# Complex calculations before computers **URL:** \ **Category:** Factual Questions\ **Created:** [November 8, 2010, 5:06pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926 "2010-11-08T17:06:08Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![mozchron](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@mozchron](https://boards.straightdope.com/u/mozchron)\ **Post date:** [November 8, 2010, 7:53pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/21 "2010-11-08T19:53:00Z")
> [@Kevbo](#): > > On days when I yell at kids to get off my lawn, I worry that kids today are not only not learning how to do calculations, they are also failing to learn how calculations _work_. A-F\*cking-men! Preach it, brother (or sister)! I can’t tell you how many times I have had a student put a patently absurd answer for a calculation on a test (like “find the volume of a sphere with r = 12 inches”. Student’s Answer: 1,000,000 m^2).\* I point out that this can’t be, the units don’t agree etc… They say “I used a calculator - that’s what it said!” No thought process behind blindly punching buttons. They don’t understand enough to even begin to notice that maybe, just maybe, something is wrong. Drives me batshit. \*example made up, but I have seen actual examples as egregious. ---
**Author:** ![Markxxx](https://avatars.discourse-cdn.com/v4/letter/m/5daacb/32.png) [@Markxxx](https://boards.straightdope.com/u/Markxxx)\ **Post date:** [November 8, 2010, 7:55pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/22 "2010-11-08T19:55:05Z")
> [@aceplace57](#): > > I’ve wondered about the people that calculated the first log tables. The work must have been so tedious. > . But remember what else did people have to do? Before radio, TV, movies, records, people would get up, go to work, come home and it was dark. Oil to light the room was expensive, books were scarce unless the urban area had any kind of a library, and that’s assuming one could read. Even if you could read and write, it was probably only a bit so you could get by. After all writing your name and reading the bible (the only book most people had) was all you needed to do. So after work you could knit or make wood work or go to the bar and get drunk or calculate logs or algorithms. Kind of like the way people do crosswords or jumbles now to kill time. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [November 8, 2010, 7:57pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/23 "2010-11-08T19:57:12Z")
Actually, LaGrange predicted that there could be a gravity well so deep even light couldn’t escape, anticipating Schwarzwald by a couple centuries. The first log tables were constructed by Napier to 7 places. It is fascinating how he set it up for hand computation. The function he computed was essentially ln(-x). Why - I can’t say, but the natural log was much easier than the common log, although not as useful. A man named Briggs first realized that base 10 logs were better for use and he and Napier began computing them until Napier died and the process was finished by Briggs. Napier conceived of the idea of a point moving from the point -1 on the x-axis to the point 0 in such a way that its speed was proportional to its distance from the origin. In modern terms, we would say that he was solving the differential equation y’ = -1/y (that’s where the ln came from). To simplify, let me replace x by -x. Now he began with f(1) = 0. Good start. Also to simplify notation, let me abbreviate 10^{-7} by b. Then f(1-b) is -b to a good approximation. You get that by subtracting .0000001 from 0. f((1-b)^2) = 2f(1-b). Now here was the clever part. Multiplication by 1-b can be replaced by subtraction; namely x(1-b) =x-xb and the nature of b is that xb is gotten by moving x over by 7 digits. In this way he computed ln of 1-b, (1-b)^2,…,(1-b)^{10}. At this point, he interpolated to replace them by 1-b, 1-2b, …, 1-10b. He then repeated this procedure with 1-10b to get to 1-100b in tens but using the first result had got all 1-nb for n = 1…100. He then repeated to get all 1-nb for b up to 1000. And then to 10,000 or so till he had covered the interval down to 1/2 (actually up to -1/2, but no matter). Then he he used ln (x/2) = ln(x) + ln(1/2) to get it all into the interval [1/4,1/2] and so on. Yes it was a massive project, calculating 10,000,000 points, but it was all done by hand and with no multiplication (the point, after all, of logs). ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [November 8, 2010, 8:10pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/24 "2010-11-08T20:10:35Z")
> [@Hari\_Seldon](#): > > Actually, LaGrange predicted that there could be a gravity well so deep even light couldn’t escape, anticipating Schwarzwald by a couple centuries. To me that is the difference between speculating about a rocket that could take you to the moon and building a rocket that could take you to the moon. Big difference. ---
**Author:** ![RTFirefly](https://avatars.discourse-cdn.com/v4/letter/r/c77e96/32.png) [@RTFirefly](https://boards.straightdope.com/u/RTFirefly)\ **Post date:** [November 8, 2010, 8:31pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/25 "2010-11-08T20:31:48Z")
> [@Thudlow\_Boink](#): > > Slide rules and logarithm tables were standard equipment for people who had to do lots of relatively complicated calculations (e.g. engineers) more than about thirty years ago. > > (I vaguely recall reading science fiction stories by the likes of Robert Heinlein where someone piloted a spaceship with the help of a slide rule.) That would have been Andy “Slipstick” Libby. ---
**Author:** ![ChrisBooth12](https://avatars.discourse-cdn.com/v4/letter/c/e95f7d/32.png) [@ChrisBooth12](https://boards.straightdope.com/u/ChrisBooth12)\ **Post date:** [November 8, 2010, 8:33pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/26 "2010-11-08T20:33:35Z")
I always heard sliderules were quicker because with a lot of things you do not need the ammount of sig figs calculators give you, for example knowing Pi to 4 or 5 decimal places is enough to build a giant circle think LHC without tolerance issues ---
**Author:** ![Cartooniverse](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cartooniverse/32/3084_2.png) [@Cartooniverse](https://boards.straightdope.com/u/Cartooniverse)\ **Post date:** [November 8, 2010, 8:37pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/27 "2010-11-08T20:37:36Z")
> [@mozchron](#): > > \*example made up, but I have seen actual examples as egregious. These are the same teenagers who cannot spell or define “egregious” on a test. > [@Hari Seldon](#): > > A man named Briggs first realized that base 10 logs were better for use and he and Napier began computing them until Napier died and the process was finished by Briggs. The first confirmed math fatality. :eek: 😃 ---
**Author:** ![Max\_Torque](https://avatars.discourse-cdn.com/v4/letter/m/e9bcb4/32.png) [@Max\_Torque](https://boards.straightdope.com/u/Max_Torque)\ **Post date:** [November 8, 2010, 9:22pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/28 "2010-11-08T21:22:29Z")
> [@ChrisBooth12](#): > > I always heard sliderules were quicker because with a lot of things you do not need the ammount of sig figs calculators give you, for example knowing Pi to 4 or 5 decimal places is enough to build a giant circle think LHC without tolerance issues That’s one of my favorite little factoids, which I think of every time I hear about calculating the squillionth digit of pi: Pi to eleven decimal places is sufficient to calculate any circle that will fit on the Earth to an accuracy of less than one millimeter. Pi to 39 decimal places is sufficient to calculate a circle the size of the known universe with an accuracy equal to the diameter of a hydrogen atom. [Wikicite](http://en.wikipedia.org/wiki/Pi#Decimal_representation). ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 8, 2010, 9:39pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/29 "2010-11-08T21:39:53Z")
> [@](#): > > \*example made up, but I have seen actual examples as egregious. OK, non-made-up example. The class I’m TAing had a test last week, and one of the questions ended with “What is the speed of the proton?”. Answers ranged from 10[sup]-43[/sup] m/s to 10[sup]30[/sup] m/s. Another one: I was grading a homework problem once about an airplane flying around in a circle, and the students were to find the radius of the circle. One student got an answer that worked out to 10 parsecs. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 8, 2010, 11:25pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/30 "2010-11-08T23:25:04Z")
There’s an interesting [story](http://unimodular.net/blog/?p=68) about an economic model from the 50s that required the inversion of a 24 x 24 matrix. Back then, that was at the limits of computational technology; nowadays, doing it 10000 times takes a little under three seconds on my PC. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [November 8, 2010, 11:28pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/31 "2010-11-08T23:28:51Z")
> [@ultrafilter](#): > > There’s an interesting [story](http://unimodular.net/blog/?p=68) about an economic model from the 50s that required the inversion of a 24 x 24 matrix. Back then, that was at the limits of computational technology; nowadays, doing it 10000 times takes a little under three seconds on my PC. Yeesh. Doing a 4x4 by hand is tiring enough. I’m glad I am not a computer. ETA: Actually, I think the biggest one I ever did by hand was a 3x3. I did 4x4’s with Mathematica though. ---
**Author:** ![bonzer](https://avatars.discourse-cdn.com/v4/letter/b/45deac/32.png) [@bonzer](https://boards.straightdope.com/u/bonzer)\ **Post date:** [November 8, 2010, 11:46pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/32 "2010-11-08T23:46:33Z")
For this sort of topic David Alan Grier’s history _When Computers Were Human_ (Princeton, 2005) really can’t be recommended highly enough. It’s unfortunately somewhat patchy - in other words, excellent on what it covers, but hardly an entirely comprehensive history of the subject - but it’s extremely good as a broadish overview of the issues involved, while engagingly drilling down into some specifics of how projects were organised. ---
**Author:** ![Ruminator](https://avatars.discourse-cdn.com/v4/letter/r/b9bd4f/32.png) [@Ruminator](https://boards.straightdope.com/u/Ruminator)\ **Post date:** [November 8, 2010, 11:56pm UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/33 "2010-11-08T23:56:26Z")
> [@Chronos](#): > > One student got an answer that worked out to 10 parsecs. The student simply forgot to leave out the frame of reference in his answer. If he imagined viewing the airplane from a galaxy far far away, it’s 10 parsecs. I hope you cut him some slack and gave him partial credit. ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [November 9, 2010, 12:13am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/34 "2010-11-09T00:13:25Z")
> [@Max\_Torque](#): > > That’s one of my favorite little factoids, which I think of every time I hear about calculating the squillionth digit of pi: Pi to eleven decimal places is sufficient to calculate any circle that will fit on the Earth to an accuracy of less than one millimeter. Pi to 39 decimal places is sufficient to calculate a circle the size of the known universe with an accuracy equal to the diameter of a hydrogen atom. [Wikicite](http://en.wikipedia.org/wiki/Pi#Decimal_representation). I never liked this factoid. Most calculations are not trivially multiplying one number by another number. They are repeated multiplies and additions often with a lot of non linear operation thrown in. If you are adding up a lot of really small numbers to get something large you need a lot of decimal places otherwise the final answer will not be very accurate. ---
**Author:** ![sleestak](https://avatars.discourse-cdn.com/v4/letter/s/919ad9/32.png) [@sleestak](https://boards.straightdope.com/u/sleestak)\ **Post date:** [November 9, 2010, 12:19am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/35 "2010-11-09T00:19:19Z")
> [@BrotherCadfael](#): > > Richard Feynman, in the “Los Alamos From Below” chapter of _Surely you’re joking Mr. Feynman_, described the process by which the Manhattan Project computed the necessary calculations for the first nuclear bombs. To someone from today, it is a fascinating glimpse of a wholly alien mindset. > > They set up a room full of people with huge mechanical calculators. Each person was assigned to a single operation. One person would add two numbers, and pass the calculation on to the next person, who would multiply them by something else. The next person would take the square root and pass it on, and so forth. A single calculation could take months. > > Feynman restructured the calculations to proceed in parallel. Each individual calculation still took two months, but they were able to turn out one finished calculation every two weeks. > > The book describes how they set up error checking procedures, which were essentially subroutines that recalculated intermediate results when necessary. Occasionally, an error would be made within the error correcting subroutine, and they had to set up procedures to deal with that. > > Today this could all be handled on a high-end programmable calculator in minutes. > > Truly, Feynman had a dizzying intellect. I was going to mention this. However, a slight correction. Feynman setup the program but if I recall correctly his students figured out how to correct for errors. I remember that Feynman describes walking in and seeing all kinds of different colored cards running through the system and the students just told him not to worry about it, they had it handled even though Feynman was freaking. Apparently, the students decided that when they found an error they could cut out the error and continue on with the calculation but would use different color cards for the ‘fixed’ calculation. Then they found another error and added another color. And so on. And, yes, Feynman\* was the bomb. Slee \*A friend of mine, Dick, passed away earlier this year. Dick was a physicist and a damned good one. He was extremely smart and quite a character. He had this thing about Feynman. Dick and Feynman worked together at one point, QED stuff I think. Apparently Feynman got Dick with one of his famous practical jokes because _every_ time you’d mention Feynman around Dick, Dick would go off on a rant about how Feynman wasn’t _that_ smart. Dick was a pretty mellow guy but mentioning Feynman really set him off. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 9, 2010, 12:28am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/36 "2010-11-09T00:28:47Z")
> [@](#): > > I never liked this factoid. Most calculations are not trivially multiplying one number by another number. They are repeated multiplies and additions often with a lot of non linear operation thrown in. If you are adding up a lot of really small numbers to get something large you need a lot of decimal places otherwise the final answer will not be very accurate. If your calculation will fail due to roundoff error from using a value of pi to 39 decimal places, then your calculation is set up incorrectly, and will fail no matter how many decimal places you have. I’m hard-pressed to think of any application where 8 or 10 digits wouldn’t be enough. Oh, there are exceptions, of course, like the infamous “tangent of 10^100” question which stumped Feynman, but you have to contrive those. That would never come up naturally. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [November 9, 2010, 12:48am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/37 "2010-11-09T00:48:31Z")
> [@Kevbo](#): > > This will overflow after “only” 96 years of operation, by which time it will not be my problem. (but it will warn the operator if/when it gets close…cause after Y2K I am anal that way.) On behalf of that yet-to-be-born operator: Bless you!! (I still have a copy of the memo I wrote as a programmer in 1989, warning my boss about the problems our software would have ten years hence. Of course, he was more concerned about the current customers than the future ones.) ---
**Author:** ![appleciders](https://avatars.discourse-cdn.com/v4/letter/a/85f322/32.png) [@appleciders](https://boards.straightdope.com/u/appleciders)\ **Post date:** [November 9, 2010, 12:50am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/38 "2010-11-09T00:50:08Z")
> [@Thudlow\_Boink](#): > > Slide rules and logarithm tables were standard equipment for people who had to do lots of relatively complicated calculations (e.g. engineers) more than about thirty years ago. > > (I vaguely recall reading science fiction stories by the likes of Robert Heinlein where someone piloted a spaceship with the help of a slide rule.) Robert Heinlein describes in one of his essays solving a Hohman orbit problem where the ship performed a gravity assist maneuver on a planet. He and his wife each covered a table with butcher paper and solved the problem by hand; this way, they could check each other’s work. It took them three days to solve it, and this work represents one paragraph in _Space Cadet_. That’s dedication. ---
**Author:** ![sevenwood](https://avatars.discourse-cdn.com/v4/letter/s/6de8d8/32.png) [@sevenwood](https://boards.straightdope.com/u/sevenwood)\ **Post date:** [November 9, 2010, 1:22am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/39 "2010-11-09T01:22:16Z")
> [@Thudlow\_Boink](#): > > Slide rules and logarithm tables were standard equipment for people who had to do lots of relatively complicated calculations (e.g. engineers) more than about thirty years ago. Ah, yes, the memories. When I entered engineering school as a freshman back in '66 the first big decision we had to make was whether to purchase a Decilon or a K&H slide rule. We engineering students proudly attached their cases to our belts and wore them dagger-style all day. And never really understood why the Arts & Sciences students thought of us as geeks. ---
**Author:** ![BrotherCadfael](https://avatars.discourse-cdn.com/v4/letter/b/977dab/32.png) [@BrotherCadfael](https://boards.straightdope.com/u/BrotherCadfael)\ **Post date:** [November 9, 2010, 1:22am UTC](https://boards.straightdope.com/t/complex-calculations-before-computers/559926/40 "2010-11-09T01:22:33Z")
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