Any problem I could solve with a calculator I could solve with a pencil and paper. It’ll just take longer. Calculating nth-roots and logarithms by hand is frustratingly laborious, but it doesn’t require anything more than multiplication and division.
I’m probably good up through most single-variable calculus. Past that point I’d probably need a reference book along with a pencil and paper.
Oh, screw you. I can’t do logs without a book by hand, nor any kind of root>2. Man/Woman up – are you a savant or ain’t you? It would take me all day to take a natural log of any kind, and I think I speak for most people who aren’t Mr. Spock.
OK, maybe I’m just jealous.
carpentry in nonmetric countries
It frequently comes up when you want to spread N holes, joists, etc. evenly over span X. If N is three or seven or six, etc. then the decimals are messy (how man places do you want to use?). It is often easier to calculate with fractions then decimalize as the last step if needed (frequently in machine shop work).
I have also had many arguments when I use fractional hole sizes on a print that is otherwise decimal dimensioned… I want the machinist to use a 9/16" bit, damnit.
To elaborate on what johnpost said, not just carpentry but virtually everything having to do with woodworking. Lumber is sold fractionally, drill bits are fractional sizes, roof slope is expressed as a ratio. You cannot escape fractions in furniture making, woodworking, carpentry, cabinetmaking, or related professions.
I’m not a savant. I am good at math. To tell you the truth, it’s been many years since I tried to calculate the logarithm of some arbitrary number by hand. It might take me all day, too. But I know how (to approximate them using an algorithm like Newton’s method). And it doesn’t take anything more than mechanical operations to actually get the answer once you know the method.
Cool! Thanks!
To be clear, there was a typo. We learned trigonometry, but I NEVER did it without consulting a calculator; not even in high school could we have done it by hand.
To Astro:
Europeans, Mexico, and Canada use Metric, but we Americans are still holding on to Standard measurements with our fingernails.
If all the goods in the fabric store are sold by yardage, if the patterns identify the amount of fabric you need in yards, it helps a helluva lot to stay in the same format.
Then, of course, should you decide to work in all-Metric measurements, you’d need to convert Standard to Metric, and you’d use…a ratio.
~VOW
Now I feel like an idiot – I didn’t even know there was an algorithm for getting logs and base-e logs except from a book of tables. I did disclaim I don’t do arithmetic like a pro, but I love math (metamath! number theory!), but there’s always something in the works, like a monkey-wrench ([muttersunderbreath]like you[/muttersunderbreath])!
I just had an epiphany.
I’ve known since my own elementary school days that when multiplying fractions, the numerators and denominators could cross-cancel. But I just realized why:
When you multiply fractions like 35/16 * 8/7, you’re really saying 35 * 1/16 * 8 * 1/7. And the associative (?) property of multiplication applies: you can do long strands of multiplication in any order you want. Change the order to 35 * 1/7 * 8 * 1/16, add some parentheses so you have (35 * 1/7) * (8 * 1/16), and it’s easy-peasy.
Anyway, yeah. I had to wave my hands in the air as though I were clutching at the numbers, but I could do that in my head. I never really grokked trig, though.
Didn’t “Peter Griffin” on that cartoon show say “fractions is hard”? Just saying. There are pretty definite rules and techniques, even if you’re not “Peter Griffin” the object on that show.
ETA like your breakthrough, above poster – yay math! Actually, pretty clever, I guess. Maybe I’ll use that shortcut sometime.
It’s not easy. Obviously it’s possible (or there would be no tables of trig ratios, nor could a calculator do it), but apart from a few standard angles it gets messy very quickly.
And the same comment applies here. In principle I know the method but in practice I know why compiling books of logs in the pen-and-paper days was a task for a team of expert mathematicians and took a mighty long time.
[Obi-Wan]You’ve taken the first step into a larger universe.[/Obi-Wan] The way I like to put it is that a fraction is just a division sum you are carrying around until you need to work it out. The thing on top is what you multiply by, the thing on the bottom is what you divide by; and if in one fraction you’re multiplying by something, and in another fraction you’re dividing by it, they cancel each other out (since multiplying and dividing by the same thing just gets you back where you started).
What Malacandra just said. There was a reason all the trig problems in high school, or calculus problems involving trig functions, generally involved taking sines and cosines of multiples of pi/6 or pi/4 radians (30 or 45 degrees, respectively). Everything else was exceedingly messy, and the only way to get y-values back in the pre-calculator era was to use the trig tables in the back of the book.
Logs were even worse.
When I was teaching this stuff in the mid to late 1980s, it was the last gasp of the days where you couldn’t just require your students to have calculators, despite the fact that they could get a TI-30, IIRC (which had the sin, cos, tan, ln, and log-10 buttons, as well as lots of other useful stuff) for about $20 if they watched the ads, so you had to teach them how to use the tables.
Which meant you had to teach motherfucking interpolation, which wasn’t exactly a complicated concept, but it was fundamentally a detour, it took time away from the stuff you really wanted to spend more time on, and always seemed like it was just one idea too many for most of the students’ minds to juggle.
When I was born, JFK was alive.
I can still multiply fractions, solve trig problems, differentiate polynomial equations, integrate, and even explain Baye’s Rule.
Well, I play with money at work quite a bit, so I’m used to doing math problems involving fractions and decimels. The first math problem in this thread does not strike me as particularly challenging. I would, however, start to run into problems trying to do some of the things I “learned” in college algebra. Quadratics–I thought they were kind of fun when I was in school, so I could probably figure out one again is somebody would give me a quick refresher. Some of the other stuff–logarithms, that hypotenuse stuff, the algebra of matrices–forget about it. I never really did get the hang of logs, and although I did well enough in all the rest, I’ve forgotten how to do most of it. In truth, I never really understood why I had to learn some of the things the Department of Mathematics said I had to learn. It was just “learn this formula to get a good grade so you can meet your undergrad requirements and never mind who does this kind of math in real life and why.” So I memorized the formula, took the test, forgot about the formula, memorized the next formula, etc., got an A on my final, and moved on to the coursework that actually seemed to pertain to what I wanted to be when I grew up. I’ve never done, to my knowledge, a real trig or calculus problem. I suppose I’ll eventually have the opportunity if my kids have to take such courses in school and run into difficulties (that, or rent a tutor).
In the above post, hypotenuse should read asymptote. For some reason those two words are stored in the same mental bin, and I grabbed the wrong one. Sorry. Duh!
I’m cracking up at myself, reading some of the postings discussing fractions! I was wincing at the word “cancel.”
I had a FANTASTIC teacher for my accelerated math classes in high school. She could really break the problems down so we could see the steps, and then figure things out for ourselves.
One thing she was INSISTENT upon, though, was never, ever, EVER use the word “cancel.”
In her early years as a math teacher, she had to teach the remedial classes. And explaining fractions and multiplication or division to remedial students was definitely challenging. She said with the word “cancel,” kids thought they had free rein to just start crossing numbers out, bam, bam, bam.
She made a point, thereafter, to use the word, “reduce.” It curtailed somewhat the gung-ho slashing of numbers in numerators and denominators, and it gave enough pause to THINK that you’re doing an actual arithmetic process, rather than eliminating superfluous numbers that are just in the way.
~VOW
What, you mean you can’t simplify the fraction 16/64 by just cancelling the 6’s in the numerator and the denominator? It works for me: 16/64 = 1/4.
That’s the easy one. More impressive: 1999999/9999995, cancel the 9s, = 1/5. 
I would do that kind of problem in my head, unless I was sitting in front of my computer. I don’t know about you, but there is a black hole that swallows calculators in my house. I just keep an app for WolframAlpha on my start page.
http://www.wolframalpha.com/input/?i=1+1%2F7+multiplied+by+2+3%2F16
I often find it is good to keep rough approximations in my head. I can always do the exact calculations if you need them later. 25 mm to the inch or 5 microradians to an arc-second or 400,000 km to the moon are often good enough for quick calculations.