When I did A level maths, we came across a technique that our maths teacher told us was used for working out roots before calculators were introduced. I’ve forgotten the exact technique, but I remember thinking that it was a particularly clever method. He also said that without calculators, mathematicians were forced to come up with other, similarly clever techniques in order to work out roots, logs etc.
So, what were some of the techniques used? How exactly were the logs that were recorded in log books calculated? What about roots? Cube roots etc.
I know one method. It’s kind of awkward, but it works.
Say you want the square root of 43. You probably know that 6x6 is 36, and 7x7 is 49, and that 43 is in between the two. So you pick a number in between and divide it into 43
43/6.5=6.16
6.5+6.16=12.66
12.66/2=6.33
You now repeat the process with 6.33, dividing it into 43, finding the result adding the two together and dividing the sum by 2.
You keep on repeating until you are as many decimal points into the root as you wish.
My old fashioned 6th grade math teacher taught us how to do square roots and cube roots by hand. The procedure looked a little similar to division and breaking down the problem into a series of steps starting with the most significant digits.
BTW does anyone know where you can get log tables these days? I’m teaching a kid about logs and I wanted him to go through some of the hard grind I did.
But no-one seems to make them any more, I could only find rudimentary ones on the web, and there are none in my local second hand bookstores (for obvious reasons - why would they buy them?).
Heh, why don’t you just create your own log tables? Just use a built-in function in a spreadsheet or many other programs. But if you must see a table in print, just go to the library and look for a copy of the CRC Standard Mathematical Tables and Formulae. It’s in its 31st edition. The copy on my shelf is the 17th edition. Betcha the log values are the same.
I believe square roots were originally computed using Newton-Raphson style iteration. And I believe they still are in calculator algorithms internally.
Heh, of course, the spreadsheet has its own log table buried in the program. Just like human calculators used to do, the spreadsheet uses the log table and interpolation to calculate any given log you want.
On another note, if you take calculus, a lot of techniques for calulating logs and other functions become much more clear.
Finally, Omphaloskeptic, your impressive coding didn’t seem to work in Firefox for me (though I may have screwed up Java at some point previously).
The hand method of calculating square roots (which I *can’t find my ()@# copy of right now :mad: ) is based on the algebraic formula (A+B)[sup]2[/sup] = A[sup]2[/sup]+2AB+B[sup]2[/sup]. First you start with the part you know e.g. the square root of 43 is 6-something, so in that example 6 would be A. Then you find the closest value for B than you can, add that to what you previously had and take the combined value as the new A for the next iteration. I presume that logs and trig values can be derived by their own algebraic equations. Other values such as Pi can be calcuated as the sum of a series.