{"id":1080,"date":"2021-08-18T20:31:37","date_gmt":"2021-08-18T20:31:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1080"},"modified":"2022-01-26T22:54:43","modified_gmt":"2022-01-26T22:54:43","slug":"scientific-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/scientific-notation\/","title":{"raw":"Scientific Notation","rendered":"Scientific Notation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"padding-left: 30px;\">Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Convert from scientific to standard notation<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Convert standard notation to scientific notation<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe exponential expression [latex]b^{n}[\/latex], where the exponent [latex]n[\/latex] is a positive integer, is used to represent the repeated product of [latex]n[\/latex] factors of the base [latex]b[\/latex].\u00a0 For example\r\n<p style=\"text-align: center;\">[latex]5^{3}=5 \\cdot 5 \\cdot 5 = 125[\/latex].<\/p>\r\nIf the exponent is not a positive integer, we can use the following properties of exponents to evaluate the expression.\r\n<p style=\"text-align: center;\">For [latex]b \\neq 0[\/latex],\r\n[latex]b^{-n}= \\frac{1}{b^{n}}[\/latex]\r\n[latex]b^{0}=1[\/latex]<\/p>\r\nA shorthand method of writing very small and very large numbers is called <strong>scientific notation<\/strong>, in which we express numbers in terms of exponents of 10. A number is said to be in scientific notation if it is written in the form\r\n<p style=\"text-align: center;\">[latex]a \\times 10^{n}[\/latex] where [latex]1 \\leq |a| &lt; 10[\/latex].<\/p>\r\nThe number [latex]a[\/latex] can be positive or negative, but will always have exactly one nonzero digit to the left of the decimal point. For example, [latex]3.2 \\times 10^{3}[\/latex] and [latex]9.01 \\times 10^{-3}[\/latex] are in scientific notation. To rewrite these numbers in standard notation, we can evaluate the powers of [latex]10[\/latex] and perform the multiplication.\r\n<p style=\"text-align: center;\">[latex]3.2 \\times 10^{3} = 3.2 \\times 1000 = 3,200[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]9.01 \\times 10^{-3} = 9.01 \\times \\frac{1}{10^{3}} = \\frac{9.01}{1000} = 0.00901[\/latex]<\/p>\r\nIn both of these examples, the decimal point moved [latex]3[\/latex] places.\u00a0In converting [latex]3.2 \\times 10^{3}[\/latex] to [latex]3,200[\/latex], we multiply by [latex]10[\/latex] three times, moving the decimal point [latex]3[\/latex] places to the right.\u00a0 In converting [latex]9.01 \\times 10^{-3}[\/latex] to [latex]0.00901[\/latex], we divided by [latex]10[\/latex] three times, moving the decimal point [latex]3[\/latex] places to the left.\r\n<h2>Converting from Scientific to Standard Notation<\/h2>\r\nTo convert a number in scientific notation to standard notation, move the decimal point n places to the right if [latex]n[\/latex] is positive, or [latex]|n|[\/latex] places to the left if [latex]n[\/latex] is negative. Add zeros as needed.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Converting Scientific Notation to Standard Notation<\/h3>\r\nConvert each number in scientific notation to standard notation.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]-5.38\\times {10}^{7}[\/latex]<\/li>\r\n \t<li>[latex]1.005\\times {10}^{-4}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"928563\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"928563\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li style=\"text-align: left;\">To convert [latex]-5.38 \\times 10^{7}[\/latex] to standard notation, first note that the exponent on [latex]10[\/latex] is [latex]7[\/latex]. Since the exponent is positive, we need to move the decimal point [latex]7[\/latex] to the right.\r\n[latex]-5.38 \\times 10^{7}=-53,000,000[\/latex]<\/li>\r\n \t<li>To convert [latex]1.005 \\times 10^{-4}[\/latex] to standard notation, first note that the exponent on [latex]10[\/latex] is [latex]-4[\/latex]. Since the exponent is negative, we need to move the decimal point [latex]4[\/latex] places to the left.\r\n[latex]1.005 \\times 10^{-4} = 0.0001005[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert each number in scientific notation to standard notation.\r\n<ol>\r\n \t<li>[latex]7.03\\times {10}^{5}[\/latex]<\/li>\r\n \t<li>[latex]-8.16\\times {10}^{11}[\/latex]<\/li>\r\n \t<li>[latex]-3.9\\times {10}^{-13}[\/latex]<\/li>\r\n \t<li>[latex]8\\times {10}^{-6}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"655272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"655272\"]\r\n<ol>\r\n \t<li>[latex]703,000[\/latex]<\/li>\r\n \t<li>[latex]-816,000,000,000[\/latex]<\/li>\r\n \t<li>[latex]-0.00000000000039[\/latex]<\/li>\r\n \t<li>[latex]0.000008[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Converting from Standard to Scientific Notation<\/h2>\r\nTo write a number in scientific notation, we reverse this process. To begin, move the decimal point to the right of the first nonzero digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places [latex]n[\/latex] that you moved the decimal point. Multiply the decimal number by 10 raised to a power of [latex]n[\/latex]. If you moved the decimal left as in a very large number, [latex]n[\/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[\/latex] is negative.\r\n\r\nFor example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225857\/CNX_CAT_Figure_01_02_001.jpg\" alt=\"The number 2,780,418 is written with an arrow extending to another number: 2.780418. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 6 places left.\" \/>\r\n\r\nWe obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.\r\n<div style=\"text-align: center;\">[latex]2,780,418 = 2.780418\\times {10}^{6}[\/latex]<\/div>\r\nWorking with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225859\/CNX_CAT_Figure_01_02_002.jpg\" alt=\"The number 0.00000000000047 is written with an arrow extending to another number: 00000000000004.7. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 13 places right.\" \/>\r\n\r\nBe careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.\r\n<div style=\"text-align: center;\">[latex]0.00000000000047 = 4.7\\times {10}^{-13}[\/latex]<\/div>\r\nWatch the following video to see more examples of writing numbers in scientific notation.\r\n\r\nhttps:\/\/youtu.be\/fsNu3AdIgdk\r\n\r\nCalculators sometimes represent values in scientific notation in a slightly different format.\u00a0Try multiplying\r\n<p style=\"text-align: center;\">[latex]500,000 \\times 600,000[\/latex]<\/p>\r\nOn some calculators your answer may read 3E11.\u00a0 This means the same thing as\r\n<p style=\"text-align: center;\">[latex]3 \\times 10^{11} = 300,000,000,000[\/latex].<\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"padding-left: 30px;\">Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Convert from scientific to standard notation<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Convert standard notation to scientific notation<\/li>\n<\/ul>\n<\/div>\n<p>The exponential expression [latex]b^{n}[\/latex], where the exponent [latex]n[\/latex] is a positive integer, is used to represent the repeated product of [latex]n[\/latex] factors of the base [latex]b[\/latex].\u00a0 For example<\/p>\n<p style=\"text-align: center;\">[latex]5^{3}=5 \\cdot 5 \\cdot 5 = 125[\/latex].<\/p>\n<p>If the exponent is not a positive integer, we can use the following properties of exponents to evaluate the expression.<\/p>\n<p style=\"text-align: center;\">For [latex]b \\neq 0[\/latex],<br \/>\n[latex]b^{-n}= \\frac{1}{b^{n}}[\/latex]<br \/>\n[latex]b^{0}=1[\/latex]<\/p>\n<p>A shorthand method of writing very small and very large numbers is called <strong>scientific notation<\/strong>, in which we express numbers in terms of exponents of 10. A number is said to be in scientific notation if it is written in the form<\/p>\n<p style=\"text-align: center;\">[latex]a \\times 10^{n}[\/latex] where [latex]1 \\leq |a| < 10[\/latex].<\/p>\n<p>The number [latex]a[\/latex] can be positive or negative, but will always have exactly one nonzero digit to the left of the decimal point. For example, [latex]3.2 \\times 10^{3}[\/latex] and [latex]9.01 \\times 10^{-3}[\/latex] are in scientific notation. To rewrite these numbers in standard notation, we can evaluate the powers of [latex]10[\/latex] and perform the multiplication.<\/p>\n<p style=\"text-align: center;\">[latex]3.2 \\times 10^{3} = 3.2 \\times 1000 = 3,200[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]9.01 \\times 10^{-3} = 9.01 \\times \\frac{1}{10^{3}} = \\frac{9.01}{1000} = 0.00901[\/latex]<\/p>\n<p>In both of these examples, the decimal point moved [latex]3[\/latex] places.\u00a0In converting [latex]3.2 \\times 10^{3}[\/latex] to [latex]3,200[\/latex], we multiply by [latex]10[\/latex] three times, moving the decimal point [latex]3[\/latex] places to the right.\u00a0 In converting [latex]9.01 \\times 10^{-3}[\/latex] to [latex]0.00901[\/latex], we divided by [latex]10[\/latex] three times, moving the decimal point [latex]3[\/latex] places to the left.<\/p>\n<h2>Converting from Scientific to Standard Notation<\/h2>\n<p>To convert a number in scientific notation to standard notation, move the decimal point n places to the right if [latex]n[\/latex] is positive, or [latex]|n|[\/latex] places to the left if [latex]n[\/latex] is negative. Add zeros as needed.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Converting Scientific Notation to Standard Notation<\/h3>\n<p>Convert each number in scientific notation to standard notation.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]-5.38\\times {10}^{7}[\/latex]<\/li>\n<li>[latex]1.005\\times {10}^{-4}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q928563\">Show Answer<\/span><\/p>\n<div id=\"q928563\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li style=\"text-align: left;\">To convert [latex]-5.38 \\times 10^{7}[\/latex] to standard notation, first note that the exponent on [latex]10[\/latex] is [latex]7[\/latex]. Since the exponent is positive, we need to move the decimal point [latex]7[\/latex] to the right.<br \/>\n[latex]-5.38 \\times 10^{7}=-53,000,000[\/latex]<\/li>\n<li>To convert [latex]1.005 \\times 10^{-4}[\/latex] to standard notation, first note that the exponent on [latex]10[\/latex] is [latex]-4[\/latex]. Since the exponent is negative, we need to move the decimal point [latex]4[\/latex] places to the left.<br \/>\n[latex]1.005 \\times 10^{-4} = 0.0001005[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert each number in scientific notation to standard notation.<\/p>\n<ol>\n<li>[latex]7.03\\times {10}^{5}[\/latex]<\/li>\n<li>[latex]-8.16\\times {10}^{11}[\/latex]<\/li>\n<li>[latex]-3.9\\times {10}^{-13}[\/latex]<\/li>\n<li>[latex]8\\times {10}^{-6}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q655272\">Show Solution<\/span><\/p>\n<div id=\"q655272\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]703,000[\/latex]<\/li>\n<li>[latex]-816,000,000,000[\/latex]<\/li>\n<li>[latex]-0.00000000000039[\/latex]<\/li>\n<li>[latex]0.000008[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Converting from Standard to Scientific Notation<\/h2>\n<p>To write a number in scientific notation, we reverse this process. To begin, move the decimal point to the right of the first nonzero digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places [latex]n[\/latex] that you moved the decimal point. Multiply the decimal number by 10 raised to a power of [latex]n[\/latex]. If you moved the decimal left as in a very large number, [latex]n[\/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[\/latex] is negative.<\/p>\n<p>For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225857\/CNX_CAT_Figure_01_02_001.jpg\" alt=\"The number 2,780,418 is written with an arrow extending to another number: 2.780418. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 6 places left.\" \/><\/p>\n<p>We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.<\/p>\n<div style=\"text-align: center;\">[latex]2,780,418 = 2.780418\\times {10}^{6}[\/latex]<\/div>\n<p>Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21225859\/CNX_CAT_Figure_01_02_002.jpg\" alt=\"The number 0.00000000000047 is written with an arrow extending to another number: 00000000000004.7. An arrow tracking the movement of the decimal point runs underneath the number. Above the number a label on the number reads: 13 places right.\" \/><\/p>\n<p>Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.<\/p>\n<div style=\"text-align: center;\">[latex]0.00000000000047 = 4.7\\times {10}^{-13}[\/latex]<\/div>\n<p>Watch the following video to see more examples of writing numbers in scientific notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples:  Write a Number in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fsNu3AdIgdk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Calculators sometimes represent values in scientific notation in a slightly different format.\u00a0Try multiplying<\/p>\n<p style=\"text-align: center;\">[latex]500,000 \\times 600,000[\/latex]<\/p>\n<p>On some calculators your answer may read 3E11.\u00a0 This means the same thing as<\/p>\n<p style=\"text-align: center;\">[latex]3 \\times 10^{11} = 300,000,000,000[\/latex].<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1080\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/li><li>Examples: Write a Number in Scientific Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/fsNu3AdIgdk\">https:\/\/youtu.be\/fsNu3AdIgdk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\">https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\"},{\"type\":\"cc\",\"description\":\"Examples: Write a Number in Scientific Notation\",\"author\":\"James Sousa 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