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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#Precursors to the electronic 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

$\cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}$

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)

January 31

Sums with Roman numerals

Can someone explain how the Romans did sums using their Roman numerals? I know they did it somehow. And what about the Greeks, Jews, etc., who used letters to denote numbers? 94.10.23.218 (talk) 13:52, 31 January 2016 (UTC)

Roman numerals#Modern use says

Roman numerals however proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been by abacus).

Loraof (talk) 18:22, 31 January 2016 (UTC)

Well, let's take 1888×2:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
MMDDCCCCCCLLXXXXXXVVIIIIII

Five I's = V:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
  MMDDCCCCCCLLXXXXXXVV V I

Two V's = X:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
  MMDDCCCCCCLLXXXXXX X V I

Five X'x = L:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
    MMDDCCCCCCLL L X X V I

Two L's = C:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
    MMDDCCCCCC C L X X V I

Five C's = D:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
      MMDD D C C L X X V I

Two D's = M:

             MDCCCLXXXVIII
            +MDCCCLXXXVIII
            --------------
      MM M D C C L X X V I

I imagine somebody good at doing math in Roman numerals could do a lot of the combining in their head, just as we can do the carrying in our heads. StuRat (talk) 18:22, 31 January 2016 (UTC)

We do have the article Roman abacus, but presumably there were procedures which were historically taught for doing calculations without such a device. A quick web search finds several examples of direct calculations, but none that I found indicate what was done historically. -- ToE 18:33, 31 January 2016 (UTC)

Thanks, everyone. Could someone answer the second part of the question, how did other ancient peoples, (such as the Greeks, the Maya, Chinese etc.) do their sums before the Hindu numerals came into use? 90.213.129.1 (talk) 20:42, 2 February 2016 (UTC)

February 2

Computation of trig function before computers

What exactly were the steps, or the algorithm to find out the values of the logarithm, sine or cosine tables? Back then There were like 1000 values for each table, but if you did not have the table, was it really impossible to do these calculations? 186.146.10.154 (talk) 12:18, 2 February 2016 (UTC) (posted by SemanticMantis (talk) 16:07, 2 February 2016 (UTC))SemanticMantis (talk) 16:15, 2 February 2016 (UTC)

Trigonometric_tables gives some info on how they were computed. Let us know if there's something in there you don't understand. SemanticMantis (talk) 16:49, 2 February 2016 (UTC)

The earliest method would just be to construct the triangles and measure the values directly, perhaps interpolating to find the in-between values. StuRat (talk) 17:03, 2 February 2016 (UTC)

See Taylor series. The method of computing the values in the log, sine, or cosine tables were essentially the same as the method that the computer uses behind the scenes. If you did not have the table and were not skilled in doing the laborious pencil-and-paper Taylor series, it was impossible to do the calculations accurately, although there were estimation techniques. If you did not have the table and were skilled in the laborious pencil-and-paper calculation, I suppose (but am guessing) that you could contract with a publisher to develop their version of the table. Robert McClenon (talk) 18:25, 2 February 2016 (UTC)