You don't think so? I've got a calculator from 1996 laying around
somewhere that can solve just about any calculus problem by typing in
solve( and then the problem. Same for algebra, trig, or any other
branch of math you care to name. I don't imagine that they've gotten
less powerful over time.
But that's the extreme case- I know a lot of people who can't do long
division, and don't care to know how because they have a calculator.
But then when they don't have a calculator handy, they're lost. That
would indicate to me that they don't understand the math, they just
know how to operate a calculator.
Perhaps they aid in some, but calculators don't "solve the calculus
problems" I've seen. That's just it; they are an aid, not "the
answer." Solution of problems in calculus involves a pretty thorough
knowledge of calculus and all that precedes it, and most of that is
done with the calculator we're born with.
That's what I mean.
All said and done, the calculator is a tremendous asset *after* an
understanding of basic principles of the subject it is supposed to
assist. Same with woodworking [thought I'd throw that in.] The tools
don't make the master woodworker, but good tools sure help. Lots of
people have much better tools than some superb craftsmen stuck with
less, but all they do is build stepstools. The calculator [ or CAD
program or whatever] is good in the right hands and next to useless in
the wrong ones, and really not all that necessary, as you pointed out.
Not true at all. You can have any device you want, that will give you any
answer you want, but it will just sit there and act stupid if you don't know
what to ask it. Even then, you have to now if it is giving you the right
answer. Take the segmented circle problem under discussion. There is no way
that any calculator is going to figure that out for you. You have to know
what you are wanting to do, lay it out and devise a formula (whether it is a
standard one or one you devised yourself). Only then, fed the correct
formula, is that calculator of any use and all it is then doing is replacing
the trig tables that yo feel are better. When ariving at and asnswer, before
commiting time and wood to that number, you will want to turn that formula
around, work it backwards and see if it is still correct. Many mistakes can
be found this way. The calculator is a dumb slave to the human brain
commanding it. If you don't know what you are doing, that device will do
Prometheus (in firstname.lastname@example.org) said:
| But that's the extreme case- I know a lot of people who can't do
| long division, and don't care to know how because they have a
| calculator. But then when they don't have a calculator handy,
| they're lost. That would indicate to me that they don't understand
| the math, they just know how to operate a calculator.
Hmm. I studied math through an advanced course in partial differential
equations - and, when it comes to solving even simple trig problems,
I'm one of those who're lost without a calculator.
I suppose I could do a little (lot of?) review and calculate my own
trig function values:
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
[ Anyone who did this in their head for sin(18 degrees = pi/10
radians) to solve Burt's problem can pat themselves on the back and
disregard the remainder of this post. All of those who evaluated /pi/
with a series approximation have a surplus of brain cells and a
serious need to "get a life".]
Ah. Glad I'm not alone :-)
It's a Good Thing, IMO, to understand the math - but I don't think
it's bad to not understand the math. Mathematics *and* calculators are
Burt has a formula and, presumably, a calculator - and can grind out
whatever chord lengths he wants. He doesn't /need/ to understand the
mathematics. It wouldn't hurt if he did, but all the understanding in
the world won't produce any better results.
DeSoto, Iowa USA
Nobody needs to understand math. Well, that's just...
Hm... Hum... What to say?
I fart in your general direction. Your mother was a hamster and your
father smelled of elderberries.
I despise your comment as quoted above, and the attitude that allowed
it to be verbalized/typalized.
It's, simply, despicable. IMO.
No more to say on this topic, except "SOHCAHTOA".
~ Stay Calm... Be Brave... Wait for the Signs ~
Dave Balderstone (in
|| Burt has a formula and, presumably, a calculator - and can grind
|| out whatever chord lengths he wants. He doesn't /need/ to
|| understand the mathematics. It wouldn't hurt if he did, but all
|| the understanding in the world won't produce any better results.
| Nobody needs to understand math. Well, that's just...
...not what I said at all. :-)
Not all people can be all things; and everyone has different aptitudes
and talents to develop. It took me a long time to internalize that not
everyone needed to become a programmer in order to use a computer to
good advantage. The same applies to mathematics.
| Hm... Hum... What to say?
| I fart in your general direction. Your mother was a hamster and your
| father smelled of elderberries.
Well, ok - but the smell hasn't reached Iowa yet. In case you hadn't
noticed, the prevailing wind is from the southwest this time of year -
I would caution you against over-exerting yourself. :-D
My mother would have smiled and assured you that she did her best to
be a proper /Scottish/ hamster. From what I've been told, I think my
father would have grinned and said that he'd been told worse. (I wish
he'd had a chance to become elder-than-26.)
| I despise your comment as quoted above, and the attitude that
| allowed it to be verbalized/typalized.
| It's, simply, despicable. IMO.
| No more to say on this topic, except "SOHCAHTOA".
I had to go to Google for that one. Neat. I wish you'd told me about
that a half century ago, when I was struggling to keep all of those
straight in my head.
DeSoto, Iowa USA
You didn't have that? Ah, that's just criminal. When one of the
other guys at work found out that I was helping one of our co-workers
learn trig, he screwed up his face a bit, thought about it for a
minute, and then came up with "Do you mean SOHCAHTOA?" It was
hilarious. Useful acronym, though.
The struggle was worth it. Those who memorise the "trick" never
really understand the main ideas. You are a lot better off having
learned to look for the sides and angles and ratios in each working
problem. You might not realise it, but you gained a lot more insight,
never mind the struggle.
*snort* who needs a series approximation?? I've had the numerical value of
pi, out to _20_ decimal places, memorized for more than 35 years.
I have, however, *rarely* needed more than 6-place accuracy for same.
Now, the numerical value for 'cornbread', THAT's a different matter. <grin>
Note; I also used to have a handful of common log values memorized.
And a dozen or so trig values. With that, and a 'half-angle' formula, you
can do faily impressive pencil-and-paper 'approximtions'.
Robert Bonomi (in email@example.com) said:
| *snort* who needs a series approximation?? I've had the numerical
| value of pi, out to _20_ decimal places, memorized for more than 35
| I have, however, *rarely* needed more than 6-place accuracy for
| Now, the numerical value for 'cornbread', THAT's a different
| matter. <grin>
| Note; I also used to have a handful of common log values memorized.
| And a dozen or so trig values. With that, and a 'half-angle'
| formula, you can do faily impressive pencil-and-paper
Lucky you! I've never been able to memorize stuff like that; but
somehow managed to remember basic formulas like the series
approximations. I can remember thinking early on that the actual
numbers weren't as important as the relationships that produced them.
Now I see 'em as two ends of the same stick that different people feel
comfortable grasping in different places.
Heh heh. Just remembered the physics prof who got repeated cases of
the heebeegeebies because I started nearly all problem solutions with
"F = ma" and derived whatever I needed from that. It was my first real
clue that we're not all wired alike.
Pencil and paper works - but (IMO) there are better tools like slide
rule, calculator, and computer to make calculations faster and easier.
A side note: a couple of years ago I wrote a tiny/fast sine/cosine
subroutine that divided a quadrant (quarter circle) into 256 parts and
used a table of 256 16-bit numerators and a common denominator of
65535. The subroutine folded all angles into the first quadrant and
interpolated to produce sine and cosine values accurate to +/-
0.0000005; it made me wish I could memorize the table.
Cornbread is good. I have the formula around here somewhere...
DeSoto, Iowa USA
See, somehow I knew you saw it my way- even if you don't agree with
the calculator bit, you just proved you're math nerd. Reminds me of
my TI-OS program for simulating synthetic division. Worked great, but
everyone who saw it thought I was nuts.
Okay, I fold. I still like the trig table in the context I needed it
for though. Let me fill it in a little, so I'm not completely nuts to
you all. We're working these problems on the motor cover of a huge
vertical CNC bandsaw amid tons of coolant and steel swarf. The
employer provides regular four-function calculators- not scientific
versions. A calculator only lasts a couple of weeks, on average. A
trig table lasts for 6 months or better once it's laminated (if the
fraction-decimal charts are anything to go by), and it allows us to
solve the problems in question without having to buy a much more
expensive calculator every couple of weeks because some dummy knocked
it into the coolant or the keys got jammed up by tiny steel chips.
The guy I'm teaching appreciated it as well- even though he is aware
that calculators that will evaluate sin/cos/tan values exist, he's got
4 kids at home that (evidently) like to break things, so he doesn't
have one of them.
Same logic applies to calipers- Sure, it's easier to use a dial or
digital caliper when measuring, but I still use a vernier. It's not
because it's inherantly better, it's just better for the environment
I'm using it in, and lasts a heck of a lot longer!
I suppose there's an argument for just dropping in here again and
asking if the application changes, but to me it's one of those Give a
man a fish V. Teach a man to fish senarios. And, if the equation is
not being used properly because of some difference in the problem, it
can lead to some pretty large errors pretty quickly because he didn't
know how to double-check it. As another poster noted, you can have
any machine you want, but it won't do anything if you don't know what
to put in it.
I see your point, though.
That argument is always good. If you use a vernier constantly, you'll
be just as good with that as someone with an electronic instrument.
That doesn't mean everyone should dash out and buy an abacus.
OK, if people promise not to throw things ...I used to teach math, and
at years end would show others how to use their calculator
effectively, or how to use a spreadsheet or suitable program, or make
up one for them if they felt the need but didn't know how. I used a
slide-rule myself. Why? I could use all of the above and more, but
had used the slide-rule so often that it was second nature; much
easier for me in the long run, if not for the others.
Moral: Use whatever works! Just build the bloody thing. It's
woodworking, not brain surgery.
The length of a side of an "n-ogon" inscribed in a circle is:
If you consider the angle out from the center of the circle, to the ends of
the section (which is called the 'chord') it's easily remembered as:
"twice the sine of half the angle".
How to confuse people -- note that you scale the above by the radius of
the circle. *BUT* there is that little '2x' factor sitting in front of
things. 2x the radius is the diameter. so you can use
and seriously confuse the spectators.
Robert Bonomi (in firstname.lastname@example.org) said:
| The length of a side of an "n-ogon" inscribed in a circle is:
I /almost/ hate to do this to you, but the length of a side of an
"n-gon" inscribed in a circle of radius r is:
DeSoto, Iowa USA
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