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Category : 6th Class

 Whole Numbers

• Counting numbers are called natural numbers. Natural numbers are represented by N

\[\therefore \]N\[=\]{1,2,3,4,.........}

The least natural number is 1.

The greatest natural number does not exist as there is no end for natural numbers.

• Natural numbers along with 0 are called whole numbers.

Whole numbers are represented by W.

\[\therefore \]W= {0, 1, 2, 3, 4,……}

The least whole number is 0.

The greatest whole number does not exist as there is no end for whole numbers.

• Successor: A number which comes immediately after a given number is called its successor.

Successor = given number + 1

• Predecessor: A number which comes immediately before a given number is called its

Predecessor= given number – 1

• Number line: A line on which we locate numbers at equal intervals.

• Properties of Addition of Whole Numbers:

(i) Closure property: If 'a' and 'b' are any two whole numbers, then a + b is also a whole number.

e.g., 5 and 13 are whole numbers. Their sum 5 + 13 = 18 is also a whole number.

Hence whole numbers are closed under addition.

(ii) Commutative property: If 'a' and 'b' are any two whole numbers, then a + b = b + a. The order of addition of two whole numbers does not affect their sum.

e.g., 3+11 =14=11+3

Hence whole numbers satisfy the commutative property under addition.

(iii) Associative property: If 'a', 'b' and 'c' are any three whole numbers, then

(a + b) + c = a + (b + c) = (a + c) + b

e.g., (2 + 3) + 4 = 9 = 2 + (3 + 4) = (2 + 4) + 3

Hence whole numbers satisfy the associative property under addition.

(iv) Additive identity: If 'a' is any whole number then a+0=0+a=a. Since the sum is the same as the number considered, 0 is called the identity element for addition of whole numbers.

e.g., 9+0=0+9=9

Note: For subtraction of whole numbers closure property. Commutative. Property, identity and associative properties are not applicable.

• Properties of Subtraction of Whole Numbers:

(i) If 'a' and 'b' are two whole numbers such that a > b (or) a = b then a - b is a whole number. If a < b, then a - b is not a whole number.

(ii) If 'a' and 'b' are two whole numbers such that a\[\ne \] b, then a - b\[\ne \] b - a.

(iii) If 'a' is any whole number, then a - 0 = a, but 0 - a  is not a whole number.

(iv) If 'a', 'b' and 'c' are three whole numbers such that a\[\ne \] b \[\ne \]c, then (a - b) - c is not equal to a - (b - c).

Note: For subtraction of whole numbers closure property, commutative property, identity and associative properties are not applicable.

• Properties of Multiplication of Whole Numbers:

(i) Closure property: If 'a' and 'b' are any two whole numbers, then a x b is also a whole number.

e.g.. For whole numbers 3 and 5, their product 3 \[\times \] 5=15 is also a whole number.

Hence whole numbers are closed under multiplication.

(ii)   Commutative property: If 'a' and 'b' are any two whole numbers, then a \[\times \] b = b \[\times \] a.

The order of multiplication of two whole numbers does not affect their product. Hence whole numbers satisfy the commutative property under multiplication.

e.g., 4\[\times \]12 = 48 =12 \[\times \]4

(iii) Associative property: If 'a', 'b' and 'c' are any three whole numbers, then

(a \[\times \]b)\[\times \]c=a\[\times \](b\[\times \]c) = (a\[\times \]c)\[\times \]b.

e.g., (3\[\times \]4)\[\times \]5 = 60 = 3\[\times \](4\[\times \]5) = (3\[\times \]5)\[\times \]4

Hence whole numbers satisfy the associative property under multiplication.

(iv) Multiplicative identity: If 'a' is any whole number, then a\[\times \]1 =1\[\times \]a = a.

Since the product is the same as the number considered, 1 is called the identity element for multiplication of whole numbers.

e.g., 7\[\times \]1 =1\[\times \]7 =7

(v) Multiplicative property of 0: If 'a' is any whole number, then a \[\times \] Q= O\[\times \] a = 0.

• Distributive law of multiplication over addition: If 'a', 'b' and c' are any three whole numbers, then ax (b + c) = (ax b) + (ax c). Thus, multiplication is distributed over addition of whole numbers.

e.g., 5\[\times \](6 + 7) = (5\[\times \]6) + (5\[\times \]7) = 65

• Distributive law of multiplication over subtraction: If 'a', 'b' and 'c' are whole numbers, then a \[\times \] (b - c)\[=\] (a \[\times \] b) - (a \[\times \] c). Thus, multiplication is distributed over subtraction of whole numbers.

e.g., 2\[\times \](4 - 3)\[=\] (2\[\times \]4) - (2\[\times \]3) \[=\] 2

• Properties of Division of Whole Numbers:

(i) If 'a' and 'b' (non-zero) are whole numbers, then a\[\div \]b is not always a whole number

(ii) If \['a'\] is the dividend, \['b'\](Where b \[\ne \]0) is divisor, \['q'\] is the quotient and \['r'\]is the remainder. Then a\[=\] bq +r.

Note: (i) Division by zero is not defined.

(ii) Zero divided by any whole number is always zero.

• If the product of two whole numbers is Zero. Then either of the two numbers of both the numbers are Zero.

• In other words, If \['a'\] and \['b'\]are any two whole numbers and a\[\times \]b\[=\]0, then either a =0 or b=0 or both a \[=\] 0 and b \[=\] 0.