numbers like 3,5, and 7 then according to associative law
( 3×5)×7 = 3×(5×7)
15×7 = 3×35
105 = 105
In associative law 1st we multiply two numbers and then their result is multiply by 3rd number.
Distributive law of multiplication over addition:
ضرب کا قانون تقسیمی بلحاظ جمع
Consider three whole numbers 3,4,5 then the distributive law of multiplication over addition of Whole numbers as
3×(4+5) = (3×4) + ( 3×5)
3×9 = 12 + 15
27 = 27 L.H.S = R.H.S
Distributive law of multiplication over subtraction:
ضرب کا قانون تقسیمی بلحاظ تفریق
Consider three whole numbers 3,4,5 then the distributive law of multiplication over subtraction of Whole numbers as
3×(4-5) = (3×4) - ( 3×5)
3×(-1) = 12 - 15
-3 = -3 L.H.S = R.H.S
Multiplicative Identity:
ضربی ذاتی عنصر
Observing the following examples
1×4 = 4 or 7× 1 = 7 we note that the product of any whole number with 1 is always the whole number itself. So 1 is called multiplicative Identity.
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