{"id":2981,"date":"2023-01-09T22:17:31","date_gmt":"2023-01-09T22:17:31","guid":{"rendered":"http:\/\/mclements.net\/blogWP\/?p=2981"},"modified":"2023-01-13T17:46:51","modified_gmt":"2023-01-13T17:46:51","slug":"slide-rule-past-the-basics","status":"publish","type":"post","link":"http:\/\/mclements.net\/blogWP\/index.php\/2023\/01\/09\/slide-rule-past-the-basics\/","title":{"rendered":"Slide Rules: Past the Basics"},"content":{"rendered":"<p>For background, here is <a href=\"http:\/\/mclements.net\/blogWP\/index.php\/2023\/01\/12\/slide-rules\/\">part 1<\/a>. It introduces slide rules and covers basic usage with C, D and CI scales. Here we cover the A, B and K scales for squares, square roots, cubes and cube roots.<\/p>\n<h3>A Scale: Squares and Square Roots<\/h3>\n<p>The A scale shows the square of each number on the D scale. Conversely, the D scale shows the square root of each number on the A scale.<\/p>\n<p>Below the cursor shows that 3 on the D scale matches 9 on the A scale.<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3006\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3.jpg\" alt=\"\" width=\"2052\" height=\"2100\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3.jpg 2052w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3-293x300.jpg 293w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3-768x786.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ3-1001x1024.jpg 1001w\" sizes=\"auto, (max-width: 2052px) 100vw, 2052px\" \/><\/a><\/p>\n<p>Note that as the D scale goes from 1 to 10, the A scale goes from 1 to 100. This means every single-digit number in the A scale has a corresponding double-digit number further up the scale. That is: 1 and 10, 2 and 20, 3 and 30, etc.<\/p>\n<p>For example, below the cursor is on the A scale 30, showing the square root of 30 is 5.48:<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3007\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30.jpg\" alt=\"\" width=\"2127\" height=\"2169\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30.jpg 2127w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30-294x300.jpg 294w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30-768x783.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squ30-1004x1024.jpg 1004w\" sizes=\"auto, (max-width: 2127px) 100vw, 2127px\" \/><\/a><\/p>\n<p>So, squares are easy: just find the number on D scale and look to the corresponding number on A scale. But going the opposite direction, square roots, becomes ambiguous. What if you want the square root of a number that is not in the range of 1 to 100? For example:<\/p>\n<ul>\n<li>If you want the square root of 400, on the A scale do you use 4 or 40?<\/li>\n<li>If you want the square root of 4000,\u00a0 on the A scale do you use 4 or 40?<\/li>\n<li>If you want the square root of 0.5, on the A scale do you use 5 or 50?<\/li>\n<\/ul>\n<p>First, observe that all of the single-digit numbers in the A scale occupy the first half, all the double-digit numbers the second half, and the split (the value 10) is exactly the mid-point of the scale. Thus, you can think of the A scale has having 2 halves. So the question boils down to knowing which half to use.<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3008\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts.jpg\" alt=\"\" width=\"2272\" height=\"2264\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts.jpg 2272w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts-150x150.jpg 150w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts-300x300.jpg 300w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts-768x765.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squParts-1024x1020.jpg 1024w\" sizes=\"auto, (max-width: 2272px) 100vw, 2272px\" \/><\/a><\/p>\n<p>The answer is based on how many zeroes in the number whose root you want to know: is that number of zeroes odd or even? Adding or removing each zero (or each decimal point shift left or right) shifts back and forth between the 2 halves of the scale.<\/p>\n<p>For example, consider the number 4.<\/p>\n<ul>\n<li>The square root of 4 is 2 (low side of A scale).<\/li>\n<li>The square root of 40 is 6.32 (high side of A scale).<\/li>\n<li>The square root of 400 is 20 (low side of A scale).<\/li>\n<li>The square root of 4000 is 63.2 (high side of A scale).<\/li>\n<\/ul>\n<p>This pattern simply repeats. And it also repeats in the opposite direction:<\/p>\n<ul>\n<li>The square root of 0.4 is 0.632 (high side of A scale).<\/li>\n<li>The square root of 0.04 is 0.2 (low side of A scale).<\/li>\n<li>The square root of 0.004 is 0.0632 (high side of A scale).<\/li>\n<\/ul>\n<h3>K Scale: Cubes and Cube Roots<\/h3>\n<p>The K scale works like the A scale. To cube any number N, find N on the D scale and look at the corresponding number on the K scale.<\/p>\n<p>For example, this shows that 2 cubed is 8 (it actually shows that 2 squared is 4, and cubed is 8):<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3010\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8.jpg\" alt=\"\" width=\"2214\" height=\"2241\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8.jpg 2214w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8-296x300.jpg 296w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8-768x777.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub8-1012x1024.jpg 1012w\" sizes=\"auto, (max-width: 2214px) 100vw, 2214px\" \/><\/a><\/p>\n<p>Yet the K scale also has the value 80, whose cube root is 4.31:<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3011\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80.jpg\" alt=\"\" width=\"2214\" height=\"2235\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80.jpg 2214w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80-150x150.jpg 150w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80-297x300.jpg 297w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80-768x775.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub80-1014x1024.jpg 1014w\" sizes=\"auto, (max-width: 2214px) 100vw, 2214px\" \/><\/a><\/p>\n<p>And the K scale also has the value 800, whose cube root is 9.28:<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3012\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800.jpg\" alt=\"\" width=\"2193\" height=\"2196\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800.jpg 2193w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800-150x150.jpg 150w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800-300x300.jpg 300w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800-768x769.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cub800-1024x1024.jpg 1024w\" sizes=\"auto, (max-width: 2193px) 100vw, 2193px\" \/><\/a><\/p>\n<p>So while the A scale splits into 2 equal parts (1-10 and 10-100), the K scale splits into 3 equal parts: 1-10, 10-100, and 100-1000. Each of these occupies exactly 1\/3 of the scale:<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3013\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts.jpg\" alt=\"\" width=\"2272\" height=\"2264\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts.jpg 2272w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts-150x150.jpg 150w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts-300x300.jpg 300w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts-768x765.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-cubParts-1024x1020.jpg 1024w\" sizes=\"auto, (max-width: 2272px) 100vw, 2272px\" \/><\/a><\/p>\n<p>As with the A scale, cubing numbers is simple. Find N on the D scale and the corresponding K scale is N cubed. Yet doing the opposite, taking the cube root, is a bit more complex. You need to know which 1\/3 of the scale to use. The procedure is a simple extension of what we did for the A scale. Yet since the A scale had only 2 parts, we thought of it as flipping back and forth. With the K scale having 3 parts, instead of flipping, each shift of the decimal point rotates forward to the next, or back to the prior part of the scale.<\/p>\n<p>For example, consider the number 8:<\/p>\n<ul>\n<li>The cube root of 8 is 2 (part 1 of\u00a0 K scale).<\/li>\n<li>The cube root of 80 is 4.31 (part 2 of K scale).<\/li>\n<li>The cube root of 800 is 9.28 (part 3 of K scale).<\/li>\n<li>The cube root of 8000 is 20 (part 1 of K scale).<\/li>\n<\/ul>\n<p>It repeats every 3rd shift of the decimal point. And it also applies in the reverse direction as numbers get smaller:<\/p>\n<ul>\n<li>The cube root of 0.8 is 0.928 (part 3 of K scale).<\/li>\n<li>The cube root of 0.08 is 0.431 (part 2 of K scale).<\/li>\n<li>The cube root of 0.008 is 0.2 (part 1 of K scale).<\/li>\n<\/ul>\n<h3>Exponents and Roots other than 2 or 3<\/h3>\n<p>Most everyday problems, especially those involving basic physics, use squares or cubes. Powers other than 2 or 3 are uncommon. Yet we do occasionally encounter them and you can solve them with a slide rule. The trick is to know some basic math rules about adding exponents.<\/p>\n<p>Suppose we some number N, say 3.5, and we want to know N to some power, say the 7th power. So how would we compute 3.5^7 with our slide rule?<\/p>\n<p>First, we split the exponent into smaller parts involving squares and cubes:<\/p>\n<pre>3.5^7 = 3.5^(2+2+3)<\/pre>\n<p>Next we split this out:<\/p>\n<pre> 3.5^(2+2+3) = 3.5^2 * 3.5^2 * 3.5^3<\/pre>\n<p>Now we have split the problem into squares and cubes so we can compute it on our slide rule:<\/p>\n<pre>3.5^2 = 12.2\r\n3.5^3 = 42.8\r\n12.2 * 12.2 * 42.8 = 150 * 42.8 = 6,430<\/pre>\n<p>Checking this with a calculator, I get 6,433.9. The slide rule gave us 3 significant figures.<\/p>\n<p>Note: you can apply this same method with roots by taking the exponents as fractions. That is, the square root of 2 is 2 to the 1\/2 power, cube root of 2 is 2 so the 1\/3 power, etc.<\/p>\n<h3>What About that B Scale?<\/h3>\n<p>The B scale is the same as the A scale, used for squares and square roots. But it corresponds with the C scale instead of the D scale. Using D-A is the same as C-B. The B scale isn&#8217;t essential, but it is sometimes useful. On the Concise model 300, the A scale is along the outer circumference and the B scale is further inside. This makes the A scale longer and more accurate.<br \/>\n<a href=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3025\" src=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB.jpg\" alt=\"\" width=\"2272\" height=\"2264\" srcset=\"http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB.jpg 2272w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB-150x150.jpg 150w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB-300x300.jpg 300w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB-768x765.jpg 768w, http:\/\/mclements.net\/blogWP\/wp-content\/uploads\/2023\/01\/sliderule-squAB-1024x1020.jpg 1024w\" sizes=\"auto, (max-width: 2272px) 100vw, 2272px\" \/><\/a><\/p>\n<p>To quantify, on this Concise model 300 the A scale has a diameter of 9.2 cm and B scale 5.9. The A scale is 9.2 \/ 5.9 = 1.56 times longer. That&#8217;s a significant difference!<\/p>\n<h3>Conclusion<\/h3>\n<p>With squares and cubes, the power of the slide rule increases and we can solve a wider variety of problems. The next step is trigonometry: sine, cosine, tangent. These come up frequently in simple physics and everyday problems. This is covered in <a href=\"http:\/\/mclements.net\/blogWP\/index.php\/2023\/01\/13\/slide-rules-trig\/\">part 3<\/a>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For background, here is part 1. It introduces slide rules and covers basic usage with C, D and CI scales. Here we cover the A, B and K scales for squares, square roots, cubes and cube roots. A Scale: Squares and Square Roots The A scale shows the square of each number on the D &hellip; <a href=\"http:\/\/mclements.net\/blogWP\/index.php\/2023\/01\/09\/slide-rule-past-the-basics\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Slide Rules: Past the Basics<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,2],"tags":[],"class_list":["post-2981","post","type-post","status-publish","format-standard","hentry","category-math","category-tech"],"_links":{"self":[{"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/posts\/2981","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/comments?post=2981"}],"version-history":[{"count":7,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/posts\/2981\/revisions"}],"predecessor-version":[{"id":3029,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/posts\/2981\/revisions\/3029"}],"wp:attachment":[{"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/media?parent=2981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/categories?post=2981"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/mclements.net\/blogWP\/index.php\/wp-json\/wp\/v2\/tags?post=2981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}