Contents
Operations on Whole Numbers:
We can perform four basic operations on whole numbers. Addition, Subtraction, Multiplication, Division.
Let us see the properties of these operations on whole numbers.
Addition:
Example: Add
1) 59, 21
2) 118, 122
Solution:
Properties of Addition on Whole Numbers:
Property 1: (Closure Property)
The sum of two whole numbers is always a whole number.
That is, for any two whole numbers a and b, a + b is always a whole number.
Verification: We have,
1) 18 + 7 = 25 is a whole number
2) 23 + 12 = 35 and 35 is a whole number.
Property 2: (Commutative law)
The addition of whole numbers is commutative, i.e, for any two whole numbers a and b, we have a + b = b + a
Verification: We have,
15 + 26 = 41 and 26 + 15 = 41
Therefore, 15 + 26 = 26 + 15.
Property 3: (Associative Law of Addition)
If a, b, c are any three whole numbers then (a + b) + c = a + (b + c).
Verification: Consider the Integers 6, 1 and 3.
We have,
(6 + 1) + 3 = (7) + 3 = 10.
And, 6 + (1 + 3) = 6 + (4) = 10.
Therefore, (6 + 1) + 3 = 6 + (1 + 3)
Property 4: (Additive Property of Zero)
If a is any whole number then a + 0 = a and 0 + a = a.
Verification: We have,
(i) 8 + 0 = 8
(ii) 235 + 0 = 235
While adding 3 or more whole numbers, we group them in such a way that the calculation becomes easier. We arrange them suitably and add.
Example: Add
1) 62 + 70 + 8
2) 43 + 62 + 17 + 38
Solution:
Subtraction:
Subtraction and Addition are inverse operations. For every addition there are two subtraction facts.
For example,
2 + 8 = 10 is an addition fact. From this, we can state two related subtraction facts. They are,
10 – 8 = 2 and 10 – 2 = 8.
Properties of Subtraction on Whole Numbers:
Property 1:
If a and b are 2 whole numbers such that a > b or a = b then a – b is a whole number;otherwise, subtraction is not possible in whole numbers.
Verification:
1) If we subtract 2 equal whole numbers, we get the whole number 0.
(9 – 9 ) = 0, (3 – 3) = 0
2) If we subtract a smaller whole number from a larger one, we always get a whole number.
(23 – 5) = 18 is a whole number
(54 – 20) = 34 is a whole number
3)Clearly, we cannot subtract larger whole number from a smaller whole number.
(20 – 54) is not defined in whole numbers.
Property 2:
For any 2 whole numbers a and b, (a – b) is not equal to (b – a)
Verification:
1) (9 – 4) = 5 but (4 – 9) is not defined in whole numbers.
2) (23 – 2) = 21 but (2 – 23) is not defined in whole numbers.
Property 3:
For any whole number a, we have: (a – 0) = a but (0 – a) is not defined in whole numbers.
Verification:
1) (23 – 0) = 23 but (0 – 23) is not defined in whole numbers.
1) (27 – 0) = 23 but (0 – 27) is not defined in whole numbers.
Property 4:
If a, b, c are any 3 whole numbers, then in general (a – b) – c is not equal to a – (b – c).
Verification:
Consider the numbers 14, 11 and 2
(14 – 11) – 2 = (3) – 2 = 1
14 – (11 – 2) = 14 – (9) = 5
(14 – 11) – 2 is not equal to 14 – (11 – 2)
Property 5:
If a, b, c are any 3 whole numbers such that a – b = c, then b + c = a.
Verification:
16 – 7 = 9, 7 + 9 = 16
Multiplication:
Multiplication is a short form of addition of the same number several times.
For example,
7 + 7 + 7 + 7 = four times 7
= 4 x 7
= 28
4 x 7 means four times seven or seven times four
In multiplication, if we use some simple and easy techniques, we can get the products easily and quickly.
Example:
Multiply 283 by 101
Solution:
Properties of Multiplication on Whole Numbers:
Property 1: (Closure Property)
The product of two whole numbers is always a whole number.
Verification: We have,
1) 16 x 5 = 80, 80 is a whole number
2) 12 x 7 = 84, 84 is a whole number
Property 2: (Commutative Law for Multiplication)
For any two whole numbers a and b, we have a x b = b x a.
Verification: We have,
40 x 5 = 200 and 5 x 40 = 200
Therefore, 40 x 5 = 5 x 40.
Property 3: (Associative Law for Multiplication)
If a, b, c are any three whole numbers then (a x b) x c = a x (b x c).
Verification: Consider the whole numbers 7, 12 and 4.
We have,
(7 x 12) x 4 = (84) x 4 = 336.
And, 7 x (12 x 4) = 7 x (48) = 336.
Therefore, (7 x 12) x 4 = 7 x (12 x 4)
Property 4: (Distributive Law for Multiplication over Addition)
If a, b, c are any three whole numbers then a x (b + c) = (a x b) + (a x c).
Verification: Consider the whole numbers 2, 12 and 19.
We have,
2 x (12 + 19) = 2 x (31) = 62.
And, 2 x 12 + 2 x 19 = 24 + 38 = 62.
Therefore, 2 x (12 + 19) = 2 x 12 + 2 x 19
Property 5: (Distributive Law for Multiplication over Subtraction)
If a, b, c are any three whole numbers then a x (b – c) = (a x b) – (a x c).
Verification: Consider the whole numbers 7, 10 and 4.
We have,
7 x (10 – 4) = 7 x 6 = 42.
And, 7 x 10 – 7 x 4 = 70 – 28 = 42.
Therefore, 7 x (10 – 4) = 7 x 10 – 7 x 4 = 70 – 28
Property 6: (Multiplicative Property of 1)
For any whole number a, we have a x 1 = 1 x a = a.
Verification: We have,
1) 11 x 1 = 1 x 11 = 11
2) 5 x 1 = 1 x 5 = 5
Property 7: (Multiplicative Property of 0)
For any whole number a, we have a x 0 = 0 x a = 0.
Verification: We have,
1) 7 x 0 = 0
2) 5 x 0 = 0
Division:
Division is the inverse operation of multiplication.
Let a and b be 2 whole numbers. Dividing a by b means finding a whole number c such that b x c = a and we write \(\frac{a}{b}\) = c.
Thus, a/b = c that implies a = b x c.
Example:
Dividing 35 by 7 is the same as finding a whole number which when multiplied by 7 gives 35.
Clearly, such a number is 5, as 7 x 5 = 35.
Let a and b be two given whole numbers such that a > b. On dividing a by b, let q be the quotient and r be the remainder.
Then we have: a = bq + r, where 0 <= r < b
This result is known as division algorithm.
Thus, dividend = (divisor x quotient) + remainder.
Properties of Division on Whole Numbers:
Property 1:
If a and b are nonzero whole numbers, then \(\frac{a}{b}\) is not always a whole number.
Verification: We have,
9 and 2 are two whole numbers, but \(\frac{9}{2}\) is not a whole number.
Property 2: (Division by Zero)
If a is a whole number, then \(\frac{a}{0}\) is meaningless.
Property 3:
If a is a nonzero whole number, then \(\frac{0}{a}\) = 0.