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Slide Rules

It's HS math:
    log(a·b) = log(a) + log(a)

Why?  Well, because every number can be written as a power of 10.

That is, for any a and b, we can find x and y such that

    a = 10x,   and
    b = 10y 


    log(a·b) = log(10x·10y) = log(10(x+y)) = x + y = log(a) + log(b)

And that's why slide rules work.  Well, that and the fact that any number can be physically represented by a length.  

This gives you the power to create a ruler bearing a log scale.  On that ruler, the left edge is marked 1, and the distance from the left edge to a given tick mark is a fraction of the rulers length equal to the log of the number written (or implied) at that tick mark.  So if the ruler is 10" long, the tick mark labeled "2" is 3.01" from the left edge (log(2) = 0.301).

Such a ruler alone lets you multiply and divide.  Put the ruler on a piece of paper and draw a line from "1" to "2"; that line is log(2) long.  Now move the "1" on the ruler to the end of that line and draw another one from "1" to "5"; that line is log(5) long.  Now use the ruler to measure the total length of the line you've drawn.  Voila, the ruler measures "10", because you've marked off a total length equal to log(2)+log(5), which is log(10), and at a length of log(10) the ruler is labeled "10".

To divide 10 by 2, draw a line from "1" to "10" and mark the line at "2".  Now measure from the mark to the end of the line and see that it's "5"!  

I think it's pretty damn awesome that there is this beautiful symmetry between multiplication of real numbers and addition of their logs.  That it can be reduced to carefully arranged scratches on wood is further amazing.  

You don't agree?  Well, it also enabled 450 years of engineering, which is to say every technological advance you take for granted.  

(As an exercise for the interested, I suggest you visit Wikipedia's Timeline of Historic Inventions.  Take a look at the kinds of things invented before 1600 and after.  To me, it looks like practically everything we think of as "technology" came out after Napier created logarithms in 1590 and Oughtred invented slide rules in 1622.)

Turning multiplication into addition and addition into slide rules gives most of us the ability to do it at least 10 times faster, I should think.  (Multiply 378 x 945 with paper and pencil and let me know how long it takes.  I can get an answer to 3 significant digits on a slide rule in less than 5 second.)  That's a big productivity gain for anyone who wants to do lots of engineering or very much science.  

Without logarithms and slide rules I suggest we would likely still be in the dark ages.  

And that's why you should pay attention in HS math class.


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