{"id":43,"date":"2015-07-30T13:32:12","date_gmt":"2015-07-30T17:32:12","guid":{"rendered":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/sigfigs\/physics-1140-notes-on-significant-figures\/"},"modified":"2016-08-23T14:17:11","modified_gmt":"2016-08-23T18:17:11","slug":"physics-1140-notes-on-significant-figures","status":"publish","type":"page","link":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/physics-1140-notes-on-significant-figures\/","title":{"rendered":"Notes on Significant Figures"},"content":{"rendered":"<p>Significant figures (also called significant digits) are <em>very important<\/em> because they imply the known precision of a number.<\/p>\n<h2>How to Count Significant Figures<\/h2>\n<ul class=\"indentedh2\">\n<li>Non-zero digits are always significant.<\/li>\n<li>Any zeros between two significant figures are significant.<\/li>\n<li>Trailing zeros in the decimal portion <em>only<\/em> are significant.<\/li>\n<\/ul>\n<h3>Examples:<\/h3>\n<ul class=\"indentedh3\">\n<li>\\(4308\\) has four significant figures.<\/li>\n<li>\\(40.05\\) has four significant figures.<\/li>\n<li>\\(470,000\\) has two significant figures.<\/li>\n<li>\\(4.00\\) has three significant figures.<\/li>\n<li>\\(0.0004\\) has one significant figure.<\/li>\n<\/ul>\n<p class=\"indentedh2\">Counting, and using, significant figures correctly becomes very easy if one uses <em>scientific notation<\/em>. Then, the numbers before the &#8220;\\(\\times 10^{\\rm{n}}\\)&#8221; are all significant. This also eliminates the vagueness of whether, for example, \\(400\\) has one, two, or three significant figures.<\/p>\n<h3>Examples with Scientific Notation:<\/h3>\n<ul class=\"indentedh3\">\n<li>\\(4\\times 10^{2}\\) has one significant figure.<\/li>\n<li>\\(4.0\\times 10^{2}\\) has two significant figures.<\/li>\n<li>\\(4.00\\times 10^{2}\\) has three significant figures, etc..<\/li>\n<\/ul>\n<p class=\"indentedh2\">The <a href=\"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/physics-1140-notes-on-measurement-uncertainties-and-error-analysis\/\">uncertainty<\/a> \\(\\delta x\\) in a measurement \\(x\\) will be presented with the measured value as \\(x\\pm\\delta x\\). Three steps for getting significant figures right in calculations involving uncertainty:<\/p>\n<ol class=\"indentedh2\">\n<li id=\"step1\">Calculate your answer, \\(x\\), ignoring uncertainty, keeping a couple more significant figures than you think you will ultimately need.<\/li>\n<li id=\"step2\">Calculate the uncertainty, \\(\\delta x\\), of the answer using the appropriate rule(s) from the section to follow on <a href=\"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/physics-1140-notes-on-measurement-uncertainties-and-error-analysis\/#ErrorPropagation\">propagating errors<\/a> in calculations. Round it to <em>one<\/em> significant figure. One exception: Keep two significant figures if the calculated uncertainty value would start with a one if written in scientific notation, as in \\(0.014\\), \\(1.7\\), \\(11\\), etc..<\/li>\n<li>Use the <em>decimal position<\/em> of the only\/last digit in the uncertainty (whose number of sig. figs. is now only one, or maybe two) to determine how many decimal places to keep in your answer.\n<p class=\"notes\"><span style=\"color: #ff0000\">The number of decimal places (not significant figures) of the answer and its uncertainty must be the same.<\/span><\/p>\n<\/li>\n<\/ol>\n<h3>Examples with Uncertainty:<\/h3>\n<p class=\"indentedh3\">If \\(\\delta x\\) turns out to be \\(0.007\\) (from <a href=\"#step2\">step two<\/a>), and your answer \\(x\\) was initially \\(1.46129\\) (from <a href=\"#step1\">step one<\/a>), then make your final statement of the result $$ x = 1.461\\pm 0.007.$$ If you use scientific notation &#8211; definitely recommended for very large or very small numbers &#8211; it is easier to present the answer and uncertainty to the same magnitude, as in $$ x = (2.52\\pm 0.04)\\times 10^{14}$$ instead of $$ x = 2.52\\times 10^{14}\\pm 4\\times 10^{12}.$$<\/p>\n<p class=\"centered\"><a href=\"http:\/\/www.mathjax.org\"> <img decoding=\"async\" title=\"Powered by MathJax\" src=\"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-content\/uploads\/sites\/105\/2015\/07\/badge.gif\" alt=\"Powered by MathJax\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Significant figures (also called significant digits) are very important because they imply the known precision of a number. How to Count Significant Figures Non-zero digits are always significant. Any zeros between two significant figures are significant. Trailing zeros in the decimal portion only are significant. Examples: \\(4308\\) has four significant figures. \\(40.05\\) has four significant [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-43","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/43","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/comments?post=43"}],"version-history":[{"count":0,"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/pages\/43\/revisions"}],"wp:attachment":[{"href":"https:\/\/courses.bowdoin.edu\/physics-1140-lab-manual\/wp-json\/wp\/v2\/media?parent=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}