{"id":10902,"date":"2026-11-02T02:04:39","date_gmt":"2026-11-02T02:04:39","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10902"},"modified":"2026-11-02T02:04:39","modified_gmt":"2026-11-02T02:04:39","slug":"properties-of-whole-numbers","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/properties-of-whole-numbers\/","title":{"rendered":"Properties Of Whole Numbers"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit<\/strong>: Number System<\/h2>\n

Chapter<\/strong>: Properties of Whole Numbers<\/h3>\n

Reference<\/strong>: \u2013 Closure Property, Commutative Property, Associative Property, Distributive Property, Identity Elements (Additive Identity and Multiplicative Identity), Zero Property, Properties of Subtraction and Division<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/p>\n

    \n
  • Closure Property of Whole Numbers<\/li>\n
  • Commutative Property of Addition and Multiplication<\/li>\n
  • Associative Property of Addition and Multiplication<\/li>\n
  • Distributive Property of Multiplication over Addition and Subtraction<\/li>\n
  • Additive Identity (0) and Multiplicative Identity (1)<\/li>\n
  • Properties That Do Not Hold for Subtraction and Division\n

    \tProperties of Whole Numbers<\/strong><\/li>\n<\/ul>\n

    \u00a0<\/p>\n

      \n
    • Closure Property<\/strong><\/li>\n<\/ul>\n

      Whole numbers are closed under addition and multiplication. Whole numbers are not closed under subtraction and division. By closure we mean-<\/p>\n

      \u00a0<\/p>\n

        \n
      • For addition –<\/strong> If a and b are any two whole numbers, then (a + b) is also a whole number.<\/li>\n<\/ul>\n

        Example-<\/strong> 2 and 3 are two whole numbers. So, 2 + 3 i.e. 5 is also a whole number.<\/p>\n

        \u00a0<\/p>\n

          \n
        • For multiplication –<\/strong> If a and b are any two whole numbers, then (a x b) is also a whole number.<\/li>\n<\/ul>\n

          Example-<\/strong> 2 and 3 are two whole numbers. So, 2 \u00d7 3 i.e. 6 is also a whole number.<\/p>\n

          \u00a0<\/p>\n

            \n
          • For subtraction –<\/strong> If a and b are any two whole numbers, then (a – b) is not necessarily a whole number.<\/li>\n<\/ul>\n

            Example-<\/strong> 2 and 3 are two whole numbers. So, 3 – 2 i.e. 1 is a whole number but 2 \u2013 3 i.e. -1 is an integer but not a whole number.<\/p>\n

            \u00a0<\/p>\n

              \n
            • For division –<\/strong> If a and b are any two whole numbers, then (a – b) is not necessarily a whole number.<\/li>\n<\/ul>\n

              Example-<\/strong> 2 and 4 are two whole numbers. So, 4 \u00f7 2 i.e. 2 is a whole number but 2 \u00f7 4 i.e. 1<\/em>2<\/em><\/p>\n

              \"\"<\/p>\n

              Source: Kapdec.com<\/p>\n<\/div>\n

              \u00a0is a fraction and not a whole number.<\/p>\n

              \u00a0<\/p>\n

                \n
              • Commutative Property<\/strong><\/li>\n<\/ul>\n

                \u00a0<\/p>\n

                  \n
                • Commutative Property of Addition-<\/strong> The sum of two whole numbers is the same, no matter in which order they are added.<\/li>\n<\/ul>\n

                  Example:<\/strong> 19 + 22 = 41<\/p>\n

                  22+ 19 = 41<\/p>\n

                  So, 19 + 22 = 22+ 19<\/strong><\/p>\n

                  \u00a0<\/p>\n

                    \n
                  • Commutative Property of Multiplication-<\/strong> The product of two whole numbers is the same, no matter in which order they are multiplied.<\/li>\n<\/ul>\n

                    Example:<\/strong> 18 \u00d7 9 = 162<\/p>\n

                    \u00a09 \u00d7 18 = 162<\/p>\n

                    So, 18 <\/strong>\u00d7 9 = 9 <\/strong>\u00d7 18<\/strong><\/p>\n

                    \u00a0<\/p>\n

                    \u00a0<\/p>\n

                      \n
                    • Subtraction and Division are not commutative <\/strong>in whole numbers.<\/li>\n<\/ul>\n

                      Examples:<\/strong><\/p>\n

                      \u00a0<\/p>\n

                        \n
                      1. 19 – 12 = 7<\/li>\n<\/ol>\n

                        12- 19 = -7<\/p>\n

                        \u00a0<\/p>\n

                        So, 19 – 12\u226012- 19<\/strong><\/p>\n

                        \u00a0<\/p>\n

                        \u00a0<\/p>\n

                          \n
                        1. 18 \u00f7 3 = 6<\/li>\n<\/ol>\n

                          3\u00f7 18 = 1<\/em>6<\/em><\/p>\n

                          \"\"<\/p>\n

                          Source: Kapdec.com<\/p>\n<\/div>\n

                          \u00a0<\/p>\n

                          So, 18 \u00f7 3 \u2260 3 \u00f7 18<\/strong><\/p>\n

                          \u00a0<\/p>\n

                          \u00a0<\/p>\n

                            \n
                          • Additive Identity<\/strong><\/li>\n<\/ul>\n

                            A whole number when added to ‘0’ remains the same. Thus, ‘0’ is called the additive identity in whole numbers. If a is any whole number, then a + 0 = a = 0 + a.<\/p>\n

                            Example-<\/strong>3 + 0 = 3 = 0 + 3<\/p>\n

                            \u00a0<\/p>\n

                              \n
                            • Multiplicative Identity<\/strong><\/li>\n<\/ul>\n

                              A whole number when multiplied by 1 remains the same. Thus, 1 is called the multiplicative identity in whole numbers. If a is any whole number, then a \u00d7 1 = a = 1 \u00d7 a.<\/p>\n

                              Example-<\/strong>3 \u00d7 1 = 3 = 1 \u00d7 3<\/p>\n

                              \u00a0<\/p>\n

                                \n
                              • Associative property<\/strong><\/li>\n<\/ul>\n

                                \u00a0<\/p>\n

                                Associative Property of Addition-<\/strong> While adding whole numbers, we can group the numbers in any order, the sum remains the same.<\/p>\n

                                If a, b and c are any two whole numbers, then (a + b) + c = a + (b + c).<\/strong><\/p>\n

                                Example-<\/strong> 22+ 8 +16 = (22+ 8) + 16 = 30+ 16 = 46<\/p>\n

                                22+ 8 +16 = 22+(8 + 16) = 22+24 = 46<\/p>\n

                                So,\u00a0 (22 + 8) + 16 = 22 + (8 + 16) = 46<\/strong><\/p>\n

                                \u00a0<\/p>\n

                                Associative property of multiplication- <\/strong>While multiplying whole numbers, we can group them in any order, the product remains the same.<\/p>\n

                                If a, b and c are any two whole numbers, then (a \u00d7 b) \u00d7 c = a \u00d7 (b \u00d7 c).<\/strong><\/p>\n

                                \u00a0<\/p>\n

                                Example-<\/strong><\/p>\n

                                60 \u00d7 8 \u00d7 100 = (60 \u00d7 8) \u00d7 100 = 480 \u00d7 100 = 48000<\/p>\n

                                60 \u00d7 8 \u00d7 100 = 60 \u00d7 (8 \u00d7 100) = 600 \u00d7 800 = 48000<\/p>\n

                                60 \u00d7 8 \u00d7 100 = (60 \u00d7 100) \u00d7 8 = 6000 \u00d7 8 = 48000<\/strong><\/p>\n

                                \u00a0<\/p>\n

                                \u00a0<\/p>\n

                                  \n
                                • Distributive Property<\/strong><\/li>\n<\/ul>\n

                                  \u00a0<\/p>\n

                                  Distributive Property of Multiplication over Addition-<\/strong> 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.<\/p>\n

                                  If a, b and c are any two whole numbers, then a (b + c) = a \u00d7 b + a \u00d7 c<\/strong>.<\/p>\n

                                  Example- \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>24 X 125 = 24 X (100 + 20 + 5)<\/p>\n

                                  \u00a0 = 24 X 100 + 24 X 20 + 24 X 5<\/p>\n

                                  \u00a0 = 2400 + 480 + 120<\/p>\n

                                  \u00a0 = 3000<\/p>\n

                                  \u00a0<\/p>\n

                                  Distributive Property of Multiplication over Subtraction-<\/strong> 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 number.<\/p>\n

                                  If a, b and c are any two whole numbers, then a(b\u2013c) = a\u00d7b \u2013 a\u00d7c.<\/strong><\/p>\n

                                  Example- \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>18 X 98 = 18 X (100 – 2)<\/p>\n

                                  = (18 X 100) \u2013 (18 X 2)<\/p>\n

                                  \u00a0\u00a0 = 1800-36<\/p>\n

                                  =1764<\/p>\n

                                    \n
                                  • Multiplication by Zero<\/strong><\/li>\n<\/ul>\n

                                    Product of a whole number by zero is equal to zero.<\/p>\n

                                    If a is any whole number, then a \u00d7 0 = 0 = 0 \u00d7 a.<\/strong><\/p>\n

                                    Example-<\/strong>5 \u00d7 0 = 0<\/p>\n

                                    \u00a0<\/p>\n

                                    \u00a0<\/p>\n

                                      \n
                                    • Properties of Division <\/strong><\/li>\n<\/ul>\n

                                      \u00a0<\/p>\n

                                        \n
                                      1. Division by 0:<\/strong> Division by 0 is not possible.<\/li>\n<\/ol>\n

                                        If a is any whole number, then a \u00f7 0 is not defined.<\/strong><\/p>\n

                                        Example:<\/strong> 51,841 \u00f7 0 = Not Possible<\/p>\n

                                        \u00a0<\/p>\n

                                        \u00a0<\/p>\n

                                          \n
                                        1. Division by 1:<\/strong> If a number is divided by 1, the result is the number itself.<\/li>\n<\/ol>\n

                                          If a is any whole number, then a \u00f7 1 = a.<\/strong><\/p>\n

                                          Example: <\/strong>8,59,283 \u00f7 1 = 8,59,283<\/p>\n

                                          \u00a0<\/p>\n

                                          \u00a0<\/p>\n

                                            \n
                                          1. Division by number itself:<\/strong> If a number is divided by itself, the result is always 1.<\/li>\n<\/ol>\n

                                            If a is any non-zero whole number, then a \u00f7 a = 1.<\/strong><\/p>\n

                                            Example:<\/strong> 4,55,993 \u00f7 4,55,993 = 1<\/p>\n

                                            \u00a0<\/p>\n

                                            \u00a0<\/p>\n

                                              \n
                                            1. Dividing 0:<\/strong> If 0 is divided by any number (except 0), the result is always 0.<\/li>\n<\/ol>\n

                                              If a is any non-zero whole number, then 0 \u00f7 a = 0.<\/strong><\/p>\n

                                              Example: <\/strong>0 \u00f7 4,93,661 = 0<\/p>\n

                                                \n
                                              1. Division Algorithm-<\/strong><\/li>\n<\/ol>\n

                                                Dividend = Divisor X Quotient + Remainder<\/strong><\/p>\n

                                                This equation is used to check division.<\/p>\n

                                                Example- <\/strong>725 \u00f7 16<\/p>\n

                                                \u00a0\u00a0\u00a0 16) 725 (45<\/p>\n

                                                \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 -64<\/p>\n

                                                \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 085<\/p>\n

                                                \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 –\u00a0 80<\/p>\n

                                                \u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a005<\/p>\n

                                                Divisor = 16<\/p>\n

                                                Quotient = 45<\/p>\n

                                                Remainder = 5<\/p>\n

                                                So,<\/p>\n

                                                Divisor X Quotient + Remainder<\/p>\n

                                                = 16 X 45 + 5 = 720 + 5 = 725 = Dividend<\/p>\n

                                                \u00a0<\/p>\n

                                                <\/p>\n