{"id":1031,"date":"2009-08-09T12:26:59","date_gmt":"2009-08-09T02:26:59","guid":{"rendered":"http:\/\/www.somethinkodd.com\/oddthinking\/?p=1031"},"modified":"2009-08-09T12:26:59","modified_gmt":"2009-08-09T02:26:59","slug":"breaking-the-slide-rules","status":"publish","type":"post","link":"https:\/\/www.somethinkodd.com\/oddthinking\/2009\/08\/09\/breaking-the-slide-rules\/","title":{"rendered":"Breaking the (Slide) Rules"},"content":{"rendered":"

I haven’t used a slide rule in ages, and even then it was as a toy. I am too young to have ever needed to use one in anger.<\/p>\n

I’ve been trying to remember enough high school maths to reinvent the concept from first principles, without actually checking on Wikipedia to remind me how they work – just to prove to myself I still have the concepts clear enough in my mind to explain them to someone else.<\/p>\n

My Attempt to Explain Slide Rules To Myself<\/h4>\n

Let’s make up a function f(x). Let’s make it injective, and call its inverse f-1<\/sup>(x), so x = f-1<\/sup>(f(x))<\/p>\n

Now, let’s go into the physical world. Take two straight sticks. Mark one end of each as the origin, and at every point which is x millimetres from the origin, write down the value f(x). <\/p>\n

Okay, not every point, but frequently enough that you cover the important values of f(x).<\/div>\n

Now take two arbitrary values, g1<\/sub> and g2<\/sub>, and find where they are written on the sticks. Line up the origin of the second stick with the position of g1<\/sub> on the first stick. Hold them parallel, oriented appropriately, and read off the number next to the position of g2<\/sub> on the second stick.<\/p>\n

What have you done, in mathematical terms?<\/p>\n

Well, lining up the sticks is a rapid way to do addition. What are you adding? The distance in millimetres between the origin of the first stick and the location of g2<\/sub> on the second stick is f-1<\/sup>(g1<\/sub>) + f-1<\/sup>(g2<\/sub> ).<\/p>\n

But, you aren’t measuring it in millimetres, you are measuring it using the first stick, so the number you read off is actually f(f-1<\/sup>(g1<\/sub>) + f-1<\/sup>(g1<\/sub>)).<\/p>\n

Now, with a traditional slide rule, we would use the power function for f.<\/p>\n

e.g. f(x) = 10x<\/sup>, f-1<\/sup>(x) = log10<\/sub>(x).<\/p>\n

That makes a slide rule a quick way to perform the following operation:<\/p>\n

10log (g1<\/sub>) + log (g2<\/sub>)<\/sup><\/p>\n

which is another way of saying g1<\/sub> × g2<\/sub>.<\/p>\n

More excitingly, division can be performed by aligning the sticks differently, but I am skimming over that here.<\/div>\n

My Sudden Realisation<\/h4>\n

Power\/Log functions aren’t the only possible functions that slide rules could be used for.<\/p>\n

If you let f(x) = sqrt, and f-1<\/sup>(x) = x2<\/sup>, then what you get is the square root of the sum of two squares. You could make a special slide rule just to work out the length of the hypotenuse of a right-angled triangle!<\/p>\n

I have no doubt that this has been known for years, and such slide rules probably even existed, but I am happy to have discovered it myself!<\/sub><\/p>\n","protected":false},"excerpt":{"rendered":"

In which Julian uses a modern computer (capable of billions of multiplications a second) as a notebook to record how to make a slide-rule (capable of performing a few per minute).<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_s2mail":"","footnotes":""},"categories":[28],"tags":[],"class_list":["post-1031","post","type-post","status-publish","format-standard","hentry","category-doubleplus-geek"],"_links":{"self":[{"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/posts\/1031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/comments?post=1031"}],"version-history":[{"count":5,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/posts\/1031\/revisions"}],"predecessor-version":[{"id":1036,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/posts\/1031\/revisions\/1036"}],"wp:attachment":[{"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/media?parent=1031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/categories?post=1031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.somethinkodd.com\/oddthinking\/wp-json\/wp\/v2\/tags?post=1031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}