What's the point of rationalizing?

[nehpest]

Sure, gfauxpas. The algorithm yakk described will terminate in a finite number of steps - the algorithm being one that solves "given an integer n, find the n^th digit in the decimal expansion of sqrt(3)/3." This algorithm will always terminate, usually after some number of steps depending on n.

[Xanthir]

No, you're still misunderstanding Yakk.

The algorithms he's referring to *do* terminate in a finite number of steps. They take as input the digit you want, do some finite amount of work, and then output that digit.

There's nothing infinite here. The algorithm itself, though, represents a real number, in all its infinite-digit glory. But it never *outputs* an infinitely long string, or takes infinitely many steps to do some work.

[gmalivuk]

what's the point of using more digits, if I can't use enough digits to get a true statement?

Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.

Surely you can think of some way in which it is better to use 3.14159 for pi instead of just 3, right?

[gfauxpas]

what's the point of using more digits, if I can't use enough digits to get a true statement?

Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.

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.

[gmalivuk]

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".

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]

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".

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.

How about "the standard that doesn't lead to contradictions and falsehoods"?

edit: I should explain myself. Just as historically the standard was different before electronic calculators, the standard was different before Nicolas Bourbaki. I don't want to be tied down to obsolete standards when math is ideally going forwards in sophistication.

[antonfire]

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.

[gfauxpas]

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.

Sounds perfectly reasonable, and I found some reasons in the thread to be good reasons. But some of the reasons I did not like, namely those that were implying that real word applications are more important or are better somehow than rigorous proofs.

[heyitsguay]

gfauxpas, you seem like someone who is very interested in math but fairly new to rigorous mathematical proofs. That's ok, we were all there at one point. However, in time you'll come to understand them in a broader context of "doing" mathematics (or at least ought to, if you want to be good at it). It seems like right now, you are treating math culture (conventions, like putting expressions in a standard form, are a part of this) as if it is entirely centered around proofs, but it isn't, and for better or for worse your opinion will not change that.

And there are good reasons for this, even if they do not seem apparent to you now, and not all of the reasons concern themselves with their utility for proofs. But it doesn't really matter. I hope you keep pursuing mathematics, and do lots of proofs, and get into arguments like these so that you come to understand the math community. For one thing, as you get into more advanced mathematics (i.e. beyond a first-year undergrad analysis or algebra course or what have you), these issues go away. Nobody will care about these things, but it's because those classes are dealing with a group of students different from the group that's taking courses where talk of "rationalizing" is even necessary. Really, what you are arguing about is ultimately a nonissue to working mathematicians, and hopefully you'll come to see it that way yourself.

[gfauxpas]

heyitsguay thank you so much for your answer. I really am not trying to be annoying or anything but I think my questions might come across as argumentative. But I'm not arguing for the same of arguing, I'm just trying to figure things out. Your comment was presented really well.

[PM 2Ring]

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.

Code: Select all

#! /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,


[gfauxpas]

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 x³-(1/3)?

[PM 2Ring]

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 x³-(1/3)?

Thanks. I guess it would: Newton's method is pretty quick for extracting roots. I'm getting a bit too tired to muck around with an integer version right now, but here's a simple version I just whipped up in bc, which shows how quickly it converges:

Code: Select all

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


[gfauxpas]

Thanks everyone for explaining things to me. I found another "simplification" that my teacher likes that I'd like explanation for.

\sqrt{\frac 4 {x^2} - 2x} should be written as:

\frac{\sqrt{4-2x^3}}{|x|}

Anyone know why the second one is better? Seeing as my teacher's intent was not to make grading easier, she said, but rather to encourage us to reduce complexity in our work.

And yes heyitsguay I know it doesn't make a difference to working mathematicians, but if I have to do it to avoid getting points off I'd like to know why I'm doing it.

[gmalivuk]

Well it seams a lot easier for me to see when the second expression is positive, negative, zero, or imaginary.

[gfauxpas]

Well it seams a lot easier for me to see when the second expression is positive, negative, zero, or imaginary.

Isn't the expression always positive and real anyway? Because sqrt(x^2) = |x| is false in C, I thought, and also sqrt(x/y) = sqrt(x)/sqrt(y) also only works for positive reals? Sorry my knowledge of complex numbers is weak, but it's a good answer, thanks.

edit: nm was confusing argument with image, whoops

[Talith]

In the second one, you can, at a glance, see that the stuff under the radical will be positive if and only if x³ > 2, ie when x > cuberoot(2). In the first, it's when (4/x²) > 2x ; which isn't something you can really visualise without either drawing a graph or, as your teacher I think has been hinting at, reducing it to a more manageable form, as in the second.

oh and to add, the reason you'd want to know when the stuff under the radical is positive or not, is because you want to know if your solution lies in the real numbers or the strictly complex numbers.

[gfauxpas]

Thanks gmalivuk, talith, that's a good answer.

[erik542]

what's the point of using more digits, if I can't use enough digits to get a true statement?

Except, we don't live in an infinite-precision universe. We deal instead with approximations, and treat closer approximations as generally better.

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.

I'd rather have 3.14159 for pi when a machinist needs to make a thingy. I'd rather have pi for pi when I'm doing math / physics. If I see 8-x^2 floating around, I'd rather have sqrt(8); most otherwise 2sqrt(2) is just easier.

To me this scenario depends on the context. If you're getting counted off for this in an ODE class, they're just being anal. If you're getting counted off in precal, then it's a bit more fair game.

[Dason]

If I see 8-x^2 floating around, I'd rather have sqrt(8); most otherwise 2sqrt(2) is just easier.

I hope you don't just replace equations with a zero of said equation very often.

[gorcee]

ITT: rationalizing rationalizing!

The whole thing about rationalizing, in the grand scheme of things, is kind of meh. Yeah, you see things like 1/sqrt(2pi sigma) all over the place in more advanced math, but on the other hand, if you had sqrt(2)/2, you would almost never convert that to 1/sqrt(2) and leave it as that (for an interim calculation, maybe). Although there are reasons to write some things in certain ways, if that information was of practical use to you, you'd quickly learn to do it that way, in that context, anyways.

I suspect, however, that the sticking point of rationalization in high-school level precalc is that most teenagers have shitty handwriting, and it's way more of a pain in the ass to try to figure out if someone meant to write 1/sqrt(2) or, say, 1/72, when the fraction bar thingie and the radical merge together, and when you have to grade 25 papers.

[Dopefish]

I recently had to deal with evaluating

\lim_{n\rightarrow\infty}(\sqrt{n^2-n}-n)

and thought of this thread, as it's a case where you essentially need to un-rationalise things if you want to be able to see what the limit is (hint:it's not 0). What's more, this sort of limit could conceivably come up in a first year calc class, so you don't need to be particularly advanced before rationlizing can make things harder rather then easier.

That said, if the prof/TA has given you a standard to follow, you should absolutely follow it, as they likely have a reason for it, even if it's simply for convenience rather then some mathematically significant reason.