{"id":40,"date":"2017-12-14T21:24:08","date_gmt":"2017-12-14T21:24:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/expressing-numbers\/"},"modified":"2022-12-08T20:03:31","modified_gmt":"2022-12-08T20:03:31","slug":"expressing-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/expressing-numbers\/","title":{"raw":"1.2 Handling Numbers in Science","rendered":"1.2 Handling Numbers in Science"},"content":{"raw":"
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Learning Objective<\/h3>\r\nBy the end of this section, you will be able to:\r\n
    \r\n \t
  • Learn to express numbers in scientific notation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n

    Throughout this course, you will frequently encounter very large and very small numbers.\u00a0 Handling these numbers can be quiet cumbersome, especially when performing calculations.<\/p>\r\n\r\n<\/div>\r\nFor example, later on in this course we will be working with Avogadro's number (the number of elementary entities in 1 mole of a substance).\r\n

    Avogadro's number:\u00a0 602,200,000,000,000,000,000,000<\/span><\/p>\r\nAvogadro's number, as show above, is expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as Avogadro's number, or for very small numbers, such as 0.000000419, using standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position. In order to make it easy to handle large or small numbers scientist use a method called scientific notation<\/strong>. Regardless of the magnitude, all numbers can be expressed in the form of scientific notation, N \u00d7 10n<\/sup><\/em>, where N is a number between 1 and 9, and n<\/em> is either a positive or negative whole number.\r\n

    Avogadro's number in scientific notation form:\u00a0 6.022 \u00d7 1023<\/sup><\/p>\r\n\r\n

    \r\n

    Scientific notation<\/span> is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:<\/p>\r\n\r\n

    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]\\large 10^{0}=1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{1}=10[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{2}=10\\times10=100[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{3}=10\\times10\\times10=1000[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{3}=10\\times10\\times10\\times10=10000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

    The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent. (Scientific notation is sometimes called exponential notation<\/em>.) The exponent\u2019s value is equal to the number of zeros in the number expressed in standard notation.<\/p>\r\n

    Small numbers can also be expressed in scientific notation but with negative exponents:<\/p>\r\n\r\n

    \r\n\r\n\r\n\r\n\r\n\r\n\r\n
    [latex]\\large 10^{-1}=\\frac{1}{10}=0.1[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{-2}=\\frac{1}{100}=0.01[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{-3}=\\frac{1}{1000}=0.001[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\large 10^{-4}=\\frac{1}{10000}=0.0001[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

    Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.<\/p>\r\n

    A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,<\/p>\r\n

    79,345 = 7.9345 \u00d7 10,000 = 7.9345 \u00d7 104<\/sup><\/span><\/span><\/p>\r\n

    Thus, the number in scientific notation is 7.9345 \u00d7 104<\/sup>. For small numbers, the same process is used, but the exponent for the power of 10 is negative:<\/p>\r\n

    [latex]\\large 0.000411 = 4.11\\times \\frac{1}{10000} = 4.11\\times 10^{-4}[\/latex]<\/p>\r\n

    Typically, the extra zero digits at the end or the beginning of a number are not included.<\/p>\r\nAnother way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.\u00a0 Examples of scientific notation are listed in Table 1.\r\n\r\n<\/div>\r\n

    \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 1. Examples of Scientific Notation<\/th>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    453<\/td>\r\n4.53 \u00d7 10<\/span>2<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    0.0022<\/td>\r\n2.2 \u00d7 10<\/span>\u22123<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    80031575<\/td>\r\n8.0031575 \u00d7 10<\/span>7<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    700.1<\/td>\r\n7.001 \u00d7 10<\/span>2<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    0.334<\/td>\r\n3.34 \u00d7 10<\/span>\u22121<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    50000<\/td>\r\n5 \u00d7 10<\/span>4<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n
    0.00000000000065<\/td>\r\n2.2 \u00d7 10<\/span>\u221213<\/sup><\/td>\r\n\u00a0<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
    \r\n

    Example 1: Writing Numbers in scientific notation\r\n<\/strong><\/h3>\r\n

    As of 2022, the United States population was estimated to be 338,977,986 people.\u00a0 What is this population expressed in scientific notation?<\/p>\r\n[reveal-answer q=\"36756211\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"36756211\"]\r\n\r\nThe decimal is understood to be at the end of the number, 338,977,986., to express the number in scientific notation we move the decimal to the left until only one digit is in front of the decimal.\u00a0 Since the decimal is moved a total of eight decimal places to the left, the exponent is a positive eight.\r\n\r\n3.38977986\u00a0\u00d7 108<\/sup> people\r\n\r\n[\/hidden-answer]\r\n

    Check Your Learning<\/strong><\/h4>\r\n

    Express the following numbers in scientific notation.<\/p>\r\n\r\n

      \r\n \t
    1. 306,000<\/li>\r\n \t
    2. 0.00884<\/li>\r\n \t
    3. 2,760,000<\/li>\r\n \t
    4. 0.000000559<\/li>\r\n \t
    5. 23,070<\/li>\r\n \t
    6. 0.0009706<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"367562\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367562\"]\r\n
        \r\n \t
      1. The number 306,000 is 3.06 times 100,000, or 3.06 times 105<\/sup>. In scientific notation, the number is 3.06 \u00d7 105<\/sup>.<\/li>\r\n \t
      2. The number 0.00884 is 8.84 times 1\/1,000, which is 8.84 times 10\u22123<\/sup>. In scientific notation, the number is 8.84 \u00d7 10\u22123<\/sup>.<\/li>\r\n \t
      3. The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 106<\/sup>. In scientific notation, the number is written as 2.76 \u00d7 106<\/sup>. Note that we omit the zeros at the end of the original number.<\/li>\r\n \t
      4. The number 0.000000559 is 5.59 times 1\/10,000,000, which is 5.59 times 10\u22127<\/sup>. In scientific notation, the number is written as 5.59 \u00d7 10\u22127<\/sup>.<\/li>\r\n \t
      5. The number 23,070 is 2.307 times 10,000, or 2.307 times 104<\/sup>. In scientific notation, the number is 2.307 \u00d7 104<\/sup>.<\/li>\r\n \t
      6. The number 0.0009706 is 9.706 times 1\/10,000, or 9.706 times 10-4<\/sup>. In scientific notation, the number is 9.706 \u00d7 10-4<\/sup>.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n
        \r\n
        \r\n

        Example 2: Converting Scientific notation to standard form\r\n<\/strong><\/h3>\r\n

        The atomic radius of gold (Au) is 1.46\u00a0\u00d7 10-8<\/sup>\u00a0cm.\u00a0 What is the radius of gold expressed in standard form?<\/p>\r\n[reveal-answer q=\"367562111\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367562111\"]\r\n\r\nSince the exponent is to the negative 8, the decimal point should be moved to the left 8 decimals.\u00a0 Add placeholder zeros before the coefficient as needed.\r\n\r\n0.0000000146\u00a0cm\r\n\r\n[\/hidden-answer]\r\n

        Check Your Learning<\/strong><\/h4>\r\n

        Express the following numbers in standard form:<\/p>\r\n\r\n

          \r\n \t
        1. 1.84 \u00d7 10\u22125<\/sup><\/li>\r\n \t
        2. 5.4 \u00d7 103<\/sup><\/li>\r\n \t
        3. 7.311 \u00d7 10\u22129<\/sup><\/li>\r\n \t
        4. 9.10 \u00d7 106<\/sup><\/li>\r\n \t
        5. 1.00 \u00d7 101<\/sup><\/li>\r\n \t
        6. 8 \u00d7 10\u22124<\/sup><\/li>\r\n<\/ol>\r\n[reveal-answer q=\"36756223\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"36756223\"]\r\n
            \r\n \t
          1. 0.0000184<\/li>\r\n \t
          2. 5400<\/li>\r\n \t
          3. 0.000000007311<\/li>\r\n \t
          4. 9,100,000<\/li>\r\n \t
          5. 10.0<\/li>\r\n \t
          6. 0.0008<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nMany quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator.<\/strong> Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.\r\n
            \r\n
            \r\n

            Key Takeaways<\/h3>\r\n
              \r\n \t
            • Standard notation expresses a number normally.<\/li>\r\n \t
            • Scientific notation expresses a number as a coefficient times a power of 10.<\/li>\r\n \t
            • The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
              \r\n

              Exercises<\/h3>\r\n
              \r\n

              1. Express these numbers in scientific notation.<\/p>\r\n\r\n<\/div>\r\n

              a) \u00a056.9<\/p>\r\n

              b) \u00a0563,100<\/p>\r\n

              c) \u00a00.0804<\/p>\r\n

              d) \u00a00.00000667<\/p>\r\n\r\n

              \r\n

              2. \u00a0Express these numbers in scientific notation.<\/p>\r\n

              a)\u00a0 890,000<\/p>\r\n

              b) \u00a0602,000,000,000<\/p>\r\n

              c) \u00a00.0000004099<\/p>\r\n

              d) \u00a00.000000000000011<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              3. \u00a0Express these numbers in scientific notation.<\/p>\r\n

              a) \u00a00.00656<\/p>\r\n

              b) \u00a065,600<\/p>\r\n

              c) \u00a04,567,000<\/p>\r\n

              d) \u00a00.000005507<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              4. \u00a0Express these numbers in scientific notation.<\/p>\r\n

              a) \u00a065<\/p>\r\n

              b)\u00a0 321.09<\/p>\r\n

              c) \u00a00.000077099<\/p>\r\n

              d) \u00a00.000000000218<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              5. \u00a0Express these numbers in standard notation.<\/p>\r\n

              a) \u00a01.381 \u00d7 105<\/sup><\/p>\r\n

              b) \u00a05.22 \u00d7 10\u22127<\/sup><\/p>\r\n

              c) \u00a09.998 \u00d7 104<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              6. \u00a0Express these numbers in standard notation.<\/p>\r\n

              a) \u00a07.11 \u00d7 10\u22122<\/sup><\/p>\r\n

              b) \u00a09.18 \u00d7 102<\/sup><\/p>\r\n

              c) \u00a03.09 \u00d7 10\u221210<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              7. \u00a0Express these numbers in standard notation.<\/p>\r\n

              a) \u00a08.09 \u00d7 100<\/sup><\/p>\r\n

              b) \u00a03.088 \u00d7 10\u22125<\/sup><\/p>\r\n

              c)\u00a0 4.239 \u00d7 102<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              8. \u00a0Express these numbers in standard notation.<\/p>\r\n

              a) \u00a02.87 \u00d7 10\u22128<\/sup><\/p>\r\n

              b) \u00a01.78 \u00d7 1011<\/sup><\/p>\r\n

              c) \u00a01.381 \u00d7 10\u221223<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\r\n

              a) \u00a072.44 \u00d7 103<\/sup><\/p>\r\n

              b) \u00a09,943 \u00d7 10\u22125<\/sup><\/p>\r\n

              c) \u00a0588,399 \u00d7 102<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              10. \u00a0These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\r\n

              a) \u00a00.000077 \u00d7 10\u22127<\/sup><\/p>\r\n

              b) \u00a00.000111 \u00d7 108<\/sup><\/p>\r\n

              c) \u00a0602,000 \u00d7 1018<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              11. \u00a0These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\r\n

              a) \u00a0345.1 \u00d7 102<\/sup><\/p>\r\n

              b) \u00a00.234 \u00d7 10\u22123<\/sup><\/p>\r\n

              c) \u00a01,800 \u00d7 10\u22122<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              12. \u00a0These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\r\n

              a) \u00a08,099 \u00d7 10\u22128<\/sup><\/p>\r\n

              b) \u00a034.5 \u00d7 100<\/sup><\/p>\r\n

              c) \u00a00.000332 \u00d7 104<\/sup><\/p>\r\n\r\n<\/div>\r\n

              \r\n

              13. \u00a0Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\r\n

              a) \u00a0123,456.78<\/p>\r\n

              b) \u00a098,490<\/p>\r\n

              c) \u00a00.000000445<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              14. \u00a0Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\r\n

              a) \u00a00.000552<\/p>\r\n

              b) \u00a01,987<\/p>\r\n

              c) \u00a00.00000000887<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              15. \u00a0Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\r\n

              a) \u00a0456 \u00d7 (7.4 \u00d7 108<\/sup>) = ?<\/p>\r\n

              b) \u00a0(3.02 \u00d7 105<\/sup>) \u00f7 (9.04 \u00d7 1015<\/sup>) = ?<\/p>\r\n

              c) \u00a00.0044 \u00d7 0.000833 = ?<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              16. \u00a0Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\r\n

              a) \u00a098,000 \u00d7 23,000 = ?<\/p>\r\n

              b) \u00a098,000 \u00f7 23,000 = ?<\/p>\r\n

              c) \u00a0(4.6 \u00d7 10\u22125<\/sup>) \u00d7 (2.09 \u00d7 103<\/sup>) = ?<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              17. \u00a0Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\r\n

              a) \u00a045 \u00d7 132 \u00f7 882 = ?<\/p>\r\n

              b) [(6.37 \u00d7 104<\/sup>) \u00d7 (8.44 \u00d7 10\u22124<\/sup>)] \u00f7 (3.2209 \u00d7 1015<\/sup>) = ?<\/p>\r\n\r\n<\/div>\r\n

              \r\n

              18. \u00a0Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\r\n

              a) \u00a0(9.09 \u00d7 108<\/sup>) \u00f7 [(6.33 \u00d7 109<\/sup>) \u00d7 (4.066 \u00d7 10\u22127<\/sup>)] = ?<\/p>\r\n

              b) \u00a09,345 \u00d7 34.866 \u00f7 0.00665 = ?<\/p>\r\n[reveal-answer q=\"367563\"]Show Answer to Select Questions[\/reveal-answer]\r\n[hidden-answer a=\"367563\"]\r\n\r\n1.\r\na) \u00a05.69 \u00d7 101<\/sup>\r\nb) \u00a05.631 \u00d7 105<\/sup>\r\nc) \u00a08.04 \u00d7 10\u22122<\/sup>\r\nd) \u00a06.67 \u00d7 10\u22126<\/sup>\r\n\r\n3.\r\na) \u00a06.56 \u00d7 10\u22123<\/sup>\r\nb) \u00a06.56 \u00d7 104<\/sup>\r\nc) \u00a04.567 \u00d7 106<\/sup>\r\nd) \u00a05.507 \u00d7 10\u22126<\/sup>\r\n\r\n5.\r\na) 138,100\r\nb) \u00a00.000000522\r\nc) \u00a099,980\r\n\r\n7.\r\na) \u00a08.09\r\nb) 0.00003088\r\nc)\u00a0 423.9\r\n\r\n9.\r\na) \u00a07.244 \u00d7 104<\/sup>\r\nb) \u00a0 9.943 \u00d7 10\u22122<\/sup>\r\nc) \u00a05.88399 \u00d7 107<\/sup>\r\n\r\n11.\r\na)\u00a0\u00a03.451 \u00d7 104<\/sup>\r\nb) \u00a02.34 \u00d7 10\u22124<\/sup>\r\nc) \u00a01.8 \u00d7 101<\/sup>\r\n

              \r\n
              \r\n\r\n13.\r\na) \u00a01.2345678 \u00d7 105<\/sup>\r\nb) \u00a09.849 \u00d7 104<\/sup>\r\nc)\u00a0\u00a04.45 \u00d7 10\u22127<\/sup>\r\n\r\n15.\r\na)\u00a03.3744 \u00d7 1011<\/sup>\r\nb)\u00a03.3407 \u00d7 10\u221211<\/sup>\r\nc)\u00a03.665 \u00d7 10\u22126<\/sup>\r\n\r\n17.\r\na) \u00a06.7346 \u00d7 100<\/sup>\r\nb) 1.6691 \u00d7 10\u221214<\/sup>\r\n\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n
              \r\n
              \r\n

              Glossary<\/h2>\r\nScientific Notation: <\/strong>Numerical expression of a large or small number using a coefficient (1 through 9, N) followed by a power of 10 raised to a positive\/negative whole number (n). Scientific notation form: N \u00d7 10n<\/sup>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
              <\/div>","rendered":"
              \n
              \n

              Learning Objective<\/h3>\n

              By the end of this section, you will be able to:<\/p>\n

                \n
              • Learn to express numbers in scientific notation.<\/li>\n<\/ul>\n<\/div>\n

                Throughout this course, you will frequently encounter very large and very small numbers.\u00a0 Handling these numbers can be quiet cumbersome, especially when performing calculations.<\/p>\n<\/div>\n

                For example, later on in this course we will be working with Avogadro’s number (the number of elementary entities in 1 mole of a substance).<\/p>\n

                Avogadro’s number:\u00a0 602,200,000,000,000,000,000,000<\/span><\/p>\n

                Avogadro’s number, as show above, is expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as Avogadro’s number, or for very small numbers, such as 0.000000419, using standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position. In order to make it easy to handle large or small numbers scientist use a method called scientific notation<\/strong>. Regardless of the magnitude, all numbers can be expressed in the form of scientific notation, N \u00d7 10n<\/sup><\/em>, where N is a number between 1 and 9, and n<\/em> is either a positive or negative whole number.<\/p>\n

                Avogadro’s number in scientific notation form:\u00a0 6.022 \u00d7 1023<\/sup><\/p>\n

                \n

                Scientific notation<\/span> is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:<\/p>\n

                \n\n\n\n\n\n\n\n
                [latex]\\large 10^{0}=1[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{1}=10[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{2}=10\\times10=100[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{3}=10\\times10\\times10=1000[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{3}=10\\times10\\times10\\times10=10000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the exponent. (Scientific notation is sometimes called exponential notation<\/em>.) The exponent\u2019s value is equal to the number of zeros in the number expressed in standard notation.<\/p>\n

                Small numbers can also be expressed in scientific notation but with negative exponents:<\/p>\n

                \n\n\n\n\n\n\n
                [latex]\\large 10^{-1}=\\frac{1}{10}=0.1[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{-2}=\\frac{1}{100}=0.01[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{-3}=\\frac{1}{1000}=0.001[\/latex]<\/td>\n<\/tr>\n
                [latex]\\large 10^{-4}=\\frac{1}{10000}=0.0001[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.<\/p>\n

                A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the coefficient. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,<\/p>\n

                79,345 = 7.9345 \u00d7 10,000 = 7.9345 \u00d7 104<\/sup><\/span><\/span><\/p>\n

                Thus, the number in scientific notation is 7.9345 \u00d7 104<\/sup>. For small numbers, the same process is used, but the exponent for the power of 10 is negative:<\/p>\n

                [latex]\\large 0.000411 = 4.11\\times \\frac{1}{10000} = 4.11\\times 10^{-4}[\/latex]<\/p>\n

                Typically, the extra zero digits at the end or the beginning of a number are not included.<\/p>\n

                Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.\u00a0 Examples of scientific notation are listed in Table 1.<\/p>\n<\/div>\n

                \n\n\n\n\n\n\n\n\n\n\n\n
                Table 1. Examples of Scientific Notation<\/th>\n<\/th>\n<\/tr>\n<\/thead>\n
                453<\/td>\n4.53 \u00d7 10<\/span>2<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                0.0022<\/td>\n2.2 \u00d7 10<\/span>\u22123<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                80031575<\/td>\n8.0031575 \u00d7 10<\/span>7<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                700.1<\/td>\n7.001 \u00d7 10<\/span>2<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                0.334<\/td>\n3.34 \u00d7 10<\/span>\u22121<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                50000<\/td>\n5 \u00d7 10<\/span>4<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n
                0.00000000000065<\/td>\n2.2 \u00d7 10<\/span>\u221213<\/sup><\/td>\n\u00a0<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                \n

                Example 1: Writing Numbers in scientific notation
                \n<\/strong><\/h3>\n

                As of 2022, the United States population was estimated to be 338,977,986 people.\u00a0 What is this population expressed in scientific notation?<\/p>\n

                Show Answer<\/span><\/p>\n
                \n

                The decimal is understood to be at the end of the number, 338,977,986., to express the number in scientific notation we move the decimal to the left until only one digit is in front of the decimal.\u00a0 Since the decimal is moved a total of eight decimal places to the left, the exponent is a positive eight.<\/p>\n

                3.38977986\u00a0\u00d7 108<\/sup> people<\/p>\n<\/div>\n<\/div>\n

                Check Your Learning<\/strong><\/h4>\n

                Express the following numbers in scientific notation.<\/p>\n

                  \n
                1. 306,000<\/li>\n
                2. 0.00884<\/li>\n
                3. 2,760,000<\/li>\n
                4. 0.000000559<\/li>\n
                5. 23,070<\/li>\n
                6. 0.0009706<\/li>\n<\/ol>\n
                  Show Answer<\/span><\/p>\n
                  \n
                    \n
                  1. The number 306,000 is 3.06 times 100,000, or 3.06 times 105<\/sup>. In scientific notation, the number is 3.06 \u00d7 105<\/sup>.<\/li>\n
                  2. The number 0.00884 is 8.84 times 1\/1,000, which is 8.84 times 10\u22123<\/sup>. In scientific notation, the number is 8.84 \u00d7 10\u22123<\/sup>.<\/li>\n
                  3. The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 106<\/sup>. In scientific notation, the number is written as 2.76 \u00d7 106<\/sup>. Note that we omit the zeros at the end of the original number.<\/li>\n
                  4. The number 0.000000559 is 5.59 times 1\/10,000,000, which is 5.59 times 10\u22127<\/sup>. In scientific notation, the number is written as 5.59 \u00d7 10\u22127<\/sup>.<\/li>\n
                  5. The number 23,070 is 2.307 times 10,000, or 2.307 times 104<\/sup>. In scientific notation, the number is 2.307 \u00d7 104<\/sup>.<\/li>\n
                  6. The number 0.0009706 is 9.706 times 1\/10,000, or 9.706 times 10-4<\/sup>. In scientific notation, the number is 9.706 \u00d7 10-4<\/sup>.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
                    \n
                    \n

                    Example 2: Converting Scientific notation to standard form
                    \n<\/strong><\/h3>\n

                    The atomic radius of gold (Au) is 1.46\u00a0\u00d7 10-8<\/sup>\u00a0cm.\u00a0 What is the radius of gold expressed in standard form?<\/p>\n

                    Show Answer<\/span><\/p>\n
                    \n

                    Since the exponent is to the negative 8, the decimal point should be moved to the left 8 decimals.\u00a0 Add placeholder zeros before the coefficient as needed.<\/p>\n

                    0.0000000146\u00a0cm<\/p>\n<\/div>\n<\/div>\n

                    Check Your Learning<\/strong><\/h4>\n

                    Express the following numbers in standard form:<\/p>\n

                      \n
                    1. 1.84 \u00d7 10\u22125<\/sup><\/li>\n
                    2. 5.4 \u00d7 103<\/sup><\/li>\n
                    3. 7.311 \u00d7 10\u22129<\/sup><\/li>\n
                    4. 9.10 \u00d7 106<\/sup><\/li>\n
                    5. 1.00 \u00d7 101<\/sup><\/li>\n
                    6. 8 \u00d7 10\u22124<\/sup><\/li>\n<\/ol>\n
                      Show Answer<\/span><\/p>\n
                      \n
                        \n
                      1. 0.0000184<\/li>\n
                      2. 5400<\/li>\n
                      3. 0.000000007311<\/li>\n
                      4. 9,100,000<\/li>\n
                      5. 10.0<\/li>\n
                      6. 0.0008<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                        Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator.<\/strong> Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.<\/p>\n

                        \n
                        \n

                        Key Takeaways<\/h3>\n