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Kewangji wrote:Someone told me I need to stop being so arrogant. Like I'd care about their plebeian opinions.
Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.gfauxpas wrote:what's the point of using more digits, if I can't use enough digits to get a true statement?
gmalivuk wrote:Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.gfauxpas wrote:what's the point of using more digits, if I can't use enough digits to get a true statement?
Surely you can think of some way in which it is better to use 3.14159 for pi instead of just 3, right?
Except that, as mentioned repeatedly before, there is an additional benefit from consistency in itself. And if you're going to pick one standard, it may as well be the standard that has existed since before fast electronic symbolic mathing was a thing.gfauxpas wrote:I'm not saying there isn't merit to using approximations, I'm saying that there are benefits either way and it seems silly to pick one way and say that it's "better".
gmalivuk wrote:Except that, as mentioned repeatedly before, there is an additional benefit from consistency in itself. And if you're going to pick one standard, it may as well be the standard that has existed since before fast electronic symbolic mathing was a thing.gfauxpas wrote:I'm not saying there isn't merit to using approximations, I'm saying that there are benefits either way and it seems silly to pick one way and say that it's "better".
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
antonfire wrote:People don't pick one way and say it is better. People pick one way and say do it this way, it makes it easier to grade.
Hopefully, everybody here is pretty familiar with the fact that the "standard" way you're told to write things down in math class is not always the best way to write things down. You wanted reasons why you might want to consistently rationalize the denominator, and you got them. Does this mean you have to rationalize the denominator all the time every time? No, and I hope you don't. Should bring your expressions to the form whoever is looking at your work prefers it in? Yeah, because if you don't you're kind of a dick.
#! /usr/bin/env python
''' Calculate sqrt(3), digit by digit, using Newton's method.
Tested against bc to 10,000 digits '''
import sys
n = len(sys.argv) > 1 and int(sys.argv[1]) or 50
#sqrt(3) ~= x / sqrt(b)
x, b = 17, 100
print x / 10.0,
for i in xrange(n-1):
d = 15 * b // x - 5 * x
x = 10 * x + d
b *= 100
print d,
PM 2Ring wrote:Inspired by an example in this thread, here's some Python code that generates a list of decimal digits of sqrt(3). The number of digits desired is given as a command line argument, default = 50. It's not particularly fast, but it works.
gfauxpas wrote:PM 2Ring wrote:Inspired by an example in this thread, here's some Python code that generates a list of decimal digits of sqrt(3). The number of digits desired is given as a command line argument, default = 50. It's not particularly fast, but it works.
This is cool. Would the program run as well if it was finding the roots of x^{3}-(1/3)?
scale=60;y=1/3;x=0.7;for(i=0;i<6;i++){x+=(y-x^3)/(3*x*x);x}
.693424036281179138321995464852607709750566893424036281179139
.693361280031056608187877105285456470527287087157008654175118
.693361274350634751380697005708477537870834137886053422306891
.693361274350634704843352274785964918961099171363629685977591
.693361274350634704843352274785961795445935113457754036565864
.693361274350634704843352274785961795445935113457754036565864
gmalivuk wrote:Well it seams a lot easier for me to see when the second expression is positive, negative, zero, or imaginary.
gfauxpas wrote:gmalivuk wrote:Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.gfauxpas wrote:what's the point of using more digits, if I can't use enough digits to get a true statement?
Surely you can think of some way in which it is better to use 3.14159 for pi instead of just 3, right?
Definitely. Can you think of some way in which it is better to use π instead of 3.14159? How about when doing proofs?
I'm not saying there isn't merit to using approximations, I'm saying that there are benefits either way and it seems silly to pick one way and say that it's "better". And if I am doing a proof by the way, I view the environment I'm in as an infinite precision universe. If I'm doing applied math, it's different.
I apologize for giving the impression that I was being snarky or obnoxious, I wasn't trying to. I will have to read those posts about algorithms several times before I get it, but thanks for explaining them to me.
erik542 wrote:If I see 8-x^2 floating around, I'd rather have sqrt(8); most otherwise 2sqrt(2) is just easier.
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