Properties of Whole Numbers
Closure property Whole numbers are closed under addition and multiplication. Whole numbers are not closed under subtraction and division. If a and b are any two whole numbers, then a+b, axb are also whole numbers. Commutative property The sum of two whole numbers is the same, no matter in which order they are added. This is called the commutative property of addition. The product of two whole numbers is the same, no matter in which order they are multiplied. This is called the commutative property of multiplication. Subtraction and division are not commutative in whole numbers. If a and b are any two whole numbers, then a+b = b+a and a×b = b×a. Additive Identity A whole number added to '0' remains unchanged. Thus, '0' is called the additive identity in whole numbers. If a is any whole number, then a + 0 = a = 0 + a. Multiplicative Identity A whole number multiplied by 1 remains unchanged. Thus, 1 is called the multiplicative identity in whole numbers. If a is any whole number, then a × 1 = a = 1 × a. Associative property While adding whole numbers, we can group the numbers in any order. This is called the associative property of addition. While multiplying whole numbers, we can group them in any order. This is called the associative property of multiplication. 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 The product of a whole number with the sum of the two other whole numbers is equal to the sum of the products of the whole number with other two whole numbers. This is called the distributive property of multiplication over addition. If a, b and c are any two whole numbers, then a(b+c) = a×b + a×c. The product of a whole number with the difference of the two other whole numbers is equal to the difference of the products of the whole number with other two whole numbers. This is called the distributive property of multiplication over subtraction. If a, b and c are any two whole numbers, then a(b–c) = a×b – a×c. Multiplication by zero Product of a whole number by zero is equal to zero. If a is any whole number, then a × 0 = 0 = 0 × a. Division by zero Division of a whole number by 0 is not defined. If a is any whole number, then a ÷ 0 is not defined. If a is any whole number, then a ÷ 1 = a. If a is any non-zero whole number, then a ÷ a = 1. If a is any non-zero whole number, then 0 ÷ a = 0.

#### Summary

Closure property Whole numbers are closed under addition and multiplication. Whole numbers are not closed under subtraction and division. If a and b are any two whole numbers, then a+b, axb are also whole numbers. Commutative property The sum of two whole numbers is the same, no matter in which order they are added. This is called the commutative property of addition. The product of two whole numbers is the same, no matter in which order they are multiplied. This is called the commutative property of multiplication. Subtraction and division are not commutative in whole numbers. If a and b are any two whole numbers, then a+b = b+a and a×b = b×a. Additive Identity A whole number added to '0' remains unchanged. Thus, '0' is called the additive identity in whole numbers. If a is any whole number, then a + 0 = a = 0 + a. Multiplicative Identity A whole number multiplied by 1 remains unchanged. Thus, 1 is called the multiplicative identity in whole numbers. If a is any whole number, then a × 1 = a = 1 × a. Associative property While adding whole numbers, we can group the numbers in any order. This is called the associative property of addition. While multiplying whole numbers, we can group them in any order. This is called the associative property of multiplication. 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 The product of a whole number with the sum of the two other whole numbers is equal to the sum of the products of the whole number with other two whole numbers. This is called the distributive property of multiplication over addition. If a, b and c are any two whole numbers, then a(b+c) = a×b + a×c. The product of a whole number with the difference of the two other whole numbers is equal to the difference of the products of the whole number with other two whole numbers. This is called the distributive property of multiplication over subtraction. If a, b and c are any two whole numbers, then a(b–c) = a×b – a×c. Multiplication by zero Product of a whole number by zero is equal to zero. If a is any whole number, then a × 0 = 0 = 0 × a. Division by zero Division of a whole number by 0 is not defined. If a is any whole number, then a ÷ 0 is not defined. If a is any whole number, then a ÷ 1 = a. If a is any non-zero whole number, then a ÷ a = 1. If a is any non-zero whole number, then 0 ÷ a = 0.

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