# 6th Class Mathematics Whole Numbers

Whole Numbers

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.

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##### Notes - Whole Numbers

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