{"id":7283,"date":"2021-06-08T21:55:30","date_gmt":"2021-06-08T21:55:30","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/expressing-numbers\/"},"modified":"2021-09-21T22:25:48","modified_gmt":"2021-09-21T22:25:48","slug":"expressing-numbers","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/expressing-numbers\/","title":{"raw":"Expressing Numbers","rendered":"Expressing Numbers"},"content":{"raw":"[latexpage]\r\n
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

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  1. Learn to express numbers properly.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nQuantities have two parts: the number and the unit. The number tells \u201chow many.\u201d It is important to be able to express numbers properly so that the quantities can be communicated properly.\r\n\r\n[pb_glossary id=\"8125\"]Standard notation[\/pb_glossary]\u00a0is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.\r\n\r\n[pb_glossary id=\"8126\"]Scientific notation[\/pb_glossary]\u00a0is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Table 2.1 Powers of 10<\/caption>\r\n
    100<\/sup><\/td>\r\n= 1<\/td>\r\n<\/tr>\r\n
    101<\/sup><\/td>\r\n= 10<\/td>\r\n<\/tr>\r\n
    102<\/sup><\/td>\r\n= 100 = 10 \u00d7 10<\/td>\r\n<\/tr>\r\n
    103<\/sup><\/td>\r\n= 1,000 = 10 \u00d7 10 \u00d7 10<\/td>\r\n<\/tr>\r\n
    104<\/sup><\/td>\r\n= 10,000 = 10 \u00d7 10 \u00d7 10 \u00d7 10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nand so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the [pb_glossary id=\"8127\"]exponent[\/pb_glossary]. (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.\r\n\r\nSmall numbers can also be expressed in scientific notation but with negative exponents:\r\n\r\n\r\n\r\n\r\n\r\n
    Table 2.2 Powers of Negative 10<\/caption>\r\n
    10\u22121<\/sup><\/td>\r\n[latex]= 0.1 = \\dfrac{1}{10}[\/latex]<\/td>\r\n<\/tr>\r\n
    10\u22122<\/sup><\/td>\r\n[latex]= 0.01 = \\dfrac{1}{100}[\/latex]<\/td>\r\n<\/tr>\r\n
    10\u22123<\/sup><\/td>\r\n[latex]= 0.001 = \\dfrac{1}{1,000}[\/latex]<\/td>\r\n<\/tr>\r\n
    10\u22124<\/sup><\/td>\r\n[latex]= 0.0001 = \\dfrac{1}{10,000}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nand so forth. 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.\r\n\r\nA 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 [pb_glossary id=\"8128\"]coefficient[\/pb_glossary]. 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,\r\n

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

    [latex]0.000411 = 4.11\\times \\dfrac{1}{10,000} = 4.11 \u00d7 10^{-4}[\/latex]<\/p>\r\nTypically, the extra zero digits at the end or the beginning of a number are not included.\r\n

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    Example 2.1<\/p>\r\n\r\n<\/header>\r\n

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    Problems<\/h1>\r\nExpress these numbers in scientific notation.\r\n
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    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<\/ol>\r\n

      Solutions<\/h2>\r\n
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      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 [latex]\\frac{1}{1,000}[\/latex], 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 [latex]\\frac{1}{10,000,000}[\/latex], 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<\/ol>\r\n

        Test Yourself<\/h1>\r\nExpress these numbers in scientific notation.\r\n
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        1. 23,070<\/li>\r\n \t
        2. 0.0009706<\/li>\r\n<\/ol>\r\n

          Answers<\/h2>\r\n
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          1. 2.307 \u00d7 104<\/sup><\/li>\r\n \t
          2. 9.706 \u00d7 10\u22124<\/sup><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\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.\r\n\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. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.\r\n
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            Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

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