{"id":794,"date":"2016-06-01T20:49:09","date_gmt":"2016-06-01T20:49:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=794"},"modified":"2019-07-24T21:12:35","modified_gmt":"2019-07-24T21:12:35","slug":"read-writing-scientific-notation-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-writing-scientific-notation-2\/","title":{"raw":"Writing Scientific Notation","rendered":"Writing Scientific Notation"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define decimal and scientific notation<\/li>\r\n \t<li>Convert between scientific and decimal notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Convert Between Scientific and Decimal Notation<\/h2>\r\nBefore we can convert between scientific and decimal notation, we need to know the difference between the two.\u00a0<strong>S<\/strong><b>cientific notation <\/b>is used by\u00a0scientists, mathematicians, and engineers when they are working with very large or very small numbers.\u00a0Using exponential notation, large and small numbers can be written in a way that is easier to read.\r\n\r\nWhen a number is written in scientific notation, the <b>exponent<\/b> tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think\u00a0about it.\r\n<table style=\"width: 668px;\">\r\n<thead>\r\n<tr>\r\n<td style=\"width: 41px;\">Word<\/td>\r\n<td style=\"width: 320px;\">How many thousands<\/td>\r\n<td style=\"width: 125px;\">Number<\/td>\r\n<td style=\"width: 138px;\">Scientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 41px;\">million<\/td>\r\n<td style=\"width: 320px;\">[latex]1000 x 1000[\/latex] = a thousand thousands<\/td>\r\n<td style=\"width: 125px;\">[latex]1,000,000[\/latex]<\/td>\r\n<td style=\"width: 138px;\">\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 41px;\">billion<\/td>\r\n<td style=\"width: 320px;\">[latex](1000 x 1000) x 1000[\/latex] = a thousand millions<\/td>\r\n<td style=\"width: 125px;\">[latex]1,000,000,000[\/latex]<\/td>\r\n<td style=\"width: 138px;\">\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 41px;\">trillion<\/td>\r\n<td style=\"width: 320px;\">[latex](1000 x 1000 x 1000)[\/latex] x 1000 = a thousand billions<\/td>\r\n<td style=\"width: 125px;\">[latex]1,000,000,000,000[\/latex]<\/td>\r\n<td style=\"width: 138px;\">\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n1 billion can be written as\u00a0[latex]1,000,000,000[\/latex] or represented as\u00a0[latex]10^9[\/latex]. How would\u00a0[latex]2[\/latex] billion be\u00a0represented? Since\u00a0[latex]2[\/latex] billion is\u00a0[latex]2[\/latex] times\u00a0[latex]1[\/latex]\u00a0billion, then\u00a0[latex]2[\/latex] billion can be written as [latex]2\\times10^9[\/latex].\r\n\r\nA\u00a0light year is the number of miles light travels in one year, about\u00a0[latex]5,880,000,000,000[\/latex]. That is a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number.\u00a0Using scientific notation, the distance is [latex]5.88\\times10^{12}[\/latex]\u00a0miles. The exponent of\u00a0[latex]12[\/latex] tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a\u00a0hydrogen atom, which is about 0.00000005 mm, and in scientific notation is\u00a0[latex]5\\times10^{-8}[\/latex]\u00a0mm. In this case, the [latex]-8[\/latex] tells us how many places to count to the right of the decimal.\r\n\r\nOutlined in the box below are some important conventions of scientific notation format.\r\n<div class=\"textbox shaded\">\r\n<h3>Scientific Notation<\/h3>\r\nA positive number is written in scientific notation if it is written as [latex]a\\times10^{n}[\/latex]\u00a0where the coefficient <i>a<\/i>\u00a0is [latex]1\\leq{a}&lt;10[\/latex], and <i>n <\/i>is an integer.\r\n\r\n<\/div>\r\nLook at the numbers below. Which of the numbers is written in scientific notation?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><b>Number<\/b><\/td>\r\n<td><b>Scientific Notation?<\/b><\/td>\r\n<td><b>Explanation<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1.85\\times10^{-2}[\/latex]<\/td>\r\n<td>yes<\/td>\r\n<td>[latex]1\\leq1.85&lt;10[\/latex]\r\n\r\n[latex]-2[\/latex] is an integer<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>[latex] \\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0.82\\times10^{14}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>\u00a0[latex]0.82[\/latex] is not [latex]\\geq1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]10\\times10^{3}[\/latex]<\/td>\r\n<td>no<\/td>\r\n<td>[latex]10[\/latex] is not &lt;[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.\r\n<table style=\"height: 97px; width: 543px;\">\r\n<tbody>\r\n<tr style=\"height: 43px;\">\r\n<td style=\"height: 22px; width: 321.5px;\" colspan=\"2\">\r\n<p style=\"text-align: center;\"><b>Large Numbers<\/b><\/p>\r\n<\/td>\r\n<td style=\"height: 22px; width: 4.5px;\"><b>\u00a0<\/b><\/td>\r\n<td style=\"height: 22px; width: 321.5px;\" colspan=\"2\">\r\n<p style=\"text-align: center;\"><b>Small Numbers<\/b><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 172.5px;\"><b>Decimal Notation<\/b><\/td>\r\n<td style=\"height: 15px; width: 136.5px;\"><b>Scientific Notation<\/b><\/td>\r\n<td style=\"height: 15px; width: 4.5px;\"><b>\u00a0<\/b><\/td>\r\n<td style=\"height: 15px; width: 168.5px;\"><b>Decimal Notation<\/b><\/td>\r\n<td style=\"height: 15px; width: 140.5px;\"><b>Scientific Notation<\/b><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 172.5px;\">[latex]500.0[\/latex]<\/td>\r\n<td style=\"height: 15px; width: 136.5px;\">[latex]5\\times10^{2}[\/latex]<\/td>\r\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\r\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.05[\/latex]<\/td>\r\n<td style=\"width: 140.5px; height: 15px;\">[latex]5\\times10^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 172.5px;\">[latex]80,000.0[\/latex]<\/td>\r\n<td style=\"height: 15px; width: 136.5px;\">[latex]8\\times10^{4}[\/latex]<\/td>\r\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\r\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.0008[\/latex]<\/td>\r\n<td style=\"width: 140.5px; height: 15px;\">[latex]8\\times10^{-4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 172.5px;\">[latex]43,000,000.0[\/latex]<\/td>\r\n<td style=\"height: 15px; width: 136.5px;\">[latex]4.3\\times10^{7}[\/latex]<\/td>\r\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\r\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.00000043[\/latex]<\/td>\r\n<td style=\"width: 140.5px; height: 15px;\">[latex]4.3\\times10^{-7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.7969px;\">\r\n<td style=\"height: 15px; width: 172.5px;\">[latex]62,500,000,000.0[\/latex]<\/td>\r\n<td style=\"height: 15px; width: 136.5px;\">[latex]6.25\\times10^{10}[\/latex]<\/td>\r\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\r\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.000000000625[\/latex]<\/td>\r\n<td style=\"width: 140.5px; height: 15px;\">[latex]6.25\\times10^{-10}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Convert from Decimal Notation to Scientific Notation<\/h3>\r\nTo write a <i>large<\/i> number in scientific notation, move the decimal point to the left to obtain a number between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. Since moving the decimal point changes the value, you have to multiply the decimal by a power of [latex]10[\/latex] so that the expression has the same value.\r\n\r\nLet us look at an example:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longleftarrow}{180,000.}=18,000.0\\times10^{1}\\\\1,800.00\\times10^{2}\\\\180.000\\times10^{3}\\\\18.0000\\times10^{4}\\\\1.80000\\times10^{5}\\\\180,000=1.8\\times10^{5}\\end{array}[\/latex]<\/p>\r\nNotice that the decimal point was moved\u00a0[latex]5[\/latex] places to the left, and the exponent is\u00a0[latex]5[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following numbers in scientific notation.\r\n<ol>\r\n \t<li>[latex]920,000,000[\/latex]<\/li>\r\n \t<li>[latex]10,200,000[\/latex]<\/li>\r\n \t<li>[latex]100,000,000,000[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"628\"]\r\n<ol>\r\n \t<li>[latex]\\underset{\\longleftarrow}{920,000,000}[\/latex]\u00a0 \u00a0 \u00a0We will move the decimal point to the left. It helps to place it\u00a0at the end of the number and then count how many times you move it to get one number before it that is between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. Move the decimal point\u00a0[latex]8[\/latex] times to the left and you will have\u00a0[latex]9.20000000[\/latex]. Now we can replace the zeros with an exponent of [latex]8[\/latex] giving us [latex]9.2\\times10^{8}[\/latex].<\/li>\r\n \t<li>[latex]\\underset{\\longleftarrow}{10,200,000}=10,200,000.0=1.02\\times10^{7}[\/latex]. Note here how we included the\u00a0[latex]0[\/latex] and the\u00a0[latex]2[\/latex] after the decimal point. In some disciplines, you may learn about when to include both of these. \u00a0Follow instructions from your teacher on rounding rules.<\/li>\r\n \t<li>[latex]\\underset{\\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\\times10^{11}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo write a small number (between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]) in scientific notation, you move the decimal to the <i>right<\/i> and the exponent will have to be <i>negative,\u00a0<\/i>as in the following example.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longrightarrow}{0.00004}=00.0004\\times10^{-1}\\\\000.004\\times10^{-2}\\\\0000.04\\times10^{-3}\\\\00000.4\\times10^{-4}\\\\000004.\\times10^{-5}\\\\0.00004=4\\times10^{-5}\\end{array}[\/latex]<\/p>\r\nYou may notice that the decimal point was moved five places to the <i>right <\/i>until you got to the number\u00a0[latex]4[\/latex] which is between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. The exponent is [latex]\u22125[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following numbers in scientific notation.\r\n<ol>\r\n \t<li>[latex]0.0000000000035[\/latex]<\/li>\r\n \t<li>[latex]0.0000000102[\/latex]<\/li>\r\n \t<li>[latex]0.00000000000000793[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"229054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229054\"]\r\n<ol>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.0000000000035}=3.5\\times10^{-12}[\/latex]. We moved the decimal 12 times to get to a number between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]<\/li>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\r\n \t<li>[latex]\\underset{\\longrightarrow}{0.00000000000000793}=7.93\\times10^{-15}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.\r\n\r\nhttps:\/\/youtu.be\/fsNu3AdIgdk\r\n<h3>Convert from Scientific Notation to Decimal Notation<\/h3>\r\nYou can also write scientific notation as decimal notation. Recall\u00a0the number of miles that light travels in a year is [latex]5.88\\times10^{12}[\/latex], and a hydrogen atom has a diameter of [latex]5\\times10^{-8}[\/latex]\u00a0mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is<i> positive<\/i>, move the decimal point to the <i>right.<\/i> If the exponent is<i> negative<\/i>, move the decimal point to the <i>left.<\/i>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5.88\\times10^{12}=\\underset{\\longrightarrow}{5.880000000000.}=5,880,000,000,000\\\\5\\times10^{-8}=\\underset{\\longleftarrow}{0.00000005.}=0.00000005\\end{array}[\/latex]<\/p>\r\nFor each power of\u00a0[latex]10[\/latex], you move the decimal point one place. Be careful here and do not get carried away with the zeros\u2014the number of zeros after the decimal point will always be\u00a0[latex]1[\/latex] <i>less<\/i> than the exponent because it takes one power of\u00a0[latex]10[\/latex] to shift that first number to the left of the decimal.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite the following in decimal notation.\r\n<ol>\r\n \t<li>[latex]4.8\\times10{-4}[\/latex]<\/li>\r\n \t<li>[latex]3.08\\times10^{6}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"489774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489774\"]\r\n<ol>\r\n \t<li>[latex]4.8\\times10^{-4}[\/latex], the exponent is negative, so we need to move the decimal to the left. \u00a0[latex]\\underset{\\longleftarrow}{4.8\\times10^{-4}}=\\underset{\\longleftarrow}{.00048}[\/latex]<\/li>\r\n \t<li>[latex]3.08\\times10^{6}[\/latex], the exponent is positive, so we need to move the decimal to the right. \u00a0[latex]\\underset{\\longrightarrow}{3.08\\times10^{6}}=\\underset{\\longrightarrow}{3080000}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nTo help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to write your ideas before you reveal the solution.\r\n\r\n1. You are writing a number whose absolute value\u00a0is greater than 1 in scientific notation. \u00a0Will your exponent be positive or negative?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n2.You are writing a number whose absolute value is between 0 and 1 in scientific notation. \u00a0Will your exponent be positive or negative?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n3. What power do you need to put on\u00a0[latex]10[\/latex] to get a result of\u00a0[latex]1[\/latex]?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"824936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824936\"]\r\n<ol>\r\n \t<li><em>You are writing a number whose absolute value is greater than 1 in scientific notation. Will your exponent be positive or negative?<\/em> For numbers greater than\u00a0[latex]1[\/latex], the exponent on\u00a0[latex]10[\/latex] will be positive when you are using scientific notation. Refer to the table presented above:<\/li>\r\n<\/ol>\r\n<table class=\"undefined\">\r\n<thead>\r\n<tr>\r\n<td>Word<\/td>\r\n<td>How many thousands<\/td>\r\n<td>Number<\/td>\r\n<td>Scientific Notation<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>million<\/td>\r\n<td>[latex]1000 x 1000[\/latex] = a thousand thousands<\/td>\r\n<td>[latex]1,000,000[\/latex]<\/td>\r\n<td>\u00a0[latex]10^6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>billion<\/td>\r\n<td>[latex](1000 x 1000) x 1000[\/latex] = a thousand millions<\/td>\r\n<td>[latex]1,000,000,000[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>trillion<\/td>\r\n<td>[latex](1000 x 1000 x 1000) x 1000[\/latex] = a thousand billions<\/td>\r\n<td>[latex]1,000,000,000,000[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n2.\u00a0<em>You are writing a number whose absolute value<\/em>\u00a0<em>is between\u00a0<\/em>[latex]0[\/latex]<em> and <\/em>[latex]1[\/latex]<em> in scientific notation. Will your exponent be positive or negative?<\/em> We can reason that since numbers greater than\u00a0[latex]1[\/latex] will have a positive exponent, numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex] will have a negative exponent. Why are we specifying numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]? The numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex] represent amounts that are fractional. Recall that we defined numbers with a negative exponent as\u00a0[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex], so if we have [latex]10^{-2}[\/latex] we have [latex]\\frac{1}{10\\times10}=\\frac{1}{100}[\/latex] which is a number between 0 and 1.\r\n\r\n3.\u00a0<em>What power do you need to put on\u00a0<\/em>[latex]10[\/latex]<em> to get a result of\u00a0<\/em>[latex]1[\/latex]? Recall\u00a0that any number or variable with an exponent of\u00a0[latex]0[\/latex] is equal to\u00a0[latex]1[\/latex], as in this example:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1\\\\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\\\\text{ therefore }\\\\{t}^{0}=1\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We now have described the notation necessary to write all possible numbers on the number line in scientific notation.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, you will see how to convert a number written in scientific notation into decimal notation.\r\n\r\nhttps:\/\/youtu.be\/8BX0oKUMIjw\r\n<h3>Summary<\/h3>\r\nLarge and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define decimal and scientific notation<\/li>\n<li>Convert between scientific and decimal notation<\/li>\n<\/ul>\n<\/div>\n<h2>Convert Between Scientific and Decimal Notation<\/h2>\n<p>Before we can convert between scientific and decimal notation, we need to know the difference between the two.\u00a0<strong>S<\/strong><b>cientific notation <\/b>is used by\u00a0scientists, mathematicians, and engineers when they are working with very large or very small numbers.\u00a0Using exponential notation, large and small numbers can be written in a way that is easier to read.<\/p>\n<p>When a number is written in scientific notation, the <b>exponent<\/b> tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think\u00a0about it.<\/p>\n<table style=\"width: 668px;\">\n<thead>\n<tr>\n<td style=\"width: 41px;\">Word<\/td>\n<td style=\"width: 320px;\">How many thousands<\/td>\n<td style=\"width: 125px;\">Number<\/td>\n<td style=\"width: 138px;\">Scientific Notation<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 41px;\">million<\/td>\n<td style=\"width: 320px;\">[latex]1000 x 1000[\/latex] = a thousand thousands<\/td>\n<td style=\"width: 125px;\">[latex]1,000,000[\/latex]<\/td>\n<td style=\"width: 138px;\">\u00a0[latex]10^6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 41px;\">billion<\/td>\n<td style=\"width: 320px;\">[latex](1000 x 1000) x 1000[\/latex] = a thousand millions<\/td>\n<td style=\"width: 125px;\">[latex]1,000,000,000[\/latex]<\/td>\n<td style=\"width: 138px;\">\u00a0\u00a0[latex]10^9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 41px;\">trillion<\/td>\n<td style=\"width: 320px;\">[latex](1000 x 1000 x 1000)[\/latex] x 1000 = a thousand billions<\/td>\n<td style=\"width: 125px;\">[latex]1,000,000,000,000[\/latex]<\/td>\n<td style=\"width: 138px;\">\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>1 billion can be written as\u00a0[latex]1,000,000,000[\/latex] or represented as\u00a0[latex]10^9[\/latex]. How would\u00a0[latex]2[\/latex] billion be\u00a0represented? Since\u00a0[latex]2[\/latex] billion is\u00a0[latex]2[\/latex] times\u00a0[latex]1[\/latex]\u00a0billion, then\u00a0[latex]2[\/latex] billion can be written as [latex]2\\times10^9[\/latex].<\/p>\n<p>A\u00a0light year is the number of miles light travels in one year, about\u00a0[latex]5,880,000,000,000[\/latex]. That is a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number.\u00a0Using scientific notation, the distance is [latex]5.88\\times10^{12}[\/latex]\u00a0miles. The exponent of\u00a0[latex]12[\/latex] tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a\u00a0hydrogen atom, which is about 0.00000005 mm, and in scientific notation is\u00a0[latex]5\\times10^{-8}[\/latex]\u00a0mm. In this case, the [latex]-8[\/latex] tells us how many places to count to the right of the decimal.<\/p>\n<p>Outlined in the box below are some important conventions of scientific notation format.<\/p>\n<div class=\"textbox shaded\">\n<h3>Scientific Notation<\/h3>\n<p>A positive number is written in scientific notation if it is written as [latex]a\\times10^{n}[\/latex]\u00a0where the coefficient <i>a<\/i>\u00a0is [latex]1\\leq{a}<10[\/latex], and <i>n <\/i>is an integer.<\/p>\n<\/div>\n<p>Look at the numbers below. Which of the numbers is written in scientific notation?<\/p>\n<table>\n<tbody>\n<tr>\n<td><b>Number<\/b><\/td>\n<td><b>Scientific Notation?<\/b><\/td>\n<td><b>Explanation<\/b><\/td>\n<\/tr>\n<tr>\n<td>[latex]1.85\\times10^{-2}[\/latex]<\/td>\n<td>yes<\/td>\n<td>[latex]1\\leq1.85<10[\/latex]\n\n[latex]-2[\/latex] is an integer<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle 1.083\\times {{10}^{\\frac{1}{2}}}[\/latex]<\/td>\n<td>no<\/td>\n<td>[latex]\\displaystyle \\frac{1}{2}[\/latex] is not an integer<\/td>\n<\/tr>\n<tr>\n<td>[latex]0.82\\times10^{14}[\/latex]<\/td>\n<td>no<\/td>\n<td>\u00a0[latex]0.82[\/latex] is not [latex]\\geq1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]10\\times10^{3}[\/latex]<\/td>\n<td>no<\/td>\n<td>[latex]10[\/latex] is not &lt;[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.<\/p>\n<table style=\"height: 97px; width: 543px;\">\n<tbody>\n<tr style=\"height: 43px;\">\n<td style=\"height: 22px; width: 321.5px;\" colspan=\"2\">\n<p style=\"text-align: center;\"><b>Large Numbers<\/b><\/p>\n<\/td>\n<td style=\"height: 22px; width: 4.5px;\"><b>\u00a0<\/b><\/td>\n<td style=\"height: 22px; width: 321.5px;\" colspan=\"2\">\n<p style=\"text-align: center;\"><b>Small Numbers<\/b><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 172.5px;\"><b>Decimal Notation<\/b><\/td>\n<td style=\"height: 15px; width: 136.5px;\"><b>Scientific Notation<\/b><\/td>\n<td style=\"height: 15px; width: 4.5px;\"><b>\u00a0<\/b><\/td>\n<td style=\"height: 15px; width: 168.5px;\"><b>Decimal Notation<\/b><\/td>\n<td style=\"height: 15px; width: 140.5px;\"><b>Scientific Notation<\/b><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 172.5px;\">[latex]500.0[\/latex]<\/td>\n<td style=\"height: 15px; width: 136.5px;\">[latex]5\\times10^{2}[\/latex]<\/td>\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.05[\/latex]<\/td>\n<td style=\"width: 140.5px; height: 15px;\">[latex]5\\times10^{-2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 172.5px;\">[latex]80,000.0[\/latex]<\/td>\n<td style=\"height: 15px; width: 136.5px;\">[latex]8\\times10^{4}[\/latex]<\/td>\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.0008[\/latex]<\/td>\n<td style=\"width: 140.5px; height: 15px;\">[latex]8\\times10^{-4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 172.5px;\">[latex]43,000,000.0[\/latex]<\/td>\n<td style=\"height: 15px; width: 136.5px;\">[latex]4.3\\times10^{7}[\/latex]<\/td>\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.00000043[\/latex]<\/td>\n<td style=\"width: 140.5px; height: 15px;\">[latex]4.3\\times10^{-7}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.7969px;\">\n<td style=\"height: 15px; width: 172.5px;\">[latex]62,500,000,000.0[\/latex]<\/td>\n<td style=\"height: 15px; width: 136.5px;\">[latex]6.25\\times10^{10}[\/latex]<\/td>\n<td style=\"width: 4.5px; height: 15px;\"><\/td>\n<td style=\"width: 168.5px; height: 15px;\">[latex]0.000000000625[\/latex]<\/td>\n<td style=\"width: 140.5px; height: 15px;\">[latex]6.25\\times10^{-10}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Convert from Decimal Notation to Scientific Notation<\/h3>\n<p>To write a <i>large<\/i> number in scientific notation, move the decimal point to the left to obtain a number between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. Since moving the decimal point changes the value, you have to multiply the decimal by a power of [latex]10[\/latex] so that the expression has the same value.<\/p>\n<p>Let us look at an example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longleftarrow}{180,000.}=18,000.0\\times10^{1}\\\\1,800.00\\times10^{2}\\\\180.000\\times10^{3}\\\\18.0000\\times10^{4}\\\\1.80000\\times10^{5}\\\\180,000=1.8\\times10^{5}\\end{array}[\/latex]<\/p>\n<p>Notice that the decimal point was moved\u00a0[latex]5[\/latex] places to the left, and the exponent is\u00a0[latex]5[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following numbers in scientific notation.<\/p>\n<ol>\n<li>[latex]920,000,000[\/latex]<\/li>\n<li>[latex]10,200,000[\/latex]<\/li>\n<li>[latex]100,000,000,000[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q628\">Show Solution<\/span><\/p>\n<div id=\"q628\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\underset{\\longleftarrow}{920,000,000}[\/latex]\u00a0 \u00a0 \u00a0We will move the decimal point to the left. It helps to place it\u00a0at the end of the number and then count how many times you move it to get one number before it that is between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. Move the decimal point\u00a0[latex]8[\/latex] times to the left and you will have\u00a0[latex]9.20000000[\/latex]. Now we can replace the zeros with an exponent of [latex]8[\/latex] giving us [latex]9.2\\times10^{8}[\/latex].<\/li>\n<li>[latex]\\underset{\\longleftarrow}{10,200,000}=10,200,000.0=1.02\\times10^{7}[\/latex]. Note here how we included the\u00a0[latex]0[\/latex] and the\u00a0[latex]2[\/latex] after the decimal point. In some disciplines, you may learn about when to include both of these. \u00a0Follow instructions from your teacher on rounding rules.<\/li>\n<li>[latex]\\underset{\\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\\times10^{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>To write a small number (between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]) in scientific notation, you move the decimal to the <i>right<\/i> and the exponent will have to be <i>negative,\u00a0<\/i>as in the following example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underset{\\longrightarrow}{0.00004}=00.0004\\times10^{-1}\\\\000.004\\times10^{-2}\\\\0000.04\\times10^{-3}\\\\00000.4\\times10^{-4}\\\\000004.\\times10^{-5}\\\\0.00004=4\\times10^{-5}\\end{array}[\/latex]<\/p>\n<p>You may notice that the decimal point was moved five places to the <i>right <\/i>until you got to the number\u00a0[latex]4[\/latex] which is between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]. The exponent is [latex]\u22125[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following numbers in scientific notation.<\/p>\n<ol>\n<li>[latex]0.0000000000035[\/latex]<\/li>\n<li>[latex]0.0000000102[\/latex]<\/li>\n<li>[latex]0.00000000000000793[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229054\">Show Solution<\/span><\/p>\n<div id=\"q229054\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\underset{\\longrightarrow}{0.0000000000035}=3.5\\times10^{-12}[\/latex]. We moved the decimal 12 times to get to a number between\u00a0[latex]1[\/latex] and\u00a0[latex]10[\/latex]<\/li>\n<li>[latex]\\underset{\\longrightarrow}{0.0000000102}=1.02\\times10^{-8}[\/latex]<\/li>\n<li>[latex]\\underset{\\longrightarrow}{0.00000000000000793}=7.93\\times10^{-15}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you are provided with examples of how to convert both a large and a small number in decimal notation to 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<h3>Convert from Scientific Notation to Decimal Notation<\/h3>\n<p>You can also write scientific notation as decimal notation. Recall\u00a0the number of miles that light travels in a year is [latex]5.88\\times10^{12}[\/latex], and a hydrogen atom has a diameter of [latex]5\\times10^{-8}[\/latex]\u00a0mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is<i> positive<\/i>, move the decimal point to the <i>right.<\/i> If the exponent is<i> negative<\/i>, move the decimal point to the <i>left.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}5.88\\times10^{12}=\\underset{\\longrightarrow}{5.880000000000.}=5,880,000,000,000\\\\5\\times10^{-8}=\\underset{\\longleftarrow}{0.00000005.}=0.00000005\\end{array}[\/latex]<\/p>\n<p>For each power of\u00a0[latex]10[\/latex], you move the decimal point one place. Be careful here and do not get carried away with the zeros\u2014the number of zeros after the decimal point will always be\u00a0[latex]1[\/latex] <i>less<\/i> than the exponent because it takes one power of\u00a0[latex]10[\/latex] to shift that first number to the left of the decimal.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Write the following in decimal notation.<\/p>\n<ol>\n<li>[latex]4.8\\times10{-4}[\/latex]<\/li>\n<li>[latex]3.08\\times10^{6}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q489774\">Show Solution<\/span><\/p>\n<div id=\"q489774\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]4.8\\times10^{-4}[\/latex], the exponent is negative, so we need to move the decimal to the left. \u00a0[latex]\\underset{\\longleftarrow}{4.8\\times10^{-4}}=\\underset{\\longleftarrow}{.00048}[\/latex]<\/li>\n<li>[latex]3.08\\times10^{6}[\/latex], the exponent is positive, so we need to move the decimal to the right. \u00a0[latex]\\underset{\\longrightarrow}{3.08\\times10^{6}}=\\underset{\\longrightarrow}{3080000}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to write your ideas before you reveal the solution.<\/p>\n<p>1. You are writing a number whose absolute value\u00a0is greater than 1 in scientific notation. \u00a0Will your exponent be positive or negative?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>2.You are writing a number whose absolute value is between 0 and 1 in scientific notation. \u00a0Will your exponent be positive or negative?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>3. What power do you need to put on\u00a0[latex]10[\/latex] to get a result of\u00a0[latex]1[\/latex]?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q824936\">Show Solution<\/span><\/p>\n<div id=\"q824936\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li><em>You are writing a number whose absolute value is greater than 1 in scientific notation. Will your exponent be positive or negative?<\/em> For numbers greater than\u00a0[latex]1[\/latex], the exponent on\u00a0[latex]10[\/latex] will be positive when you are using scientific notation. Refer to the table presented above:<\/li>\n<\/ol>\n<table class=\"undefined\">\n<thead>\n<tr>\n<td>Word<\/td>\n<td>How many thousands<\/td>\n<td>Number<\/td>\n<td>Scientific Notation<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>million<\/td>\n<td>[latex]1000 x 1000[\/latex] = a thousand thousands<\/td>\n<td>[latex]1,000,000[\/latex]<\/td>\n<td>\u00a0[latex]10^6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>billion<\/td>\n<td>[latex](1000 x 1000) x 1000[\/latex] = a thousand millions<\/td>\n<td>[latex]1,000,000,000[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]10^9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>trillion<\/td>\n<td>[latex](1000 x 1000 x 1000) x 1000[\/latex] = a thousand billions<\/td>\n<td>[latex]1,000,000,000,000[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]10^{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>2.\u00a0<em>You are writing a number whose absolute value<\/em>\u00a0<em>is between\u00a0<\/em>[latex]0[\/latex]<em> and <\/em>[latex]1[\/latex]<em> in scientific notation. Will your exponent be positive or negative?<\/em> We can reason that since numbers greater than\u00a0[latex]1[\/latex] will have a positive exponent, numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex] will have a negative exponent. Why are we specifying numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex]? The numbers between\u00a0[latex]0[\/latex] and\u00a0[latex]1[\/latex] represent amounts that are fractional. Recall that we defined numbers with a negative exponent as\u00a0[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex], so if we have [latex]10^{-2}[\/latex] we have [latex]\\frac{1}{10\\times10}=\\frac{1}{100}[\/latex] which is a number between 0 and 1.<\/p>\n<p>3.\u00a0<em>What power do you need to put on\u00a0<\/em>[latex]10[\/latex]<em> to get a result of\u00a0<\/em>[latex]1[\/latex]? Recall\u00a0that any number or variable with an exponent of\u00a0[latex]0[\/latex] is equal to\u00a0[latex]1[\/latex], as in this example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1\\\\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\\\\text{ therefore }\\\\{t}^{0}=1\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">We now have described the notation necessary to write all possible numbers on the number line in scientific notation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video, you will see how to convert a number written in scientific notation into decimal notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples:  Writing a Number in Decimal Notation When Given in Scientific Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8BX0oKUMIjw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Summary<\/h3>\n<p>Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.<\/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-794\">\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>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Examples: Write a Number in Scientific Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <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><li>Examples: Writing a Number in Decimal Notation When Given in Scientific Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8BX0oKUMIjw\">https:\/\/youtu.be\/8BX0oKUMIjw<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Write 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