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.
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.
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).
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.
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.
There’s an interesting story 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.
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.
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.
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.
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.
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.
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.)
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.
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.
