Wikipedia:Reference desk/Archives/Mathematics/2016 January 28

= January 28 =

What did people use before slide rule and calculators and computers were invented?
What did people use before slide rule and calculators and computers were invented? An abacus? Their heads? Were people smarter back in the olden days because they might have had to do a lot of mental math while nowadays people just get lazy and use a calculator? 140.254.136.157 (talk) 18:23, 28 January 2016 (UTC)


 * There were look-up tables for things like trig functions. Of course, back then, few people really needed to do that type of math.  As for whether people were "smarter" then, I'd say no, that just having to memorize more things doesn't make one smarter.  We are coming into a period now where it really isn't necessary to remember many facts at all, as they are all available online, such as right here on Wikipedia.  Not needing to do math in your head was just an earlier level of this movement from keeping all the world's knowledge in your head to storing it externally.  The invention of writing was the first step in that direction. StuRat (talk) 18:46, 28 January 2016 (UTC)


 * Also see Calculator. Loraof (talk) 18:50, 28 January 2016 (UTC)
 * Also Human computer. Loraof (talk) 18:54, 28 January 2016 (UTC)


 * And log tables→86.139.120.76 (talk) 20:09, 28 January 2016 (UTC)

Really, the question needs to be more specific. What did people use to do what sort of calculation? An abacus is great if you want to add and subtract numbers that may be several digits long, and it can help you do multiplication if you convert it to repeated additions: for example, to multiply 7,479 by 4,186, you could simply add 7,479,000 + 7,479,000 + 7,479,000 + 7,479,000 + 747,900 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 74,790 + 7,479 + 7,479 + 7,479 + 7,479 + 7,479 + 7,479, carefully keeping track of how many times you repeated each distinct term. Division would be of similar complexity, only you wouldn't know in advance how many steps there would be. And all of this would be possible even if you didn't have a positional notation like our 31,307,094 to express the answer in.

On the other hand, once people did have positional notation, they could also do calculations like multiplication and division using the long multiplication that I hope is still taught in schools today, and doing the additions mentally. And if they didn't want to work out the intermediate values directly (the ones you get by multiplying a multi-digit number by a single digit), they could use a set of Napier's bones for that purpose.

There is much more to say on this topic, but as I said, it really all depends on what sort of calculations you're talking about. --76.69.45.64 (talk) 23:32, 28 January 2016 (UTC)
 * The sort of calculations made by slide rule was: multiplication, division, square, square root, log, sin, cos, and tan. Before the slide rule these operations were done using paper and pencil and table lookup. Special formulas were designed to optimize trigonometric calculations. For example the angle A in a triangle having sides a, b and c was not computed by
 * $$\cos A = \frac{b^2+c^2-a^2}{2 b c }$$
 * but rather by
 * $$\log \tan \frac A 2 = \frac {\log (s-b) + \log (s-c)-\log s - \log (s-a)}2$$ where $$s=\frac{a+b+c}2$$
 * avoiding multiplications and using only 5 table lookups. Bo Jacoby (talk) 00:31, 29 January 2016 (UTC).


 * Recall also Mechanical calculators. --CiaPan (talk) 08:09, 31 January 2016 (UTC)


 * I think that falls outside of the scope of the question, since it was about what people did before slide rule and calculators (and computers) were invented. The slide rule had and has significant limits on its accuracy.  Mechanical and electronic calculators (and a computer is for this purpose an electronic calculator) could be made to arbitrary accuracy.  Also, calculations of transcendental functions using the Taylor series could be done to arbitrary accuracy, and, if one needed greater accuracy than provided by the slide rule, one took out the table of logarithms (or trig functions or whatever) where someone had already calculated to arbitrary accuracy.  All of these techniques except the abacus depended on the use of Arabic numerals.  The use of Arabic numerals also provided one advantage over the use of the abacus, which was that, because you did the intermediate calculations (for long multiplication) with the same quill pen as you used to record the answer, you could keep the intermediate calculations to cross-check.  A Taylor series, using Arabic numerals, permits calculations to arbitrary accuracy; they are just tedious.  Robert McClenon (talk) 22:13, 2 February 2016 (UTC)