What's the point of rationalizing?

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gfauxpas
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What's the point of rationalizing?

Postby gfauxpas » Wed Jan 04, 2012 10:32 pm UTC

My latest Calc professor is really into algebraic simplification. She'll take off points for, e.g., leaving the answer as [imath]\sqrt 8[/imath], [imath]\frac 1 { \sqrt {x}}[/imath], and [imath]e^{x \ln 2}[/imath].

What's the justification for this? I haven't got a satisfactory answer out of her, but she's not the first teacher I've had who cared.

Regarding the first, isn't the very definition of [imath]\sqrt{8}[/imath] the unique positive solution to [imath]x^2 = 8?[/imath] At least, that's how I've always looked at it. If so, [imath]2 \sqrt{2}[/imath] means "the unique positive solution to [imath]x^2 = 2[/imath] scaled by a constant multiple [imath]2[/imath]". That's more complicated!

As to the second, I heard an unsatisfactory answer. That is, if you have something like [imath]\frac 1 {\sqrt{2}}[/imath], and you wanted to evaluate it, you would end up with something like this: [imath]1.414.....\vert \overline {1.000000...}[/imath] there's no way to start the process. But if you have [imath]2\vert \overline {1.414...}[/imath], you could start evaluating it. This also seems strange to me. You can't finish the process anyway, so what's the point of starting?

Lastly, regarding [imath]2^x[/imath], I could understand her viewpoint if we're dealing with rational indices. But if we're dealing with a continuous function, [imath]2^x[/imath] is less meaningful than [imath]\exp({{x \ln 2}})[/imath], isn't it? Especially in Calculus, when base [imath]e[/imath] is so much more pleasant than other bases. So, what's the big deal?
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Re: What's the point of rationalizing?

Postby saus » Wed Jan 04, 2012 11:23 pm UTC

There are lots of ways to write the same thing, so it makes sense to have a standard way to write them. Some students might not recognize that [imath]e^{x\ln 2}[/imath] is simply [imath]2^x[/imath]. Everyone knows sqrt 2 is about 1.4, so 3*sqrt2 can be seen to be 4.2 easily, sqrt18 isnt as quick to calculate. Same with 1/sqrt(2), roughly how big is that number? well it's half of sqrt2, or about .7.

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Re: What's the point of rationalizing?

Postby gfauxpas » Wed Jan 04, 2012 11:28 pm UTC

That's a reasonable answer, thanks. But it seems to be a pointless argumewnt if it's the class that loses points for forgetting to rationalize, and the teacher who is the one who grades it. I guess I agree with you in terms of practical applications, physics etc, but in things like problem solving it seems unnecessary. You don't need to know where on the number line [imath]\sqrt 8[/imath] is to integrate from [imath]x = -\sqrt 8[/imath] to [imath]x = \sqrt 8[/imath]. But I'll think about your answer more.

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Re: What's the point of rationalizing?

Postby lightvector » Thu Jan 05, 2012 2:25 am UTC

I think many of your examples are not bad left as-is.

Particularly [imath]\frac 1 { \sqrt {x}}[/imath]. Losing points for not "simplifying" it to [imath]\frac {\sqrt {x}} { x}[/imath] is absurd. Leaving a radical in the denominator is extremely standard in mathematical notation, especially if the contents of the radical are not known to be rational. For example, a large number of the equations on http://en.wikipedia.org/wiki/Normal_distribution.

The specific details and context do matter. Often radicals are simplified by pulling squares out because they can be combined with other radicals. It's not uncommon to be able to pull out squares from many different radicals, and rationalize things so that all the numbers you happen to be working with end up as just multiples of a few common ones, particularly [imath]{ \sqrt {2}}[/imath], [imath]{ \sqrt {3}}[/imath], and [imath]{ \sqrt {5}}[/imath]. In such a context, rationalizing is very helpful. Yet, even [imath]\frac 1 { \sqrt {4}}[/imath] can be reasonable if it's part of a series [imath]\frac 1 { \sqrt {2}}, \frac 1 { \sqrt {3}}, \frac 1 { \sqrt {4}}, \frac 1 { \sqrt {5}}, ...[/imath]. Similarly, even if [imath]2^x[/imath] is nicer alone, [imath]e^{x \log 2}[/imath] is indeed better if further computations will combine it with other exponentials or involve derivatives or integrals.

Absent any specific context, as in the final answer of a generic homework problem, I think that in a reasonable number of the examples you described, your professor's answer is a little more in line with generic convention. But enforcing this by taking away points and rigidly making everyone put things into one canonical form is a little overzealous, in my opinion.

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Re: What's the point of rationalizing?

Postby Yakk » Thu Jan 05, 2012 3:54 am UTC

Mathematics is, in a sense, a game of tautologies.

"The positive solution to x^2 = 8" is the same value as "sqrt(8)", and the same value as "2 sqrt(2)".

All three are logically the same. When someone wants a mathematical answer, what they are usually asking for is "can you get this into an easier to manipulate form".

Sometimes, that form goes from "theorem" to "true" or "false". Other times, it goes from "theorem" to "proof of theorem" (in a sense, the theorem implies the proof). Other times, from x^2=8 to sqrt(8) or 2 sqrt(2).

In this case, your teacher is asking you to turn the question into a standard form. There is an infinite number of non-standard forms -- so by giving a reasonable description of some standard form, it describes (uniquely, sometimes) a form that they want the answer in.
You can't finish the process anyway, so what's the point of starting?

A real number is a mapping from an epsilon to a rational number that, in a sense, "approximates" the "real" value of the number to within epsilon.

The "start" of 2 going into sqrt(2) gives you a really brain-dead simple way to generate such an approximation (ie, a series of decimal digits), which is a real number. Now, you could also do it directly from x^2 = 1/2, but the steps required to turn that into an approximation algorithm isn't nearly as brain dead.

As they are testing your ability to use your brain to do mathematics, asking you to reach the point where the rest of the solution is brain dead seems a reasonable thing to ask for.

...

There is also the problem of making it easy for your markers. They really don't want to spend extra time double-checking that yet another permutation of the "correct answer" is the same. And as there is a huge number of "correct answers" that are logically equivalent to the actual answer (INCLUDING the original question, as noted above), they need some method to be able to shut down the whiny students who say "but, it is logically the same as the answer you want" when the student barely made it a half-step away from the original question and didn't show much in the way of understanding.

So you set an easy to understand standard. You then ask students to transform your problems into that easy to understand standard. If they fail to be able to reach that standard, you tell them (via lost marks) that they should get to that standard. Students who cannot reach that standard form can then realize that they need help in some area, instead of thinking that they are doing fine and just need to whine a bit to markers. It isn't as if reaching 2^x from e^(x ln 2) is hard, nor is it difficult to understand the "standard form" they want -- so students who do know how to bridge that gap are not all that harmed (they lose a handful of assignment marks, and maybe have to spend a few extra seconds per question munging things into a standard form). Those who are unable to go from e^(x ln 2) to 2^x learn something -- that they still don't have the mastery of the material required.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: What's the point of rationalizing?

Postby gfauxpas » Thu Jan 05, 2012 4:36 am UTC

Thank you very much lightvector, yakk. A few questions please.
lightvector wrote: Leaving a radical in the denominator is extremely standard in mathematical notation, especially if the contents of the radical are not known to be rational.]

Thanks light. What's the significance of the radicand being rational or not?
Yakk wrote:In this case, your teacher is asking you to turn the question into a standard form. There is an infinite number of non-standard forms -- so by giving a reasonable description of some standard form, it describes (uniquely, sometimes) a form that they want the answer in.

You make a good point, Yakk. But the particular example was, I forgot the exact details, but it was something like:
"Set up a definite integral describing the area of the plane region bounded by f and g. Do not evaluate the integral". And the solutions to [imath]f(x) = g(x)[/imath] were [imath]x^2 = 8[/imath]. So the answer was something along the lines of [imath]\int_{-\sqrt 8}^{\sqrt 8} p(x) \mathrm dx[/imath]. To take off a point for not simplifying the bounds of integration seems useless if I could have evaluated the integral and got a number, but the question said I shouldn't.

As to making it easier to grade, one of my Calc professors actually said he'd prefer it if we didn't simplify if it meant that it were to make less obvious how we derived our answer. For example, let's say the question was to find [imath]D_x (2\sin(\cos 2x))[/imath] and the student writes [imath]2\cdot \cos(\cos2x)) \cdot -\sin2x \cdot 2[/imath], what benefit is there to write instead [imath]-4\sin2x\cos(\cos2x))[/imath]? To make it less obvious that the student knows the chain rule?

Though you have a reasonable point, I think you're making a bit of a slippery slope argument, unless I'm misunderstanding you (likely). "If we didn't make students simplify their answer, they could just write the original problem over again".

And lastly, we spent a significant amount of time in class learning the definition of [imath]x^y \equiv e^{y \ln x}[/imath]. What was the point of that, then? Isn't that unfair, to spend a class teaching us that and take off points for converting [imath]2^x[/imath]?

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Re: What's the point of rationalizing?

Postby Yakk » Thu Jan 05, 2012 5:10 am UTC

I don't see how the inability to currently evaluate the integral matters.

Having to mark both 2 cos(cos 2x) (-sin 2x) 2 and -4 sin 2x cos(cos 2x) as correct is strictly harder than having to mark the second one as correct. Adding more forms of correct answer makes it harder to mark. Because your answer key just got twice as big, or you require that the marker resimplify the question.

Also note that getting to -4 sin 2x cos(cos 2x) without the chain rule would be a miracle. So checking for that one form is sufficient. And necessary, if you allow a range of solutions.

You seem to be under the impression that the marker approaches the question as if the student's answer was a unique flower, and puzzles out the skills of the student from a question the marker has never seen before. Rather, the marker is looking at dozens of identical questions being answered by dozens of different students -- with (in some cases) 100s, 1000s or 10000s of students who all answered the same question before. Work that can be done once (like generating an answer key) is very cheap, once amortized.

So if you can generate a question such that the answer is unlikely to be reached without knowing your stuff, and have a standardization system such that there is a unique answer form, then marking the dozens of students becomes easy. If instead you allow each student to come up with an idiosyncratic form, it means that you (as a marker) have to reverse engineer the answer and figure out if it is the correct one. On top of that, marginal cases will occur -- instead of of saying sqrt(8), they say "bounded by the two roots of x^2 = 8" -- do you mark that as full marks or not? I mean, it means the same thing. Or they say something even more vague that is even further away.

This makes marking harder than just comparing to an answer key. If it doesn't match the answer key, the marker can then check if it is identical (taking off a mark for not being in the correct form). If that doesn't work, the marker has to then go find the error in the derivation, and determine how bad it was. (even if the marker is checking the derivation in every case, knowing that the answer in the end is right does make checking the derivation easier, as you can be lazier doing it).


Why do you learn x^y = e^(ylnx)? First, because it is a very useful intermediate step that permits entire categories of manipulation that are more annoying otherwise. Why take off points for not reformatting back to 2^x in the end? Because it doesn't match the answer key, because transforming back to 2^x demonstrates an understanding that keeping it in the form e^(xln2) doesn't, because 2^x is tidier in many cases and looks more pleasing, and because the marker/teacher/professors time is worth far, far more than the students time. (There are a lot more students than markers/professors/teachers)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: What's the point of rationalizing?

Postby PM 2Ring » Thu Jan 05, 2012 5:41 am UTC

What Yakk said.

Also, the practice of reducing expressions to a standard form is a holdover from the days before electronic calculators. Obviously, it's much easier to evaluate complicated expressions when denominators are integers, or at least rational. That's why it's not such a big deal to leave things like sqrt(pi) in the denominator - getting rid of the sqrt() doesn't make it that much easier to evaluate the whole expression.

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Re: What's the point of rationalizing?

Postby Proginoskes » Thu Jan 05, 2012 7:00 am UTC

PM 2Ring wrote:Also, the practice of reducing expressions to a standard form is a holdover from the days before electronic calculators. Obviously, it's much easier to evaluate complicated expressions when denominators are integers, or at least rational. That's why it's not such a big deal to leave things like sqrt(pi) in the denominator - getting rid of the sqrt() doesn't make it that much easier to evaluate the whole expression.


Ditto. The OP should use the approximation [imath]\sqrt{3}=1.732[/imath] and calculate [imath]{\sqrt{3}\over 3}[/imath] and [imath]{1 \over \sqrt{3}}[/imath] to three decimal places BY HAND (i.e., using long division). Which calculation takes less work, gfauxpas?

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Re: What's the point of rationalizing?

Postby gfauxpas » Thu Jan 05, 2012 11:09 am UTC

Thank you for that answer Yakk, very clear. I don't quite agree with everything you said, but you explained your point well.

Proginoskes wrote:The OP should use the approximation [imath]\sqrt{3}=1.732[/imath] and calculate [imath]{\sqrt{3}\over 3}[/imath] and [imath]{1 \over \sqrt{3}}[/imath] to three decimal places BY HAND (i.e., using long division). Which calculation takes less work, gfauxpas?


I don't understand, is this a joke? Why would the approximation of the decimal expansion be any better or cleaner than leaving it the way it is? If you're doing practical applications, fine. If not, you're just adding inaccuracy.

The equation " [imath]\sqrt{3}=1.732[/imath]" is false. It's "true enough" in the sciences and in describing real world phenomenon, but it's false.

edit: I think I see. I've been looking at each problem on a test as a mini-proof each on its own, so that's why I'm uncomfortable with rounding, as it feels like I'm weakening the proof somehow. But I think you guys are saying that a math problem is kind of like "a quest to find a certain number", and when you finally do have that number you would like it to be easy to read. Am I totally off?
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Re: What's the point of rationalizing?

Postby antonfire » Thu Jan 05, 2012 12:25 pm UTC

Proginoskes wrote:
PM 2Ring wrote:Also, the practice of reducing expressions to a standard form is a holdover from the days before electronic calculators.
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?

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Re: What's the point of rationalizing?

Postby Blatm » Thu Jan 05, 2012 12:36 pm UTC

I simplify a lot of my answers because oftentimes putting answers into certain forms will make it easier to use them later, and sometimes certain forms will be familiar to me and help me figure things out more quickly about my answer. Consequently I think a lot of conventions regarding simplification are a matter of taste. For example I almost never write [imath]\frac{\sqrt{x}}{x}[/imath], and never when dealing with functions, but I'll write [imath]2\sqrt{2}[/imath] because things like [imath]\sqrt{8} + \sqrt{2}[/imath] come up often enough in my experience, and it's a lot easier to carry around [imath]3\sqrt{2}[/imath] than [imath]\sqrt{8} + \sqrt{2}[/imath], and a lot more familiar in the sense that some properties of the number are more readily apparent when written as [imath]3\sqrt{2}[/imath].

If you're not doing any math that makes it hard to work with expressions which aren't simplified then it's probably pretty hard to see the motivation behind it. If you make sure you know how to turn expressions into equivalent expressions and do it once in a while to keep it in mind, then I wouldn't worry too much. Eventually you'll find yourself in situations where one form will be much more convenient than another.

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Re: What's the point of rationalizing?

Postby gfauxpas » Thu Jan 05, 2012 4:08 pm UTC

Okay I e-mailed her the question of why she thinks it's important, but I think the topics in the thread are interesting enough to discuss on their own merit.

"gfauxpas, The reason behind rationalizing and simplifying the result is that in most cases we are dealing with extremely long and complex problems in the real world. If you see two radicals which are equivalent but one is not simplified, you might forget to combine them. This would lead to the unnecessary length and false sense of complexity. - gfauxpas's professor"

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Re: What's the point of rationalizing?

Postby Afif_D » Fri Jan 06, 2012 4:53 pm UTC

I faced similar problems. You say to the professor that there is no universally accepted rule of simplicaton of mathematical expressions . So cutting off marks is not sensible.
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Re: What's the point of rationalizing?

Postby gfauxpas » Fri Jan 06, 2012 5:04 pm UTC

Afif_D wrote:I faced similar problems. You say to the professor that there is no universally accepted rule of simplicaton of mathematical expressions . So cutting off marks is not sensible.

Though I tend to agree with you, it's not really a great defense if she has told the class what she likes.
For example, what I like to do is, if I have an equation like f(t) = 1000exp(.00053215t), then I write
f(t) = Cert, and on the side write r = .00053215 and C = 1000. But if she says "I don't like that", then I know she doesn't like it. So the universally accepted rule in Gfp's professor's math class is "Whatever Gfp's math professor has told us she likes". What do you think?

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Re: What's the point of rationalizing?

Postby Afif_D » Sat Jan 07, 2012 10:39 am UTC

Seriously , you better tell him/her to write a full thesis on that and get it accepted.
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Re: What's the point of rationalizing?

Postby Proginoskes » Sat Jan 07, 2012 7:12 pm UTC

gfauxpas wrote:
Proginoskes wrote:The OP should use the approximation [imath]\sqrt{3}=1.732[/imath] and calculate [imath]{\sqrt{3}\over 3}[/imath] and [imath]{1 \over \sqrt{3}}[/imath] to three decimal places BY HAND (i.e., using long division). Which calculation takes less work, gfauxpas?


I don't understand, is this a joke?


No, a joke is a story with a humorous punchline. Doing the exercise clarifies the point more than explanation can.

And NO CALCULATORS!

Why would the approximation of the decimal expansion be any better or cleaner than leaving it the way it is? If you're doing practical applications, fine. If not, you're just adding inaccuracy.

The equation " [imath]\sqrt{3}=1.732[/imath]" is false. It's "true enough" in the sciences and in describing real world phenomenon, but it's false.


I wasn't sure how much of TeX was supported here. Yes, I should have written [imath]\sqrt{3}\approx1.732[/imath] instead.

[ ... ]when you finally do have that number you would like it to be easy to read. Am I totally off?


That's part of the reason. Not many people can recognize the simplified form of
[math]\sqrt{16-2\sqrt{29}+2\sqrt{55-10\sqrt{29}}} + \sqrt{11+2\sqrt{29}} - \sqrt{22+2\sqrt{5}}[/math]
right away. (Andrew Bremner wrote this up on a chalkboard while teaching Number Theory, and asked if anyone knew what it equalled. 30 seconds later, someone actually got it.)

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Re: What's the point of rationalizing?

Postby gfauxpas » Sat Jan 07, 2012 11:35 pm UTC

Proginoskes wrote:
gfauxpas wrote:
Proginoskes wrote:The OP should use the approximation [imath]\sqrt{3}=1.732[/imath] and calculate [imath]{\sqrt{3}\over 3}[/imath] and [imath]{1 \over \sqrt{3}}[/imath] to three decimal places BY HAND (i.e., using long division). Which calculation takes less work, gfauxpas?


I don't understand, is this a joke?


No, a joke is a story with a humorous punchline. Doing the exercise clarifies the point more than explanation can.

And NO CALCULATORS!


Okay with the risk of sounding obtuse:
What's the point of starting to evaluate it using long division if I know I can't finish?

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Re: What's the point of rationalizing?

Postby Meteoric » Sun Jan 08, 2012 12:12 am UTC

In the real world where rulers, scales, etc don't have radicals on them but do have decimals, it's preferable to have a decimal approximation of your answer. And as PM2Ring said and two people quoted, this is partially a holdover from the days before electronic calculators, when getting that decimal approximation was far easier after rationalizing.
Don't read that and say "aha, so it's pointless now that we have electronic calculators", either. Rationalizing a denominator lets you evaluate such things in your head quickly and easily, but only if you've practiced it enough. Doing so on your homework not only gives you that practice, it helps show your teacher that you've learned it.
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Re: What's the point of rationalizing?

Postby gfauxpas » Sun Jan 08, 2012 12:19 am UTC

I think I didn't ask my question well enough. I don't see a point in using a calculator either, because the calculator can't give me a finished answer for [imath]\sqrt3[/imath] either. Sure, in practical applications it's nice, but a calculator isn't a proof. I guess it's more of a general problem that I don't understand what a non-terminating decimal expansion means. Hopefully I'll understand it more when I take Calc III.
I view long division as an algorithm, and an algorithm isn't an algorithm unless it has a finite number of steps. So the real issue is, I have no idea what long division even means if I have to stop before I'm done.

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Re: What's the point of rationalizing?

Postby skullturf » Sun Jan 08, 2012 3:56 am UTC

But if you work with the number 1.732, what takes longer: dividing 1 by 1.732, or dividing 1.732 by 3?

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Re: What's the point of rationalizing?

Postby gfauxpas » Sun Jan 08, 2012 3:13 pm UTC

skullturf wrote:But if you work with the number 1.732, what takes longer: dividing 1 by 1.732, or dividing 1.732 by 3?


For sure the first one is harder to do. But neither 1/1.732 nor 1.732/3 is a solution to x2 = 1/3.

What's easier to analyze, [imath]\frac 1 {\sqrt x}[/imath] or [imath]\frac {\sqrt x} x[/imath]? If you had to integrate, differentiate, or find composition of functions with one of the two, which would you use?

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Re: What's the point of rationalizing?

Postby skullturf » Sun Jan 08, 2012 5:00 pm UTC

gfauxpas wrote:But neither 1/1.732 nor 1.732/3 is a solution to x2 = 1/3.


Not to infinite precision, no. But they are solutions to a high degree of accuracy. And if you want more digits, then use more digits.

I'm not claiming that sqrt(3)/3 is always better than 1/sqrt(3). Just that there are reasons for preferring sqrt(3)/3 at least sometimes.

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Re: What's the point of rationalizing?

Postby Yesila » Sun Jan 08, 2012 7:11 pm UTC

gfauxpas wrote:I think I didn't ask my question well enough. I don't see a point in using a calculator either, because the calculator can't give me a finished answer for [imath]\sqrt3[/imath] either.



Sometimes one is interested in whether or not [imath]\sqrt3-1.73[/imath] is positive or negative, for instance if it is under a radical and we know that we are only interested in it if it is real. In such cases being able to compute an approximation to [imath]\sqrt3[/imath] is sufficient to let us know if we are dealing with a real or an imaginary value.

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Re: What's the point of rationalizing?

Postby gfauxpas » Sun Jan 08, 2012 7:22 pm UTC

Yesila wrote:
gfauxpas wrote:I think I didn't ask my question well enough. I don't see a point in using a calculator either, because the calculator can't give me a finished answer for [imath]\sqrt3[/imath] either.



Sometimes one is interested in whether or not [imath]\sqrt3-1.73[/imath] is positive or negative, for instance if it is under a radical and we know that we are only interested in it if it is real. In such cases being able to compute an approximation to [imath]\sqrt3[/imath] is sufficient to let us know if we are dealing with a real or an imaginary value.


Aah that's a great example of when it would be useful in analysis. Thanks!

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Re: What's the point of rationalizing?

Postby tomtom2357 » Mon Jan 09, 2012 4:57 am UTC

Suppose in an exam question they asked you to compute 1/SQRT(3) to 3 decimal places. Which is easier, solving it in its original form or converting it to SQRT(3)/3 and then solving?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

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Re: What's the point of rationalizing?

Postby Proginoskes » Mon Jan 09, 2012 9:06 am UTC

gfauxpas wrote:
skullturf wrote:But if you work with the number 1.732, what takes longer: dividing 1 by 1.732, or dividing 1.732 by 3?


For sure the first one is harder to do.


Thank you, Jesus, Buddha, Allah, Eris, Cthulhu, Satan, etc., HE FINALLY GOT THE ****ING POINT!

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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 12:32 pm UTC

Proginoskes wrote:
HE FINALLY GOT THE ****ING POINT!


I knew it was easier to calculate from the beginning, but it still didn't answer my question. Because to skullturf's answer:

skullturf wrote:
gfauxpas wrote:But neither 1/1.732 nor 1.732/3 is a solution to x2 = 1/3.

Not to infinite precision, no. But they are solutions to a high degree of accuracy. And if you want more digits, then use more digits.


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

I'm not saying there's not a use for simplifying it during applied math. My problem is that I couldn't think of a use for it during analysis and proofs, which Yesila had a great example of.

tomtom2357 wrote:Suppose in an exam question they asked you to compute 1/SQRT(3) to 3 decimal places. Which is easier, solving it in its original form or converting it to SQRT(3)/3 and then solving?


That's begging the question. "Suppose that in an exam question you're asked to not leave radicals in the denominator".

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Re: What's the point of rationalizing?

Postby Yakk » Mon Jan 09, 2012 3:19 pm UTC

sqrt(3)/3 isn't 1.732/3 -- it is a description of an algorithm for generating the digits of sqrt(3)/3.

The algorithm sqrt(3)/3 is better than the algorithm 1/sqrt(3) in many cases.

That algorithm for generating digits is actually the number sqrt(3)/3 about as much as the series of symbols sqrt(3)/3 is -- maybe more. As numbers, 1/sqrt(3) = sqrt(3)/3, but as algorithms they don't execute the same steps, so are not identical.

[x^2 = 1/3, x>0] also uniquely describes the same number as 1/sqrt(3) or 3/sqrt(3), but the algorithm to go from that to producing digits is even less practical to use in many situations.
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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 3:25 pm UTC

I thought the definition of an algorithm required that it terminate in a finite number of steps?

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Re: What's the point of rationalizing?

Postby Yakk » Mon Jan 09, 2012 3:40 pm UTC

gfauxpas wrote:I thought the definition of an algorithm required that it terminate in a finite number of steps?
An algorithm that generates a list is an algorithm that takes, as input, a number n, and returns the nth element of the list (or, sometimes, the first n elements of the list).

Such an algorithm (together with a proof of its properties) is a real number. (not the output of the algorithm -- the algorithm itself)
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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 3:46 pm UTC

Yakk wrote:
gfauxpas wrote:I thought the definition of an algorithm required that it terminate in a finite number of steps?
An algorithm that generates a list is an algorithm that takes, as input, a number n, and returns the nth element of the list (or, sometimes, the first n elements of the list).

Such an algorithm (together with a proof of its properties) is a real number. (not the output of the algorithm -- the algorithm itself)


Okay, but the definition of a list (an n-tuple) requires that it have a finite number of elements, so even according to your definition your "algorithm" can't ever produce the decimal expansion of a sqrt(3)/3.

Edit: that is to say, your algorithm, as you defined an algorithm, can't even start expanding it. There is no list of digits that it can pull the first n objects out of, because a list has a finite number of elements.

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Re: What's the point of rationalizing?

Postby Talith » Mon Jan 09, 2012 4:08 pm UTC

If you want to be picky, then you're not running the "find the decimal expansion of x" algorithm. You're running some "fine the first n digits of the decimal expansion of x" algorithm for some n in an infinite family of algorithms.

Also, the set theoretical definition of a list doesn't say anywhere that the list has to have a finite number of elements, only that the set of elements in the list is countable.

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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 4:28 pm UTC

Talith wrote:If you want to be picky, then you're not running the "find the decimal expansion of x" algorithm. You're running some "fine the first n digits of the decimal expansion of x" algorithm for some n in an infinite family of algorithms.

Also, the set theoretical definition of a list doesn't say anywhere that the list has to have a finite number of elements, only that the set of elements in the list is countable.


Oh this is interesting, where can I read about this more? I thought a list had two possible definitions:

A) An ordered pair, where each object in it can be another ordered pair.
B) A sequence where the domain is a finite subset of N.

But you're telling me there are more definitions than that?

In any event, all this stuff certainly seems more complicated than thinking about "1/√3 = x" as meaning "x2 = 1/3, x >0"

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Re: What's the point of rationalizing?

Postby LaserGuy » Mon Jan 09, 2012 5:45 pm UTC

tomtom2357 wrote:Suppose in an exam question they asked you to compute 1/SQRT(3) to 3 decimal places. Which is easier, solving it in its original form or converting it to SQRT(3)/3 and then solving?


That is a reason to know how to convert from one form to the other. That is not a reason to prefer to always leave it in one form versus the other in any given situation. It is easier to evaluate (sqrt(50))^2 than it is to evaluate (5sqrt(2))^2, for example, and it is certainly easier to differentiate 1/sqrt(x) than it is to differentiate sqrt(x)/x.

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Re: What's the point of rationalizing?

Postby Yakk » Mon Jan 09, 2012 6:47 pm UTC

gfauxpas wrote:
Yakk wrote:
gfauxpas wrote:I thought the definition of an algorithm required that it terminate in a finite number of steps?
An algorithm that generates a list is an algorithm that takes, as input, a number n, and returns the nth element of the list (or, sometimes, the first n elements of the list).

Such an algorithm (together with a proof of its properties) is a real number. (not the output of the algorithm -- the algorithm itself)
Okay, but the definition of a list (an n-tuple) requires that it have a finite number of elements, so even according to your definition your "algorithm" can't ever produce the decimal expansion of a sqrt(3)/3.

I never said that the algorithm's produced lists where the real number in question.

I said that the algorithm itself is the real number in question.
Edit: that is to say, your algorithm, as you defined an algorithm, can't even start expanding it. There is no list of digits that it can pull the first n objects out of, because a list has a finite number of elements.

So you don't understand. Ok, here goes!

I am giving a definition for "an algorithm that generates a list".

This definition is: "An algorithm that generates a list" is an algorithm that, given an index n, outputs the nth element of the list.

This is not a controversial definition.

The algorithm doesn't have the list, or have to have it -- it just has to produce the elements when asked.

If you want to be picky, then you're not running the "find the decimal expansion of x" algorithm. You're running some "fine the first n digits of the decimal expansion of x" algorithm for some n in an infinite family of algorithms.

The output of the algorithm is not the real number. The algorithm itself is the real number.

The output of the algorithm, given n, is successively more accurate rational approximations to the real number.

---

Next, only slightly more controversial, "An algorithm that lists the decimal digits of a real number is a depiction of that real number". I'm not defining "depiction of a real number" there, I'm just noting that an algorithm that, given n, returns the nth digit of a real number, is a depiction of one. And, in fact, it is a quite useful depiction -- in many senses, more useful than 1/sqrt(3) (other than the fact that 1/sqrt(3) implicitly describes such an algorithm).

More specifically, defining a real number as a function f:Q+->Q such that for any epsilon > 0, there is an n such that for all n_1, n_2 < n, |f(n_1) - f(n_2)| < epsilon, is actually pretty pedestrian. Algorithms that output decimal digits (or, rather, a rational of precision 10^ciel(log_10(n)), are examples of such a function.
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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 7:00 pm UTC

Yakk where are you getting your definitions from? This is interesting stuff I'd like to read up on it.

But in any event,

""An algorithm that generates a list" is an algorithm that, given an index n, outputs the nth element of the list."

You're using the word "algorithm"here to define "an algorithm that generates a list". What's the definition of an "algorithm"? I'm confused :(

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Re: What's the point of rationalizing?

Postby Xanthir » Mon Jan 09, 2012 7:23 pm UTC

A sequence of steps that accomplish some task?

Are you actually unaware of the definition of "algorithm", or are you being snarky? If the former, dictionaries exist. If the latter, stop.
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Re: What's the point of rationalizing?

Postby gfauxpas » Mon Jan 09, 2012 7:27 pm UTC

Xanthir wrote:A sequence of steps that accomplish some task?

Are you actually unaware of the definition of "algorithm", or are you being snarky? If the former, dictionaries exist. If the latter, stop.


I am aware of a definition of algorithm but not the one that Yakk is using. I thought an algorithm is a finite sequence of steps that accomplishes some task. I dont think that's how Yakk is using it.

edit:
m-w.com
: a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an operation; broadly : a step-by-step procedure for solving a problem or accomplishing some end especially by a computer

wikitionary:
algorithm (plural algorithms)
A precise step-by-step plan for a computational procedure that begins with an input value and yields an output value in a finite number of steps.  

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Re: What's the point of rationalizing?

Postby Yakk » Mon Jan 09, 2012 7:45 pm UTC

If you want a formal definition of "algorithm", how about a "Turing Machine program", where we use the 5-tuple definition of Turing machine? (well, with the input value n written on the tape to start).

Regardless, the algorithm definition I'm using does only run a finite sequence of steps when given its input number.

I'll repeat and clarify.

Definition: "An algorithm that generates a list" is an algorithm that, given an index n, outputs the nth element of the list.

Algorithm: A set of steps that takes an input (possibly empty) and from it, generates an output.

My definition of "An algorithm that generates a list" does not print out each element of the list. That would be an algorithm that prints a list.

Instead, it is a function that takes an index (a natural number, or positive integer), and then does a series of steps with that number as input, and then outputs the value "in the list" that corresponds to that index (0-based).

So suppose my list was:
(1,2,3,4,...)
An algorithm that generates this list might be:
A(n): return n+1
Here is another list:
(1,0,1,0,...)
And an algorithm that generates this list:
B(n): If n/2 is an integer, return 1. Otherwise, return 0.
How about this list?
(0, ., 3, 3, 3, 3, ...)
C(n): if n = 0, return "0". If n = 1, return ".". Otherwise, return 3.

How about a transcendental number?
(0, ., 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,...)

Code: Select all

D(n):
if n = 0 return 0, if n = 1, return 1.  n = n-2.
int step = 0
while n > 0
  step = step + 1
  n = n - step
if n = 0 return 1
return 0

Similar algorithms can be used to generate the digits of pi.

Nowhere in my definition do you need to actually "run the algorithm for every n". The algorithm itself is the real number, what it outputs is just digits of the real number.
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