{"id":94,"date":"2018-03-19T15:32:19","date_gmt":"2018-03-19T15:32:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-orgbiochemistry\/chapter\/expressing-numbers-scientific-notation\/"},"modified":"2018-07-30T15:34:46","modified_gmt":"2018-07-30T15:34:46","slug":"expressing-numbers-scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-monroecc-orgbiochemistry\/chapter\/expressing-numbers-scientific-notation\/","title":{"raw":"1.4 Expressing Numbers: Scientific Notation","rendered":"1.4 Expressing Numbers: Scientific Notation"},"content":{"raw":"
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Learning Objective<\/h3>\r\n
    \r\n \t
  1. Express a large number or a small number in scientific notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n

    <\/p>\r\n\r\n<\/div>\r\n \r\n

    The instructions for making a pot of coffee specified 3 scoops (rather than 12,000 grounds) because any measurement is expressed more efficiently with units that are appropriate in size. In science, however, we often must deal with quantities that are extremely small or incredibly large. For example, you may have 5,000,000,000,000 red blood cells in a liter of blood, and the diameter of an iron atom is 0.000000014 inches. Numbers with many zeros can be cumbersome to work with, so scientists use scientific notation.<\/p>\r\n\r\n<\/div>\r\n

    Scientific notation<\/span><\/span><\/strong>\u00a0is a system for expressing very large or very small numbers in a compact manner. It uses the idea that such numbers can be rewritten as a number called the\u00a0coefficient<\/strong>,\u00a0 multiplied by 10 raised to a certain power, called the exponent<\/strong>..\u00a0<\/span><\/span><\/p>\r\n

    Let us look first at numbers with large absolute values. Suppose a spacecraft is 1,500,000 miles from Mars. The number 1,500,000 can be thought of as follows:<\/p>\r\n\r\n

    \"image\"<\/div>\r\n

    That is, 1,500,000 is the same as 1.5 times 1 million, and 1 million is 10 \u00d7 10 \u00d7 10 \u00d7 10 \u00d7 10 \u00d7 10, or 106<\/sup> (which is read as \u201cten to the sixth power\u201d). Therefore, 1,500,000 can be rewritten as 1.5 times 106<\/sup>, or 1.5 \u00d7 106<\/sup>. The distance of the spacecraft from Mars can therefore be expressed as 1.5 \u00d7 106<\/sup> miles.<\/p>\r\n\r\n

    \r\n
    \r\n
    \r\n

    100<\/sup> = 1<\/p>\r\n \r\n

    101<\/sup> = 10<\/p>\r\n \r\n

    102<\/sup> = 100<\/p>\r\n \r\n

    103<\/sup> = 1,000<\/p>\r\n \r\n

    104<\/sup> = 10,000<\/p>\r\n \r\n

    and so forth<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    The convention for expressing numbers in scientific notation is to write a single nonzero first digit, a decimal point, and the rest of the digits<\/strong>, excluding any non-significant trailing zeros. This\u00a0 coefficient<\/strong>\u00a0 is followed by a multiplication sign and then by 10 raised to the power necessary to reproduce the original number.\u00a0<\/span> For a number with a large absolute value, the exponent is the number of places you have to move the decimal point to the left to make it follow the first digit, so that the coefficient is between 1 and 10:<\/span><\/p>\r\n\r\n<\/div>\r\n

    \"image\"<\/div>\r\n
    \r\n

    Example 4<\/h3>\r\n

    Express each number in scientific notation.<\/p>\r\n\r\n

      \r\n \t
    1. 67,000,000,000<\/li>\r\n \t
    2. 1,689<\/li>\r\n \t
    3. 12.6<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"938405\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"938405\"]\r\n
        \r\n \t
      1. Moving the decimal point 10 places to the left gives 6.7 \u00d7 1010.<\/li>\r\n \t
      2. The decimal point is assumed to be at the end of the number, so moving it three places to the left gives 1.689 \u00d7 103.<\/li>\r\n \t
      3. In this case, we need to move the decimal point only one place to the left, which yields 1.26 \u00d7 101.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
        \r\n
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        Skill-Building Exercise<\/h3>\r\nExpress each number in scientific notation\r\n
          \r\n \t
        1. \r\n
          \r\n

          1,492<\/p>\r\n\r\n<\/div><\/li>\r\n \t

        2. \r\n
          \r\n

          102,000,000<\/p>\r\n\r\n<\/div><\/li>\r\n \t

        3. \r\n
          \r\n

          101,325<\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

          To change scientific notation to standard notation for a number with a large absolute value, reverse the process, moving the decimal point to the right. Place zeros at the end of the number being converted, if necessary, to produce a number of the proper magnitude.<\/p>\r\n\r\n<\/div>\r\n

          \r\n

          Example 5<\/h3>\r\n

          Express each number in standard, or nonscientific, notation.<\/p>\r\n\r\n

            \r\n \t
          1. 5.27 \u00d7 104<\/sup><\/li>\r\n \t
          2. 1.0008 \u00d7 106<\/sup><\/li>\r\n<\/ol>\r\n

            Solution<\/p>\r\n[reveal-answer q=\"903539\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"903539\"]\r\n

              \r\n \t
            1. Rather than moving the decimal to the left, we move it four places to the right and add zeros to give 52,700.<\/li>\r\n \t
            2. Moving the decimal six places to the right gives 1,000,800.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
              \r\n
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              Skill-Building Exercise<\/h3>\r\n

              Express each number in standard notation.<\/p>\r\n\r\n

                \r\n \t
              1. \r\n
                \r\n

                6.98 \u00d7 108<\/sup><\/p>\r\n\r\n<\/div><\/li>\r\n \t

              2. \r\n
                \r\n

                1.005 \u00d7 102<\/sup><\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                We can also use scientific notation to express numbers whose absolute values are less than 1. For example, the number 0.006 can be expressed as follows:<\/p>\r\n\r\n

                \"image\"<\/div>\r\n
                \r\n
                \r\n

                10\u22121<\/sup> = 1\/10<\/p>\r\n \r\n

                10\u22122<\/sup> = 1\/100<\/p>\r\n \r\n

                10\u22123<\/sup> = 1\/1,000<\/p>\r\n \r\n

                10\u22124<\/sup> = 1\/10,000<\/p>\r\n \r\n

                10\u22125<\/sup> = 1\/100,000<\/p>\r\n \r\n

                and so forth<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

                Use a negative number as the power to indicate the number of places we have to move the decimal point to the right to follow the first nonzero digit. This is illustrated as follows:<\/p>\r\n\r\n

                \"image\"<\/div>\r\n

                In scientific notation, numbers with an absolute value greater than one have a positive power, while numbers with an absolute value between zero and one have a negative power.\u00a0 The coefficient is positive for positive numbers, negative for negative numbers.<\/p>\r\n\r\n

                \r\n

                Example 6<\/h3>\r\n

                Express each number in scientific notation.<\/p>\r\n\r\n

                  \r\n \t
                1. 0.000006567<\/li>\r\n \t
                2. \u22120.0004004<\/li>\r\n \t
                3. 0.000000000000123<\/li>\r\n<\/ol>\r\n \r\n\r\nSolution\r\n\r\n[reveal-answer q=\"607725\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"607725\"]\r\n
                    \r\n \t
                  1. Move the decimal point six places to the right to get 6.567 \u00d7 10\u22126.<\/li>\r\n \t
                  2. Move the decimal point four places to the right to get \u22124.004 \u00d7 10\u22124. The negative sign on the number itself does not affect how we apply the rules of scientific notation.<\/li>\r\n \t
                  3. Move the decimal point 13 places to the right to get 1.23 \u00d7 10\u221213.[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                    \r\n
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                    Skill-Building Exercise<\/h3>\r\n

                    Express each number in scientific notation.<\/p>\r\n\r\n

                      \r\n \t
                    1. \r\n
                      \r\n

                      0.000355<\/p>\r\n\r\n<\/div><\/li>\r\n \t

                    2. \r\n
                      \r\n

                      0.314159<\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n

                      As with numbers with positive powers of 10, when changing from scientific notation to standard notation, we reverse the process, moving the decimal point to the left the number of times indicated by the exponent and inserting zeros as necessary.\u00a0 It is customary to put a zero to the left of the decimal point as well.<\/p>\r\n\r\n

                      \r\n

                      Example 7<\/h3>\r\n

                      Express each number in standard notation.<\/p>\r\n\r\n

                        \r\n \t
                      1. 6.22 \u00d7 10\u22122<\/sup><\/li>\r\n \t
                      2. 9.9 \u00d7 10\u22129<\/sup><\/li>\r\n<\/ol>\r\n

                        Solution<\/p>\r\n[reveal-answer q=\"710350\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"710350\"]\r\n

                          \r\n \t
                        1. 0.0622<\/li>\r\n \t
                        2. 0.0000000099[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                          \r\n
                          \r\n

                          Skill-Building Exercise<\/h3>\r\nExpress each number in standard notation.\r\n
                            \r\n \t
                          1. \r\n
                            \r\n

                            9.98 \u00d7 10\u22125<\/sup><\/p>\r\n\r\n<\/div><\/li>\r\n \t

                          2. \r\n
                            \r\n

                            5.109 \u00d7 10\u22128<\/sup><\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\nMany calculators can express numbers in scientific notation and toggle between scientific notation and\u00a0floating point\u00a0<\/strong>(regular) notation. The methods for dealing with scientific notation differs for each calculator model, so take the time to learn how to do it properly on your calculator<\/strong>, <\/span>asking your instructor for assistance if necessary<\/em>. If you do not learn to enter scientific notation into your calculator properly, you will not get the correct final answer when performing a calculation.\r\n<\/span>\r\n\r\n<\/div>\r\n

                            \r\n
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                            Concept Review Exercises<\/h3>\r\n
                              \r\n \t
                            1. \r\n
                              \r\n

                              Why it is easier to use scientific notation to express very large or very small numbers?<\/p>\r\n\r\n<\/div><\/li>\r\n \t

                            2. \r\n
                              \r\n

                              What is the relationship between how many places a decimal point moves and the power of 10 used in changing a conventional number into scientific notation?<\/p>\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n

                              \r\n

                              Answers<\/h3>\r\n
                              \r\n

                              [reveal-answer q=\"318789\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"318789\"]1.Scientific notation is more convenient than listing a large number of zeros.<\/p>\r\n

                              2.The number of places the decimal point moves equals the power of 10\u2014positive if the decimal point moves to the left and negative if the decimal point moves to the right.[\/hidden-answer]\u00a0<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

                              \r\n
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                              Key Takeaway<\/h3>\r\n<\/div>\r\n
                              \r\n
                                \r\n \t
                              • Large or small numbers are expressed in scientific notation, which use powers of 10.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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                                Exercises<\/h3>\r\n<\/div>\r\n
                                \r\n
                                  \r\n \t
                                1. \r\n
                                  \r\n

                                  Why is scientific notation useful in expressing numbers?<\/p>\r\n\r\n<\/div><\/li>\r\n \t

                                2. \r\n
                                  \r\n

                                  What is the relationship between the power and the number of places a decimal point is moved when going from standard to scientific notation?<\/p>\r\n\r\n<\/div><\/li>\r\n \t

                                3. \r\n
                                  \r\n

                                  Express each number in scientific notation.<\/p>\r\n\r\n

                                    \r\n \t
                                  1. 0.00064<\/li>\r\n \t
                                  2. 5,230,000<\/li>\r\n \t
                                  3. \u221256,200<\/li>\r\n \t
                                  4. 0.000000000220<\/li>\r\n \t
                                  5. 1.0<\/li>\r\n<\/ol>\r\n<\/div><\/li>\r\n \t
                                  6. \r\n
                                    \r\n

                                    Express each number in scientific notation.<\/p>\r\n\r\n

                                      \r\n \t
                                    1. 678<\/li>\r\n \t
                                    2. \u22121,061<\/li>\r\n \t
                                    3. 0.000560<\/li>\r\n \t
                                    4. 0.0000003003<\/li>\r\n \t
                                    5. 100,000,000<\/li>\r\n<\/ol>\r\n<\/div><\/li>\r\n \t
                                    6. \r\n
                                      \r\n

                                      Express each number in standard form.<\/p>\r\n\r\n

                                        \r\n \t
                                      1. 6.72 \u00d7 104<\/sup><\/li>\r\n \t
                                      2. 2.088 \u00d7 10\u22124<\/sup><\/li>\r\n \t
                                      3. \u22123 \u00d7 106<\/sup><\/li>\r\n \t
                                      4. 9.98 \u00d7 10\u22127<\/sup><\/li>\r\n<\/ol>\r\n<\/div><\/li>\r\n \t
                                      5. \r\n
                                        \r\n

                                        Express each number in standard form.<\/p>\r\n\r\n

                                          \r\n \t
                                        1. 9.05 \u00d7 105<\/sup><\/li>\r\n \t
                                        2. 1.0 \u00d7 10\u22123<\/sup><\/li>\r\n \t
                                        3. 6.022 \u00d7 1023<\/sup><\/li>\r\n \t
                                        4. 8.834 \u00d7 10\u221212<\/sup><\/li>\r\n<\/ol>\r\n<\/div><\/li>\r\n \t
                                        5. \r\n
                                          \r\n

                                          Complete the following table:<\/p>\r\n\r\n

                                          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                          Incorrect Scientific Notation<\/th>\r\nCorrect Scientific Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                                          54.7 \u00d7 104<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                                          0.0066 \u00d7 103<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                                          3,078 \u00d7 100<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div><\/li>\r\n \t
                                        6. \r\n
                                          \r\n

                                          Complete the following table:<\/p>\r\n\r\n

                                          \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                          Incorrect Scientific Notation<\/th>\r\nCorrect Scientific Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                                          234.0 \u00d7 101<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                                          36 \u00d7 10\u22124<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                                          0.993 \u00d7 105<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n
                                          \r\n

                                          Answers<\/h3>\r\n[reveal-answer q=\"551491\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"551491\"]\r\n\r\n1.Scientific notation is more convenient than listing a large number of zeros.\r\n\r\n3.\r\n
                                            \r\n \t
                                          1. \r\n
                                              \r\n \t
                                            1. 6.4 \u00d7 10\u22124<\/li>\r\n \t
                                            2. 2.5.23 \u00d7 106<\/li>\r\n \t
                                            3. \u22125.62 \u00d7 104<\/li>\r\n \t
                                            4. 2.20 \u00d7 10\u221210<\/li>\r\n \t
                                            5. 1.0 \u00d7 100<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n5. 67,200 0.0002088 \u22123,000,000 0.000000998\r\n\r\n7.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                              Incorrect Scientific Notation<\/strong><\/td>\r\nCorrect Scientific Notation<\/strong><\/td>\r\n<\/tr>\r\n
                                              54.7 \u00d7 104<\/td>\r\n\u00a05.47 \u00d7 105<\/td>\r\n<\/tr>\r\n
                                              \u00a00.0066 \u00d7 103<\/td>\r\n6.6 \u00d7 100<\/td>\r\n<\/tr>\r\n
                                              3,078 \u00d7 100<\/td>\r\n\u00a03.078 \u00d7 103<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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                                              \r\n\r\n \r\n\r\n<\/div>\r\n<\/div>","rendered":"
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                                              Learning Objective<\/h3>\n
                                                \n
                                              1. Express a large number or a small number in scientific notation.<\/li>\n<\/ol>\n<\/div>\n

                                                \n<\/div>\n

                                                 <\/p>\n

                                                The instructions for making a pot of coffee specified 3 scoops (rather than 12,000 grounds) because any measurement is expressed more efficiently with units that are appropriate in size. In science, however, we often must deal with quantities that are extremely small or incredibly large. For example, you may have 5,000,000,000,000 red blood cells in a liter of blood, and the diameter of an iron atom is 0.000000014 inches. Numbers with many zeros can be cumbersome to work with, so scientists use scientific notation.<\/p>\n<\/div>\n

                                                Scientific notation<\/span><\/span><\/strong>\u00a0is a system for expressing very large or very small numbers in a compact manner. It uses the idea that such numbers can be rewritten as a number called the\u00a0coefficient<\/strong>,\u00a0 multiplied by 10 raised to a certain power, called the exponent<\/strong>..\u00a0<\/span><\/span><\/p>\n

                                                Let us look first at numbers with large absolute values. Suppose a spacecraft is 1,500,000 miles from Mars. The number 1,500,000 can be thought of as follows:<\/p>\n

                                                \"image\"<\/div>\n

                                                That is, 1,500,000 is the same as 1.5 times 1 million, and 1 million is 10 \u00d7 10 \u00d7 10 \u00d7 10 \u00d7 10 \u00d7 10, or 106<\/sup> (which is read as \u201cten to the sixth power\u201d). Therefore, 1,500,000 can be rewritten as 1.5 times 106<\/sup>, or 1.5 \u00d7 106<\/sup>. The distance of the spacecraft from Mars can therefore be expressed as 1.5 \u00d7 106<\/sup> miles.<\/p>\n

                                                \n
                                                \n
                                                \n

                                                100<\/sup> = 1<\/p>\n

                                                 <\/p>\n

                                                101<\/sup> = 10<\/p>\n

                                                 <\/p>\n

                                                102<\/sup> = 100<\/p>\n

                                                 <\/p>\n

                                                103<\/sup> = 1,000<\/p>\n

                                                 <\/p>\n

                                                104<\/sup> = 10,000<\/p>\n

                                                 <\/p>\n

                                                and so forth<\/p>\n<\/div>\n<\/div>\n

                                                The convention for expressing numbers in scientific notation is to write a single nonzero first digit, a decimal point, and the rest of the digits<\/strong>, excluding any non-significant trailing zeros. This\u00a0 coefficient<\/strong>\u00a0 is followed by a multiplication sign and then by 10 raised to the power necessary to reproduce the original number.\u00a0<\/span> For a number with a large absolute value, the exponent is the number of places you have to move the decimal point to the left to make it follow the first digit, so that the coefficient is between 1 and 10:<\/span><\/p>\n<\/div>\n

                                                \"image\"<\/div>\n
                                                \n

                                                Example 4<\/h3>\n

                                                Express each number in scientific notation.<\/p>\n

                                                  \n
                                                1. 67,000,000,000<\/li>\n
                                                2. 1,689<\/li>\n
                                                3. 12.6<\/li>\n<\/ol>\n
                                                  Show Answer<\/span><\/p>\n
                                                  \n
                                                    \n
                                                  1. Moving the decimal point 10 places to the left gives 6.7 \u00d7 1010.<\/li>\n
                                                  2. The decimal point is assumed to be at the end of the number, so moving it three places to the left gives 1.689 \u00d7 103.<\/li>\n
                                                  3. In this case, we need to move the decimal point only one place to the left, which yields 1.26 \u00d7 101.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
                                                    \n
                                                    \n
                                                    \n

                                                    Skill-Building Exercise<\/h3>\n

                                                    Express each number in scientific notation<\/p>\n

                                                      \n
                                                    1. \n
                                                      \n

                                                      1,492<\/p>\n<\/div>\n<\/li>\n

                                                    2. \n
                                                      \n

                                                      102,000,000<\/p>\n<\/div>\n<\/li>\n

                                                    3. \n
                                                      \n

                                                      101,325<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                      To change scientific notation to standard notation for a number with a large absolute value, reverse the process, moving the decimal point to the right. Place zeros at the end of the number being converted, if necessary, to produce a number of the proper magnitude.<\/p>\n<\/div>\n

                                                      \n

                                                      Example 5<\/h3>\n

                                                      Express each number in standard, or nonscientific, notation.<\/p>\n

                                                        \n
                                                      1. 5.27 \u00d7 104<\/sup><\/li>\n
                                                      2. 1.0008 \u00d7 106<\/sup><\/li>\n<\/ol>\n

                                                        Solution<\/p>\n

                                                        Show Answer<\/span><\/p>\n
                                                        \n
                                                          \n
                                                        1. Rather than moving the decimal to the left, we move it four places to the right and add zeros to give 52,700.<\/li>\n
                                                        2. Moving the decimal six places to the right gives 1,000,800.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
                                                          \n
                                                          \n

                                                          Skill-Building Exercise<\/h3>\n

                                                          Express each number in standard notation.<\/p>\n

                                                            \n
                                                          1. \n
                                                            \n

                                                            6.98 \u00d7 108<\/sup><\/p>\n<\/div>\n<\/li>\n

                                                          2. \n
                                                            \n

                                                            1.005 \u00d7 102<\/sup><\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                            We can also use scientific notation to express numbers whose absolute values are less than 1. For example, the number 0.006 can be expressed as follows:<\/p>\n

                                                            \"image\"<\/div>\n
                                                            \n
                                                            \n

                                                            10\u22121<\/sup> = 1\/10<\/p>\n

                                                             <\/p>\n

                                                            10\u22122<\/sup> = 1\/100<\/p>\n

                                                             <\/p>\n

                                                            10\u22123<\/sup> = 1\/1,000<\/p>\n

                                                             <\/p>\n

                                                            10\u22124<\/sup> = 1\/10,000<\/p>\n

                                                             <\/p>\n

                                                            10\u22125<\/sup> = 1\/100,000<\/p>\n

                                                             <\/p>\n

                                                            and so forth<\/p>\n<\/div>\n<\/div>\n

                                                            Use a negative number as the power to indicate the number of places we have to move the decimal point to the right to follow the first nonzero digit. This is illustrated as follows:<\/p>\n

                                                            \"image\"<\/div>\n

                                                            In scientific notation, numbers with an absolute value greater than one have a positive power, while numbers with an absolute value between zero and one have a negative power.\u00a0 The coefficient is positive for positive numbers, negative for negative numbers.<\/p>\n

                                                            \n

                                                            Example 6<\/h3>\n

                                                            Express each number in scientific notation.<\/p>\n

                                                              \n
                                                            1. 0.000006567<\/li>\n
                                                            2. \u22120.0004004<\/li>\n
                                                            3. 0.000000000000123<\/li>\n<\/ol>\n

                                                               <\/p>\n

                                                              Solution<\/p>\n

                                                              Show Answer<\/span><\/p>\n
                                                              \n
                                                                \n
                                                              1. Move the decimal point six places to the right to get 6.567 \u00d7 10\u22126.<\/li>\n
                                                              2. Move the decimal point four places to the right to get \u22124.004 \u00d7 10\u22124. The negative sign on the number itself does not affect how we apply the rules of scientific notation.<\/li>\n
                                                              3. Move the decimal point 13 places to the right to get 1.23 \u00d7 10\u221213.<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
                                                                \n
                                                                \n

                                                                Skill-Building Exercise<\/h3>\n

                                                                Express each number in scientific notation.<\/p>\n

                                                                  \n
                                                                1. \n
                                                                  \n

                                                                  0.000355<\/p>\n<\/div>\n<\/li>\n

                                                                2. \n
                                                                  \n

                                                                  0.314159<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n

                                                                   <\/p>\n<\/div>\n

                                                                  As with numbers with positive powers of 10, when changing from scientific notation to standard notation, we reverse the process, moving the decimal point to the left the number of times indicated by the exponent and inserting zeros as necessary.\u00a0 It is customary to put a zero to the left of the decimal point as well.<\/p>\n

                                                                  \n

                                                                  Example 7<\/h3>\n

                                                                  Express each number in standard notation.<\/p>\n

                                                                    \n
                                                                  1. 6.22 \u00d7 10\u22122<\/sup><\/li>\n
                                                                  2. 9.9 \u00d7 10\u22129<\/sup><\/li>\n<\/ol>\n

                                                                    Solution<\/p>\n

                                                                    Show Answer<\/span><\/p>\n
                                                                    \n
                                                                      \n
                                                                    1. 0.0622<\/li>\n
                                                                    2. 0.0000000099<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
                                                                      \n
                                                                      \n

                                                                      Skill-Building Exercise<\/h3>\n

                                                                      Express each number in standard notation.<\/p>\n

                                                                        \n
                                                                      1. \n
                                                                        \n

                                                                        9.98 \u00d7 10\u22125<\/sup><\/p>\n<\/div>\n<\/li>\n

                                                                      2. \n
                                                                        \n

                                                                        5.109 \u00d7 10\u22128<\/sup><\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n

                                                                        Many calculators can express numbers in scientific notation and toggle between scientific notation and\u00a0floating point\u00a0<\/strong>(regular) notation. The methods for dealing with scientific notation differs for each calculator model, so take the time to learn how to do it properly on your calculator<\/strong>, <\/span>asking your instructor for assistance if necessary<\/em>. If you do not learn to enter scientific notation into your calculator properly, you will not get the correct final answer when performing a calculation.
                                                                        \n<\/span><\/p>\n<\/div>\n

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                                                                        \n

                                                                        Concept Review Exercises<\/h3>\n
                                                                          \n
                                                                        1. \n
                                                                          \n

                                                                          Why it is easier to use scientific notation to express very large or very small numbers?<\/p>\n<\/div>\n<\/li>\n

                                                                        2. \n
                                                                          \n

                                                                          What is the relationship between how many places a decimal point moves and the power of 10 used in changing a conventional number into scientific notation?<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n

                                                                          \n

                                                                          Answers<\/h3>\n
                                                                          \n

                                                                          \n

                                                                          Show Answer<\/span><\/p>\n
                                                                          1.Scientific notation is more convenient than listing a large number of zeros.<\/p>\n

                                                                          2.The number of places the decimal point moves equals the power of 10\u2014positive if the decimal point moves to the left and negative if the decimal point moves to the right.<\/div>\n<\/div>\n

                                                                          \u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

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                                                                          \n

                                                                          Key Takeaway<\/h3>\n<\/div>\n
                                                                          \n