{"id":52,"date":"2017-12-14T21:24:34","date_gmt":"2017-12-14T21:24:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/significant-figures\/"},"modified":"2026-01-30T16:53:30","modified_gmt":"2026-01-30T16:53:30","slug":"significant-figures","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/chapter\/significant-figures\/","title":{"raw":"2.1 Significant Figures","rendered":"2.1 Significant Figures"},"content":{"raw":"<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\r\n<div id=\"ball-ch02_s03_n01\" class=\"learning_objectives editable block\">\r\n<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Apply the concept of significant figures to limit a measurement to the proper number of digits.<\/li>\r\n \t<li>Recognize the number of significant figures in a given quantity.<\/li>\r\n \t<li>Limit mathematical results to the proper number of significant figures.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h2>Uncertainty of Measurement<\/h2>\r\n<p class=\"para editable block\">If you use a calculator to evaluate the expression 337 \u00f7 217, you will get the following:<\/p>\r\n<p style=\"text-align: center;\"><span class=\"informalequation block\">337 \u00f7 217 = 1.55299539171...<\/span><\/p>\r\n<p id=\"ball-ch02_s03_p02\" class=\"para editable block\">and so on for many more digits.\u00a0 Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has <em class=\"emphasis\">twelve<\/em> digits?\u00a0 That does not make much sense from a strict numerical point of view.<\/p>\r\nConsider using a graduated cylinder to measure the volume of a liquid, as shown in Figure 2.1. The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"880\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23210939\/CNX_Chem_01_05_Measure1.jpg\" alt=\"This diagram shows a 25 milliliter graduated cylinder filled with about 20.8 milliliters of fluid. The diagram zooms in on the meniscus, which is the curved surface of the water that is visible when the graduated cylinder is viewed from the side. You make the reading at the lowest point of the curve of the meniscus.\" width=\"880\" height=\"503\" \/> Figure 2.1. To measure the volume of liquid in this graduated cylinder, you must mentally subdivide the distance between the 21 and 22 mL marks into tenths of a milliliter, and then make a reading (estimate) at the bottom of the meniscus.[\/caption]\r\n\r\nRefer to the illustration in Figure 2.1. The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is <em>certainly<\/em> greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid\u2019s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.\r\n\r\nThis concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs <em>about<\/em> 6.72 grams, with a nominal uncertainty in the measurement of \u00b1 0.01 gram. If we weigh the quarter on a more sensitive balance, we may find that its mass is 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some <strong>uncertainty<\/strong>, which depends on the device used (and the user\u2019s ability). All of the digits in a measurement, including the uncertain last digit, are called <strong>significant figures<\/strong> or <strong>significant digits<\/strong>. Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows \u201c120,\u201d then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1: uncertainty of Measurements<\/h3>\r\n<ol>\r\n \t<li>Record the following measurement to the proper number of significant figures<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212412\/151d73b2e1318e4386f1be7579009b82-1.jpg\" alt=\"image\" width=\"349\" height=\"349\" \/><\/li>\r\n \t<li>Record the following measurement to the proper number of significant figures<\/li>\r\n<\/ol>\r\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler.png\"><img class=\"wp-image-4613 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212415\/Ruler-1.png\" alt=\"Ruler\" width=\"369\" height=\"235\" \/><\/a>\r\n\r\n[reveal-answer q=\"367562\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367562\"]\r\n<ol>\r\n \t<li>The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it\u2019s at least 0.3. We will have to estimate the last place. It looks like about one-third of the way across the space, so let us estimate the hundredths place as 3. Combining the digits, we have a measurement of 4.33 psi (psi stands for \u201cpounds per square inch\u201d and is a unit of pressure, like air in a tire). We say that the measurement is reported to three significant figures.<\/li>\r\n \t<li>The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle\u2019s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant digit. We have to estimate the next place because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. Again, the measurement is reported to three significant figures.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nWhat would be the reported width of this rectangle?\r\n<p class=\"para\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Rectangle.png\"><img class=\"wp-image-4615 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212417\/Rectangle-1.png\" alt=\"Rectangle\" width=\"325\" height=\"207\" \/><\/a><\/p>\r\n[reveal-answer q=\"367563\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367563\"]\r\n\r\n0.63 cm\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\r\n<h2>How Many Significant Figures are in a Given Measurement?<\/h2>\r\n<p id=\"ball-ch02_s03_p09\" class=\"para editable block\">In many cases, you will be given a measurement. How do we know what digits are significant?\u00a0 For example, the reported population of the United States is 306,000,000.\u00a0 Does that mean that it is <em class=\"emphasis\">exactly<\/em> three hundred six million or is some estimation occurring? Nonzero digits are always significant figures.\u00a0 However, a zero may or not be significant.<\/p>\r\n<p id=\"ball-ch02_s03_p10\" class=\"para editable block\">The following conventions dictate which numbers in a reported measurement are significant and which are not significant:<\/p>\r\n\r\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 97.4033%; height: 90px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px; text-align: center;\"><strong>Significant Figures in Numbers<\/strong><\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>Examples <\/strong>(significant figures in bold)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px;\">Any nonzero digit is significant.<\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\">\u00a0<strong>13.4<\/strong> (three significant figures)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px;\">Any zeros between nonzero digits (i.e., embedded zeros) are significant<\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>404.1<\/strong> (four significant figures)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px;\">Leading zeros (zeros to the left of the first nonzero digit) are not significant.<\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\">0.0000<strong>21<\/strong> (two significant figures)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px;\">Trailing zeros (zeros at the end of a number) without a decimal point are not significant.<\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>4<\/strong>000 (1 significant figure)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 65.0913%; height: 15px;\">Trailing zeros with a decimal point are significant.<\/td>\r\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>4000.0<\/strong> (5 significant figures)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"ball-ch02_s03_p11\" class=\"para editable block\">So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them.\u00a0 The remaining six zeros simply put the 306 in the millions position.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2: Significant figures in measurements<\/h3>\r\n<p id=\"ball-ch02_s03_p12\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\r\n\r\n<ol id=\"ball-ch02_s03_l05\" class=\"orderedlist\">\r\n \t<li>36.7 m<\/li>\r\n \t<li>0.006606 s<\/li>\r\n \t<li>2,002 kg<\/li>\r\n \t<li>306,490,000 people<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"367564\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367564\"]\r\n<ol id=\"ball-ch02_s03_l06\" class=\"orderedlist\">\r\n \t<li>By rule 1, all nonzero digits are significant, so this measurement has three significant figures.<\/li>\r\n \t<li>By rule 4, the first three zeros are not significant, but by rule 2 the zero between the sixes is; therefore, this number has four significant figures.<\/li>\r\n \t<li>By rule 2, the two zeros between the twos are significant, so this measurement has four significant figures.<\/li>\r\n \t<li>The four trailing zeros in the number are not significant, but the other five numbers are, so this number has five significant figures.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\n<p id=\"ball-ch02_s03_p13\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\r\n\r\n<ol id=\"ball-ch02_s03_l07\" class=\"orderedlist\">\r\n \t<li>0.000601 m<\/li>\r\n \t<li>65.080 kg<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"367565\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"367565\"]\r\n<ol id=\"ball-ch02_s03_l08\" class=\"orderedlist\">\r\n \t<li>three significant figures<\/li>\r\n \t<li>five significant figures<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Significant Figures in Calculations<\/h2>\r\n<p id=\"ball-ch02_s03_p14\" class=\"para editable block\">How are significant figures handled in calculations?\u00a0 It depends on what type of calculation is being performed.\u00a0 If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common.\u00a0 For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column.\u00a0 We therefore limit our answer to the tenths column.<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-1.png\"><img class=\"aligncenter wp-image-4616\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212419\/Sig-Figs-1-1.png\" alt=\"Sig Figs 1\" width=\"631\" height=\"134\" \/><\/a><\/p>\r\n<p class=\"para editable block\">We drop the last digit\u2014the 1\u2014because it is not significant to the final answer.<\/p>\r\n<p id=\"ball-ch02_s03_p16\" class=\"para editable block\">The dropping of positions in sums and differences brings up the topic of rounding.\u00a0 Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-2.png\"><img class=\"wp-image-4617 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212421\/Sig-Figs-2-1.png\" alt=\"Sig Figs 2\" width=\"608\" height=\"129\" \/><\/a><\/p>\r\n\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3: Rounding<\/h3>\r\nRound the following to the indicated number of significant figures:\r\n<ol>\r\n \t<li>31.67 (to two significant figures)<\/li>\r\n \t<li>8.1649 (to three significant figures)<\/li>\r\n \t<li>0.90275 (to four significant figures)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"347710\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"347710\"]\r\n<ol>\r\n \t<li>31.67 rounds \u201cup\u201d to 32 (the dropped digit, 6, is greater than or equal to 5)<\/li>\r\n \t<li>8.1649 rounds \u201cdown\u201d to 8.16 (the dropped digit, 4, is less than 5)<\/li>\r\n \t<li>0.90275 rounds \u201cup\u201d to 0.9028 (the dropped digit, 5, and is greater than or equal to 5)<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4 id=\"fs-idm185983232\"><strong>Check Your Learning<\/strong><\/h4>\r\nRound the following to the indicated number of significant figures:\r\n<ol>\r\n \t<li>0.424 (to two significant figures)<\/li>\r\n \t<li>0.0038661 (to three significant figures)<\/li>\r\n \t<li>421.25 (to four significant figures)<\/li>\r\n \t<li>28,683.5 (to five significant figures)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"776155\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"776155\"]\r\n<ol>\r\n \t<li>0.42<\/li>\r\n \t<li>0.00387<\/li>\r\n \t<li>421.3<\/li>\r\n \t<li>28,684<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4: Addition and Subtraction with Significant Figures<\/h3>\r\nRule: When we add or subtract numbers, we should round the result to the same number of decimal places as the number with the least number of decimal places (i.e., the least precise value in terms of addition and subtraction).\r\n<ol>\r\n \t<li>Add 1.0023 g and 4.383 g.<\/li>\r\n \t<li>Subtract 421.23 g from 486 g.<\/li>\r\n<\/ol>\r\n<p id=\"fs-idm277651872\">[reveal-answer q=\"998865\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"998865\"]<\/p>\r\n\r\n<ol>\r\n \t<li>[latex]\\large\\begin{array}{r}\\\\\\underline{\\begin{array}{rr}{}&amp;1.0023\\text{ g}\\\\+&amp;4.383\\text{ g}\\end{array}}\\\\{}5.3853\\text{ g}\\end{array}[\/latex]\r\nAnswer is 5.385 g (round to the thousandths place; three decimal places)<\/li>\r\n \t<li>[latex]\\large\\begin{array}{r}\\\\\\underline{\\begin{array}{rr}{}&amp;486\\text{ g}\\\\-&amp;421.23\\text{ g}\\end{array}}\\\\{}64.77\\text{ g}\\end{array}[\/latex]\r\nAnswer is 65 g (round to the ones place; no decimal places)<\/li>\r\n<\/ol>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23210947\/CNX_Chem_01_05_SigDigits4_img1.jpg\" alt=\"Figure\u00a0A shows 1.0023 being added to 4.383 to yield the answer 5.385. 1.0023 goes to the ten thousandths place, but 4.383 goes to the thousandths place, making it the less precise of the two numbers. Therefore the answer, 5.3853, should be rounded to the thousandths, to yield 5.385. Figure\u00a0B shows 486 grams minus 421.23 grams, which yields the answer 64.77 grams. This answer should be round to the ones place, making the answer 65 grams.\" width=\"879\" height=\"188\" \/>\r\n\r\n[\/hidden-answer]\r\n<h4 id=\"fs-idm97432976\"><strong>Check Your Learning<\/strong><\/h4>\r\n<ol>\r\n \t<li>Add 2.334 mL and 0.31 mL.<\/li>\r\n \t<li>Subtract 55.8752 m from 56.533 m.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"409370\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"409370\"]\r\n<ol>\r\n \t<li>2.64 mL<\/li>\r\n \t<li>0.658 m<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"ball-ch02_s03_p20\" class=\"para editable block\">If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the <em class=\"emphasis\">least<\/em> number of significant figures has.\u00a0 So if we are dividing 23 by 448, which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures):<\/p>\r\n<p style=\"text-align: center;\"><span class=\"informalequation block\">23 \u00f7 448 = 0.051339286... = 0.051<\/span><\/p>\r\n<p id=\"ball-ch02_s03_p21\" class=\"para editable block\">The same rounding rules apply in multiplication and division as they do in addition and subtraction.<\/p>\r\n\r\n<div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5: Multiplication and Division with Significant Figures<\/h3>\r\nRule: When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant figures (the least precise value in terms of multiplication and division).\r\n<ol>\r\n \t<li>Multiply 0.6238\u00a0 by 6.6<\/li>\r\n \t<li>Divide 421.23 by 486<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"481696\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"481696\"]\r\n<ol>\r\n \t<li>[latex]\\large\\begin{array}{l}\\begin{array}{l}\\text{0.6238}\\times 6.6\\text{ }=4.11708{\\text{ }}\\rightarrow\\text{result is }4.1{\\text{ }}\\left(\\text{round to two significant figures}\\right)\\hfill \\\\ \\text{four significant figures}\\times \\text{two significant figures}\\rightarrow\\text{two significant figures answer}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\large\\begin{array}{l}\\frac{\\text{421.23 }}{\\text{486 }}=\\text{0.86728 }\\rightarrow\\text{result is 0.867 }\\left(\\text{round to three significant figures}\\right)\\\\ \\frac{\\text{five significant figures}}{\\text{three significant figures}}\\rightarrow\\text{three significant figures answer}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n<h4 id=\"fs-idp45405152\">Check Your Learning<\/h4>\r\n<ol>\r\n \t<li>Multiply 2.334 and 0.320<\/li>\r\n \t<li>Divide 55.8752 by 56.53<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"155520\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"155520\"]\r\n<ol>\r\n \t<li>0.747<\/li>\r\n \t<li>0.9884<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>As you have probably realized by now, the biggest issue in determining the number of significant figures in a value is the zero.\u00a0 Is the zero significant or not?\u00a0 One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number <em class=\"emphasis\">only if they are significant<\/em>.\u00a0 Thus, the number 8.666 \u00d7 10<sup class=\"superscript\">6<\/sup> has four significant figures.\u00a0 However, the number 8.6660 \u00d7 10<sup class=\"superscript\">6<\/sup> has five significant figures.\u00a0 That last zero is significant; if it were not, it would not be written in the coefficient.\u00a0 So when in doubt about expressing the number of significant figures in a quantity, use scientific notation and include the number of zeros that are truly significant.<\/div>\r\n<\/div>\r\n<\/div>\r\nVideo source: Significant figures by keyj (<a href=\"https:\/\/viuvideos.viu.ca\/media\/Significant+Figures\/0_t8xwe4s9\">https:\/\/viuvideos.viu.ca\/media\/Significant+Figures\/0_t8xwe4s9<\/a>)\r\n<div id=\"ball-ch02_s03_qs01\" class=\"qandaset block\">\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Key Takeaways<\/h3>\r\n<ul id=\"ball-ch02_s03_l15\" class=\"itemizedlist\">\r\n \t<li>Significant figures in a quantity indicate the number of known values plus one place that is estimated.<\/li>\r\n \t<li>There are rules for which numbers in a quantity are significant and which are not significant.<\/li>\r\n \t<li>In calculations involving addition and subtraction, limit significant figures based on the rightmost place that all values have in common.<\/li>\r\n \t<li>In calculations involving multiplication and division, limit significant figures to the least number of significant figures in all the data values.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Exercises<\/h3>\r\n<div class=\"question\">\r\n\r\n\u00a01. Express each measurement to the correct number of significant figures.\r\n<p class=\"para\" style=\"padding-left: 30px;\">a)<\/p>\r\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212426\/f85b7d0b1d3f3563a5b973ef04349df3-1.jpg\" alt=\"image\" width=\"383\" height=\"383\" \/><\/p>\r\n<p id=\"ball-ch02_s03_qs01_p2\" class=\"para\" style=\"padding-left: 30px;\">b)<\/p>\r\n<p class=\"para\" style=\"padding-left: 30px;\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-2.png\"><img class=\"alignnone wp-image-4618\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212428\/Ruler-2-1.png\" alt=\"Ruler-2\" width=\"400\" height=\"255\" \/><\/a><\/p>\r\n<p class=\"para\">2.\u00a0 Express each measurement to the correct number of significant figures.<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p class=\"para\" style=\"padding-left: 30px;\">a)<\/p>\r\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212431\/06979677f6f13c11ea559e495ebcbf85-1.jpg\" alt=\"image\" width=\"390\" height=\"390\" \/><\/p>\r\n<p class=\"para\" style=\"padding-left: 30px;\">b)<\/p>\r\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0\u00a0<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-3.png\"><img class=\"alignnone wp-image-4619\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212433\/Ruler-3-1.png\" alt=\"Ruler-3\" width=\"400\" height=\"255\" \/><\/a><\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p3\" class=\"para\">3.\u00a0 How many significant figures do these numbers have?<\/p>\r\n\r\n<\/div>\r\n<p style=\"padding-left: 40px;\">a) \u00a023<\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a023.0<\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a00.00023<\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a00.0002302<\/p>\r\n4. \u00a0How many significant figures do these numbers have?\r\n<p style=\"padding-left: 40px;\">a) \u00a05.44 \u00d7 10<sup class=\"superscript\">8<\/sup><\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a01.008 \u00d7 10<sup class=\"superscript\">\u22125<\/sup><\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a043.00<\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a00.00000013810<\/p>\r\n5. \u00a0How many significant figures do these numbers have?\r\n<p style=\"padding-left: 40px;\">a) \u00a0765,890<\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a0765,890.0<\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a01.2000 \u00d7 10<sup class=\"superscript\">5<\/sup><\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a00.0005060<\/p>\r\n\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p6\" class=\"para\">6) \u00a0How many significant figures do these numbers have?<\/p>\r\n<p style=\"padding-left: 40px;\">a) \u00a00.009<\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a00.0000009<\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a065,444<\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a065,040<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p7\" class=\"para\">7. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\r\n<p style=\"padding-left: 40px;\">a) \u00a056.0 +\u00a03.44 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a00.00665 +\u00a01.004 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a045.99 \u2212 32.8 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a045.99 \u2212 32.8 +\u00a075.02 = ?<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p8\" class=\"para\">8. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">a) \u00a01.005 +\u00a017.88 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">b) \u00a056,700 \u2212 324 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">c) \u00a0405,007 \u2212 123.3 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">d) \u00a055.5 +\u00a066.66 \u2212 77.777 = ?<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p9\" class=\"para\">9. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">a) \u00a056.7 \u00d7 66.99 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">b) \u00a01.000 \u00f7 77 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">c) \u00a01.000 \u00f7 77.0 = ?<\/p>\r\n<p class=\"para\" style=\"padding-left: 40px;\">d) \u00a06.022 \u00d7 1.89 = ?<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">10. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\r\n<p style=\"padding-left: 40px;\">a) \u00a00.000440 \u00d7 17.22 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">b) \u00a0203,000 \u00f7 0.044 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">c) \u00a067 \u00d7 85.0 \u00d7 0.0028 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">d) \u00a0999,999 \u00f7 3,310 = ?<\/p>\r\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">\u00a011. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\r\n<p style=\"padding-left: 40px;\">a) (12.4 \u2212 3.16) \u00f7 0.804 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">b)\u00a0 (45.2 \u2212 3.18) \u00d7 2.5 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">c)\u00a0 (15.60 \u00f7 4.00) \u2212 0.872 = ?<\/p>\r\n<p style=\"padding-left: 40px;\">d)\u00a0 (9.84 + 0.0065) \u00f7 3.205 = ?<\/p>\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n\r\n12. \u00a0Write the number 87,449 in scientific notation with four significant figures.\r\n\r\n13. \u00a0Write the number 0.000066600 in scientific notation with five significant figures.\r\n\r\n<\/div>\r\n<div class=\"question\">\r\n\r\n14. \u00a0Write the number 306,000,000 in scientific notation to the proper number of significant figures.\r\n\r\n15. \u00a0Write the number 0.0000558 in scientific notation with two significant figures.\r\n\r\n<\/div>\r\n[reveal-answer q=\"367568\"]Show Answers to Selected Questions[\/reveal-answer]\r\n[hidden-answer a=\"367568\"]\r\n\r\n1. a) 375 psi; b) 1.30 cm\r\n\r\n2. a) 32.5 psi; b) 0.90 cm\r\n\r\n3. a) \u00a0two; b) \u00a0three; c) \u00a0two; d) \u00a0four\r\n\r\n4. a) three; b) four; c) four; d) five\r\n\r\n5. a) \u00a0five or ambiguous; b) \u00a0seven; c) \u00a0five; d) \u00a0four\r\n\r\n6. a) one; b) one; c) five; d) four or ambiguous\r\n\r\n7. a) \u00a059.4; b) 1.011; c) 13.2; d) 88.2\r\n\r\n8. a) 18.89; b) 56376; c) 404884; d) 44.4\r\n\r\n9. a) \u00a03.80 \u00d7 10<sup class=\"superscript\">3<\/sup>; b) \u00a00.013; c) \u00a00.0130; d) \u00a011.4\r\n\r\n10. a) 0.00758 or 7.58 \u00d7 10<sup class=\"superscript\">\u2212<\/sup><sup class=\"superscript\">3<\/sup> ; b) 4.6 \u00d7 10<sup class=\"superscript\">5<\/sup>; c) 16; d) 302.1 or 3.021 \u00d7 10<sup class=\"superscript\">2<\/sup>\r\n\r\n11. a) 11; b) 1.1 \u00d7 10<sup class=\"superscript\">2<\/sup>; c) 3.03; d) 3.07\r\n\r\n12. 8.745 \u00d7 10<sup class=\"superscript\">4<\/sup>\r\n\r\n13. 6.6600 \u00d7 10<sup class=\"superscript\">\u22125<\/sup>\r\n\r\n14. 3.06 \u00d7 10<sup class=\"superscript\">8<\/sup>\r\n\r\n15. 5.6 \u00d7 10<sup class=\"superscript\">\u22125<\/sup>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>exact number:\u00a0<\/strong>number derived by counting or by definition\r\n\r\n<strong>precision:\u00a0<\/strong>how closely a measurement matches the same measurement when repeated\r\n\r\n<strong>rounding:\u00a0<\/strong>procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation\r\n\r\n<strong>significant figures:\u00a0<\/strong>(also, significant digits) all of the measured digits in a determination, including the uncertain last digit\r\n\r\n<strong>uncertainty:\u00a0<\/strong>estimate of amount by which measurement differs from true value\r\n\r\n&nbsp;","rendered":"<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\n<div id=\"ball-ch02_s03_n01\" class=\"learning_objectives editable block\">\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Apply the concept of significant figures to limit a measurement to the proper number of digits.<\/li>\n<li>Recognize the number of significant figures in a given quantity.<\/li>\n<li>Limit mathematical results to the proper number of significant figures.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Uncertainty of Measurement<\/h2>\n<p class=\"para editable block\">If you use a calculator to evaluate the expression 337 \u00f7 217, you will get the following:<\/p>\n<p style=\"text-align: center;\"><span class=\"informalequation block\">337 \u00f7 217 = 1.55299539171&#8230;<\/span><\/p>\n<p id=\"ball-ch02_s03_p02\" class=\"para editable block\">and so on for many more digits.\u00a0 Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has <em class=\"emphasis\">twelve<\/em> digits?\u00a0 That does not make much sense from a strict numerical point of view.<\/p>\n<p>Consider using a graduated cylinder to measure the volume of a liquid, as shown in Figure 2.1. The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.<\/p>\n<div style=\"width: 890px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23210939\/CNX_Chem_01_05_Measure1.jpg\" alt=\"This diagram shows a 25 milliliter graduated cylinder filled with about 20.8 milliliters of fluid. The diagram zooms in on the meniscus, which is the curved surface of the water that is visible when the graduated cylinder is viewed from the side. You make the reading at the lowest point of the curve of the meniscus.\" width=\"880\" height=\"503\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2.1. To measure the volume of liquid in this graduated cylinder, you must mentally subdivide the distance between the 21 and 22 mL marks into tenths of a milliliter, and then make a reading (estimate) at the bottom of the meniscus.<\/p>\n<\/div>\n<p>Refer to the illustration in Figure 2.1. The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is <em>certainly<\/em> greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid\u2019s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.<\/p>\n<p>This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs <em>about<\/em> 6.72 grams, with a nominal uncertainty in the measurement of \u00b1 0.01 gram. If we weigh the quarter on a more sensitive balance, we may find that its mass is 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some <strong>uncertainty<\/strong>, which depends on the device used (and the user\u2019s ability). All of the digits in a measurement, including the uncertain last digit, are called <strong>significant figures<\/strong> or <strong>significant digits<\/strong>. Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows \u201c120,\u201d then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1: uncertainty of Measurements<\/h3>\n<ol>\n<li>Record the following measurement to the proper number of significant figures<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212412\/151d73b2e1318e4386f1be7579009b82-1.jpg\" alt=\"image\" width=\"349\" height=\"349\" \/><\/li>\n<li>Record the following measurement to the proper number of significant figures<\/li>\n<\/ol>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4613 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212415\/Ruler-1.png\" alt=\"Ruler\" width=\"369\" height=\"235\" \/><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367562\">Show Answer<\/span><\/p>\n<div id=\"q367562\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it\u2019s at least 0.3. We will have to estimate the last place. It looks like about one-third of the way across the space, so let us estimate the hundredths place as 3. Combining the digits, we have a measurement of 4.33 psi (psi stands for \u201cpounds per square inch\u201d and is a unit of pressure, like air in a tire). We say that the measurement is reported to three significant figures.<\/li>\n<li>The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle\u2019s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant digit. We have to estimate the next place because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. Again, the measurement is reported to three significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>What would be the reported width of this rectangle?<\/p>\n<p class=\"para\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Rectangle.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4615 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212417\/Rectangle-1.png\" alt=\"Rectangle\" width=\"325\" height=\"207\" \/><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367563\">Show Answer<\/span><\/p>\n<div id=\"q367563\" class=\"hidden-answer\" style=\"display: none\">\n<p>0.63 cm<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\n<h2>How Many Significant Figures are in a Given Measurement?<\/h2>\n<p id=\"ball-ch02_s03_p09\" class=\"para editable block\">In many cases, you will be given a measurement. How do we know what digits are significant?\u00a0 For example, the reported population of the United States is 306,000,000.\u00a0 Does that mean that it is <em class=\"emphasis\">exactly<\/em> three hundred six million or is some estimation occurring? Nonzero digits are always significant figures.\u00a0 However, a zero may or not be significant.<\/p>\n<p id=\"ball-ch02_s03_p10\" class=\"para editable block\">The following conventions dictate which numbers in a reported measurement are significant and which are not significant:<\/p>\n<table class=\"no-lines\" style=\"border-collapse: collapse; width: 97.4033%; height: 90px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px; text-align: center;\"><strong>Significant Figures in Numbers<\/strong><\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>Examples <\/strong>(significant figures in bold)<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px;\">Any nonzero digit is significant.<\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\">\u00a0<strong>13.4<\/strong> (three significant figures)<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px;\">Any zeros between nonzero digits (i.e., embedded zeros) are significant<\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>404.1<\/strong> (four significant figures)<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px;\">Leading zeros (zeros to the left of the first nonzero digit) are not significant.<\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\">0.0000<strong>21<\/strong> (two significant figures)<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px;\">Trailing zeros (zeros at the end of a number) without a decimal point are not significant.<\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>4<\/strong>000 (1 significant figure)<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 65.0913%; height: 15px;\">Trailing zeros with a decimal point are significant.<\/td>\n<td style=\"width: 85.7147%; height: 15px; text-align: center;\"><strong>4000.0<\/strong> (5 significant figures)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"ball-ch02_s03_p11\" class=\"para editable block\">So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them.\u00a0 The remaining six zeros simply put the 306 in the millions position.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2: Significant figures in measurements<\/h3>\n<p id=\"ball-ch02_s03_p12\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n<ol id=\"ball-ch02_s03_l05\" class=\"orderedlist\">\n<li>36.7 m<\/li>\n<li>0.006606 s<\/li>\n<li>2,002 kg<\/li>\n<li>306,490,000 people<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367564\">Show Answer<\/span><\/p>\n<div id=\"q367564\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"ball-ch02_s03_l06\" class=\"orderedlist\">\n<li>By rule 1, all nonzero digits are significant, so this measurement has three significant figures.<\/li>\n<li>By rule 4, the first three zeros are not significant, but by rule 2 the zero between the sixes is; therefore, this number has four significant figures.<\/li>\n<li>By rule 2, the two zeros between the twos are significant, so this measurement has four significant figures.<\/li>\n<li>The four trailing zeros in the number are not significant, but the other five numbers are, so this number has five significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p id=\"ball-ch02_s03_p13\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n<ol id=\"ball-ch02_s03_l07\" class=\"orderedlist\">\n<li>0.000601 m<\/li>\n<li>65.080 kg<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367565\">Show Answer<\/span><\/p>\n<div id=\"q367565\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"ball-ch02_s03_l08\" class=\"orderedlist\">\n<li>three significant figures<\/li>\n<li>five significant figures<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Significant Figures in Calculations<\/h2>\n<p id=\"ball-ch02_s03_p14\" class=\"para editable block\">How are significant figures handled in calculations?\u00a0 It depends on what type of calculation is being performed.\u00a0 If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common.\u00a0 For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column.\u00a0 We therefore limit our answer to the tenths column.<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4616\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212419\/Sig-Figs-1-1.png\" alt=\"Sig Figs 1\" width=\"631\" height=\"134\" \/><\/a><\/p>\n<p class=\"para editable block\">We drop the last digit\u2014the 1\u2014because it is not significant to the final answer.<\/p>\n<p id=\"ball-ch02_s03_p16\" class=\"para editable block\">The dropping of positions in sums and differences brings up the topic of rounding.\u00a0 Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4617 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212421\/Sig-Figs-2-1.png\" alt=\"Sig Figs 2\" width=\"608\" height=\"129\" \/><\/a><\/p>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 3: Rounding<\/h3>\n<p>Round the following to the indicated number of significant figures:<\/p>\n<ol>\n<li>31.67 (to two significant figures)<\/li>\n<li>8.1649 (to three significant figures)<\/li>\n<li>0.90275 (to four significant figures)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q347710\">Show Answer<\/span><\/p>\n<div id=\"q347710\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>31.67 rounds \u201cup\u201d to 32 (the dropped digit, 6, is greater than or equal to 5)<\/li>\n<li>8.1649 rounds \u201cdown\u201d to 8.16 (the dropped digit, 4, is less than 5)<\/li>\n<li>0.90275 rounds \u201cup\u201d to 0.9028 (the dropped digit, 5, and is greater than or equal to 5)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4 id=\"fs-idm185983232\"><strong>Check Your Learning<\/strong><\/h4>\n<p>Round the following to the indicated number of significant figures:<\/p>\n<ol>\n<li>0.424 (to two significant figures)<\/li>\n<li>0.0038661 (to three significant figures)<\/li>\n<li>421.25 (to four significant figures)<\/li>\n<li>28,683.5 (to five significant figures)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q776155\">Show Answer<\/span><\/p>\n<div id=\"q776155\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>0.42<\/li>\n<li>0.00387<\/li>\n<li>421.3<\/li>\n<li>28,684<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 4: Addition and Subtraction with Significant Figures<\/h3>\n<p>Rule: When we add or subtract numbers, we should round the result to the same number of decimal places as the number with the least number of decimal places (i.e., the least precise value in terms of addition and subtraction).<\/p>\n<ol>\n<li>Add 1.0023 g and 4.383 g.<\/li>\n<li>Subtract 421.23 g from 486 g.<\/li>\n<\/ol>\n<p id=\"fs-idm277651872\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q998865\">Show Answer<\/span><\/p>\n<div id=\"q998865\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\large\\begin{array}{r}\\\\\\underline{\\begin{array}{rr}{}&1.0023\\text{ g}\\\\+&4.383\\text{ g}\\end{array}}\\\\{}5.3853\\text{ g}\\end{array}[\/latex]<br \/>\nAnswer is 5.385 g (round to the thousandths place; three decimal places)<\/li>\n<li>[latex]\\large\\begin{array}{r}\\\\\\underline{\\begin{array}{rr}{}&486\\text{ g}\\\\-&421.23\\text{ g}\\end{array}}\\\\{}64.77\\text{ g}\\end{array}[\/latex]<br \/>\nAnswer is 65 g (round to the ones place; no decimal places)<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/887\/2015\/04\/23210947\/CNX_Chem_01_05_SigDigits4_img1.jpg\" alt=\"Figure\u00a0A shows 1.0023 being added to 4.383 to yield the answer 5.385. 1.0023 goes to the ten thousandths place, but 4.383 goes to the thousandths place, making it the less precise of the two numbers. Therefore the answer, 5.3853, should be rounded to the thousandths, to yield 5.385. Figure\u00a0B shows 486 grams minus 421.23 grams, which yields the answer 64.77 grams. This answer should be round to the ones place, making the answer 65 grams.\" width=\"879\" height=\"188\" \/><\/p>\n<\/div>\n<\/div>\n<h4 id=\"fs-idm97432976\"><strong>Check Your Learning<\/strong><\/h4>\n<ol>\n<li>Add 2.334 mL and 0.31 mL.<\/li>\n<li>Subtract 55.8752 m from 56.533 m.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q409370\">Show Answer<\/span><\/p>\n<div id=\"q409370\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>2.64 mL<\/li>\n<li>0.658 m<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"ball-ch02_s03_p20\" class=\"para editable block\">If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the <em class=\"emphasis\">least<\/em> number of significant figures has.\u00a0 So if we are dividing 23 by 448, which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures):<\/p>\n<p style=\"text-align: center;\"><span class=\"informalequation block\">23 \u00f7 448 = 0.051339286&#8230; = 0.051<\/span><\/p>\n<p id=\"ball-ch02_s03_p21\" class=\"para editable block\">The same rounding rules apply in multiplication and division as they do in addition and subtraction.<\/p>\n<div>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 5: Multiplication and Division with Significant Figures<\/h3>\n<p>Rule: When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant figures (the least precise value in terms of multiplication and division).<\/p>\n<ol>\n<li>Multiply 0.6238\u00a0 by 6.6<\/li>\n<li>Divide 421.23 by 486<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q481696\">Show Answer<\/span><\/p>\n<div id=\"q481696\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\large\\begin{array}{l}\\begin{array}{l}\\text{0.6238}\\times 6.6\\text{ }=4.11708{\\text{ }}\\rightarrow\\text{result is }4.1{\\text{ }}\\left(\\text{round to two significant figures}\\right)\\hfill \\\\ \\text{four significant figures}\\times \\text{two significant figures}\\rightarrow\\text{two significant figures answer}\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\large\\begin{array}{l}\\frac{\\text{421.23 }}{\\text{486 }}=\\text{0.86728 }\\rightarrow\\text{result is 0.867 }\\left(\\text{round to three significant figures}\\right)\\\\ \\frac{\\text{five significant figures}}{\\text{three significant figures}}\\rightarrow\\text{three significant figures answer}\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h4 id=\"fs-idp45405152\">Check Your Learning<\/h4>\n<ol>\n<li>Multiply 2.334 and 0.320<\/li>\n<li>Divide 55.8752 by 56.53<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155520\">Show Answer<\/span><\/p>\n<div id=\"q155520\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>0.747<\/li>\n<li>0.9884<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div>As you have probably realized by now, the biggest issue in determining the number of significant figures in a value is the zero.\u00a0 Is the zero significant or not?\u00a0 One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number <em class=\"emphasis\">only if they are significant<\/em>.\u00a0 Thus, the number 8.666 \u00d7 10<sup class=\"superscript\">6<\/sup> has four significant figures.\u00a0 However, the number 8.6660 \u00d7 10<sup class=\"superscript\">6<\/sup> has five significant figures.\u00a0 That last zero is significant; if it were not, it would not be written in the coefficient.\u00a0 So when in doubt about expressing the number of significant figures in a quantity, use scientific notation and include the number of zeros that are truly significant.<\/div>\n<\/div>\n<\/div>\n<p>Video source: Significant figures by keyj (<a href=\"https:\/\/viuvideos.viu.ca\/media\/Significant+Figures\/0_t8xwe4s9\">https:\/\/viuvideos.viu.ca\/media\/Significant+Figures\/0_t8xwe4s9<\/a>)<\/p>\n<div id=\"ball-ch02_s03_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s03_l15\" class=\"itemizedlist\">\n<li>Significant figures in a quantity indicate the number of known values plus one place that is estimated.<\/li>\n<li>There are rules for which numbers in a quantity are significant and which are not significant.<\/li>\n<li>In calculations involving addition and subtraction, limit significant figures based on the rightmost place that all values have in common.<\/li>\n<li>In calculations involving multiplication and division, limit significant figures to the least number of significant figures in all the data values.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Exercises<\/h3>\n<div class=\"question\">\n<p>\u00a01. Express each measurement to the correct number of significant figures.<\/p>\n<p class=\"para\" style=\"padding-left: 30px;\">a)<\/p>\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212426\/f85b7d0b1d3f3563a5b973ef04349df3-1.jpg\" alt=\"image\" width=\"383\" height=\"383\" \/><\/p>\n<p id=\"ball-ch02_s03_qs01_p2\" class=\"para\" style=\"padding-left: 30px;\">b)<\/p>\n<p class=\"para\" style=\"padding-left: 30px;\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4618\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212428\/Ruler-2-1.png\" alt=\"Ruler-2\" width=\"400\" height=\"255\" \/><\/a><\/p>\n<p class=\"para\">2.\u00a0 Express each measurement to the correct number of significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p class=\"para\" style=\"padding-left: 30px;\">a)<\/p>\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212431\/06979677f6f13c11ea559e495ebcbf85-1.jpg\" alt=\"image\" width=\"390\" height=\"390\" \/><\/p>\n<p class=\"para\" style=\"padding-left: 30px;\">b)<\/p>\n<p class=\"para\" style=\"padding-left: 30px;\">\u00a0\u00a0<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-4619\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2835\/2017\/12\/14212433\/Ruler-3-1.png\" alt=\"Ruler-3\" width=\"400\" height=\"255\" \/><\/a><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p3\" class=\"para\">3.\u00a0 How many significant figures do these numbers have?<\/p>\n<\/div>\n<p style=\"padding-left: 40px;\">a) \u00a023<\/p>\n<p style=\"padding-left: 40px;\">b) \u00a023.0<\/p>\n<p style=\"padding-left: 40px;\">c) \u00a00.00023<\/p>\n<p style=\"padding-left: 40px;\">d) \u00a00.0002302<\/p>\n<p>4. \u00a0How many significant figures do these numbers have?<\/p>\n<p style=\"padding-left: 40px;\">a) \u00a05.44 \u00d7 10<sup class=\"superscript\">8<\/sup><\/p>\n<p style=\"padding-left: 40px;\">b) \u00a01.008 \u00d7 10<sup class=\"superscript\">\u22125<\/sup><\/p>\n<p style=\"padding-left: 40px;\">c) \u00a043.00<\/p>\n<p style=\"padding-left: 40px;\">d) \u00a00.00000013810<\/p>\n<p>5. \u00a0How many significant figures do these numbers have?<\/p>\n<p style=\"padding-left: 40px;\">a) \u00a0765,890<\/p>\n<p style=\"padding-left: 40px;\">b) \u00a0765,890.0<\/p>\n<p style=\"padding-left: 40px;\">c) \u00a01.2000 \u00d7 10<sup class=\"superscript\">5<\/sup><\/p>\n<p style=\"padding-left: 40px;\">d) \u00a00.0005060<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p6\" class=\"para\">6) \u00a0How many significant figures do these numbers have?<\/p>\n<p style=\"padding-left: 40px;\">a) \u00a00.009<\/p>\n<p style=\"padding-left: 40px;\">b) \u00a00.0000009<\/p>\n<p style=\"padding-left: 40px;\">c) \u00a065,444<\/p>\n<p style=\"padding-left: 40px;\">d) \u00a065,040<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p7\" class=\"para\">7. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p style=\"padding-left: 40px;\">a) \u00a056.0 +\u00a03.44 = ?<\/p>\n<p style=\"padding-left: 40px;\">b) \u00a00.00665 +\u00a01.004 = ?<\/p>\n<p style=\"padding-left: 40px;\">c) \u00a045.99 \u2212 32.8 = ?<\/p>\n<p style=\"padding-left: 40px;\">d) \u00a045.99 \u2212 32.8 +\u00a075.02 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p8\" class=\"para\">8. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">a) \u00a01.005 +\u00a017.88 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">b) \u00a056,700 \u2212 324 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">c) \u00a0405,007 \u2212 123.3 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">d) \u00a055.5 +\u00a066.66 \u2212 77.777 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p9\" class=\"para\">9. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">a) \u00a056.7 \u00d7 66.99 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">b) \u00a01.000 \u00f7 77 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">c) \u00a01.000 \u00f7 77.0 = ?<\/p>\n<p class=\"para\" style=\"padding-left: 40px;\">d) \u00a06.022 \u00d7 1.89 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">10. \u00a0Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p style=\"padding-left: 40px;\">a) \u00a00.000440 \u00d7 17.22 = ?<\/p>\n<p style=\"padding-left: 40px;\">b) \u00a0203,000 \u00f7 0.044 = ?<\/p>\n<p style=\"padding-left: 40px;\">c) \u00a067 \u00d7 85.0 \u00d7 0.0028 = ?<\/p>\n<p style=\"padding-left: 40px;\">d) \u00a0999,999 \u00f7 3,310 = ?<\/p>\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">\u00a011. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p style=\"padding-left: 40px;\">a) (12.4 \u2212 3.16) \u00f7 0.804 = ?<\/p>\n<p style=\"padding-left: 40px;\">b)\u00a0 (45.2 \u2212 3.18) \u00d7 2.5 = ?<\/p>\n<p style=\"padding-left: 40px;\">c)\u00a0 (15.60 \u00f7 4.00) \u2212 0.872 = ?<\/p>\n<p style=\"padding-left: 40px;\">d)\u00a0 (9.84 + 0.0065) \u00f7 3.205 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p>12. \u00a0Write the number 87,449 in scientific notation with four significant figures.<\/p>\n<p>13. \u00a0Write the number 0.000066600 in scientific notation with five significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p>14. \u00a0Write the number 306,000,000 in scientific notation to the proper number of significant figures.<\/p>\n<p>15. \u00a0Write the number 0.0000558 in scientific notation with two significant figures.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367568\">Show Answers to Selected Questions<\/span><\/p>\n<div id=\"q367568\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. a) 375 psi; b) 1.30 cm<\/p>\n<p>2. a) 32.5 psi; b) 0.90 cm<\/p>\n<p>3. a) \u00a0two; b) \u00a0three; c) \u00a0two; d) \u00a0four<\/p>\n<p>4. a) three; b) four; c) four; d) five<\/p>\n<p>5. a) \u00a0five or ambiguous; b) \u00a0seven; c) \u00a0five; d) \u00a0four<\/p>\n<p>6. a) one; b) one; c) five; d) four or ambiguous<\/p>\n<p>7. a) \u00a059.4; b) 1.011; c) 13.2; d) 88.2<\/p>\n<p>8. a) 18.89; b) 56376; c) 404884; d) 44.4<\/p>\n<p>9. a) \u00a03.80 \u00d7 10<sup class=\"superscript\">3<\/sup>; b) \u00a00.013; c) \u00a00.0130; d) \u00a011.4<\/p>\n<p>10. a) 0.00758 or 7.58 \u00d7 10<sup class=\"superscript\">\u2212<\/sup><sup class=\"superscript\">3<\/sup> ; b) 4.6 \u00d7 10<sup class=\"superscript\">5<\/sup>; c) 16; d) 302.1 or 3.021 \u00d7 10<sup class=\"superscript\">2<\/sup><\/p>\n<p>11. a) 11; b) 1.1 \u00d7 10<sup class=\"superscript\">2<\/sup>; c) 3.03; d) 3.07<\/p>\n<p>12. 8.745 \u00d7 10<sup class=\"superscript\">4<\/sup><\/p>\n<p>13. 6.6600 \u00d7 10<sup class=\"superscript\">\u22125<\/sup><\/p>\n<p>14. 3.06 \u00d7 10<sup class=\"superscript\">8<\/sup><\/p>\n<p>15. 5.6 \u00d7 10<sup class=\"superscript\">\u22125<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Glossary<\/h2>\n<p><strong>exact number:\u00a0<\/strong>number derived by counting or by definition<\/p>\n<p><strong>precision:\u00a0<\/strong>how closely a measurement matches the same measurement when repeated<\/p>\n<p><strong>rounding:\u00a0<\/strong>procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation<\/p>\n<p><strong>significant figures:\u00a0<\/strong>(also, significant digits) all of the measured digits in a determination, including the uncertain last digit<\/p>\n<p><strong>uncertainty:\u00a0<\/strong>estimate of amount by which measurement differs from true value<\/p>\n<p>&nbsp;<\/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-52\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Chemistry- 1st Canadian Edition . <strong>Authored by<\/strong>: Jessie A. Key and David W. Ball. <strong>Provided by<\/strong>: BCCampus. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/opentextbc.ca\/introductorychemistry\/\">https:\/\/opentextbc.ca\/introductorychemistry\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Download this book for free at http:\/\/open.bccampus.ca<\/li><li>Chemistry. <strong>Provided by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/openstaxcollege.org\">http:\/\/openstaxcollege.org<\/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>: Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get<\/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":23485,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Chemistry- 1st Canadian Edition \",\"author\":\"Jessie A. Key and David W. Ball\",\"organization\":\"BCCampus\",\"url\":\"https:\/\/opentextbc.ca\/introductorychemistry\/\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Download this book for free at http:\/\/open.bccampus.ca\"},{\"type\":\"cc\",\"description\":\"Chemistry\",\"author\":\"\",\"organization\":\"OpenStax College\",\"url\":\"http:\/\/openstaxcollege.org\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-52","chapter","type-chapter","status-publish","hentry"],"part":36,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/52","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":41,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/52\/revisions"}],"predecessor-version":[{"id":1232,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/52\/revisions\/1232"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/52\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/media?parent=52"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=52"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/contributor?post=52"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-mcc-introductorychemistry\/wp-json\/wp\/v2\/license?post=52"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}