## Seemingly ridiculous rules/conventions

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Indubitable.
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### Seemingly ridiculous rules/conventions

Not sure if this is really appropriate for the Mathematics forum or if I should place it in the Science forum instead, but it's about calculations (albeit in Physics) and I like this forum, so I'm posting here.

So, in my Physics course, when we do calculations, we get penalised for being too accurate, even if the answer we give is perfectly correct and was found using only the data provided. We tend to have to stay as accurate as the data provided to within about three significant figures.
My question is this: does anyone know why this would be the case? Is there a specific benefit to it? I just don't see why we should be penalised for accuracy, especially if we have to use the answer in the next question or something.

Also, post any other ridiculous rules by which you have to abide in this sort of situation, if you want. Just so this thread doesn't become a waste of space. Or something.

I apologise if this turns out to be an unoriginal thread, I did a quick search of the Maths and Science fora and didn't see anything similar.
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NathanielJ
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### Re: Seemingly ridiculous rules/conventions

The reason is simple: more decimal places in some problems isn't actually more accurate.

If I say to you "I measured the diameter of this circle to be 10.0 centimetres" and ask you to compute the circumference, what would you say?

You could say 31.4159265358979323 centimeters, but that's ridiculous -- you're reporting an answer that is far more "accurate" than is even possible given the information in the problem. The diameter I reported could have been rounded off from any number in the interval [9.95,10.05). So the circumference could be anywhere in the interval [9.95pi, 10.05pi) = [31.2588..., 31.573006...) centimeters. If you say that it's 31.4159265358979, you're assuming my measurement was more accurate than it is.
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Indubitable.
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### Re: Seemingly ridiculous rules/conventions

Ah, fair enough, that makes sense. Thanks.
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Timtu
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### Re: Seemingly ridiculous rules/conventions

This isn't so ridiculous, but:

I think it is kinda cute with your electrical engineers using 'j' for the imaginary unit when using 'i' for current. But that alone isn't ridiculous.
I'm wondering what happens when j is used for current. Is the imaginary unit then denoted by the letter 'k'? Funny if it was. If it isn't, I'ma introduce it.

nehpest
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### Re: Seemingly ridiculous rules/conventions

Timtu wrote:This isn't so ridiculous, but:

I think it is kinda cute with your electrical engineers using 'j' for the imaginary unit when using 'i' for current. But that alone isn't ridiculous.
I'm wondering what happens when j is used for current. Is the imaginary unit then denoted by the letter 'k'? Funny if it was. If it isn't, I'ma introduce it.

Why would we ever use j for current? That's what i is for
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vilidice
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### Re: Seemingly ridiculous rules/conventions

no, but if you have to deal with an energy output in the system you might need J for joules and then you might want to avoid confusion. More importantly what would they do if they had to use Quaternions for something? (a +bi + cj +dk = h)

snowyowl
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### Re: Seemingly ridiculous rules/conventions

So don't use quaternions. What would an electrical engineer want with quaternions anyway?
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### Re: Seemingly ridiculous rules/conventions

What I don't understand is why they don't just use I for the current, like the rest of the physics community. Then they can use i for imaginary numers, again like the rest of the physics community.
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Timtu
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### Re: Seemingly ridiculous rules/conventions

I agree with current = I and imaginary unit = i. I have certainly seen j used for current, though, but I don't know what took i's place in those cases.
Electrical engineers are just queer.
I reckon I kind find a use for quaternions in some sort of something, then it'd really mess things up notation wise.

nehpest
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### Re: Seemingly ridiculous rules/conventions

I suppose one could argue that we use lowercase i for time-domain current because we use uppercase I (or at least, [imath]$\mathbb{I}$[/imath]) for frequency-domain current. We switch back and forth between the two enough to justify having two separate characters for them.

/speculation
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Kirby
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### Re: Seemingly ridiculous rules/conventions

I tend to disagree with the first assumption that goes into significant figure rules - that the precision of a number is determined by the number of decimal places given.

To me, it makes FAR more sense to give measurements with a margin of error, and ask for an answer with a margin of error/interval of answers that the given input data would allow.

Ex. if you tell me that the radius of a circle is 0.5 inches, then I'll tell you that the circumference of that circle is pi (3.14159...) inches to however many decimal places I feel like writing down. And that would be a true statement.

If, however, you tell me that the radius of the circle is .5 inches plus or minus .01 inches, then I would give my answer as either the interval (3.078761, 3.2044245) or as 3.14159 plus or minus .062832. And that would also be a true statement.

What's wrong with this?

NathanielJ
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:Ex. if you tell me that the radius of a circle is 0.5 inches, then I'll tell you that the circumference of that circle is pi (3.14159...) inches to however many decimal places I feel like writing down. And that would be a true statement.

That would be true if the radius of the circle is exactly 0.5 inches, yes. So it depends on where that figures of 0.5 inches is coming from. If it's coming from a math textbook than you're done -- don't worry about sig figs. If, however, a guy in a lab coat comes up to you and says "the radius of this circle is 0.5 inches, find me the circumference", then you need to worry about sig figs. If he meant that the radius of the circle was 0.5000 inches, he would have said that. But he didn't, so you have to assume that the measurement can only be trusted to 1 decimal place. In other words he means 0.5 plus or minus 0.05 inches.

Kirby wrote:If, however, you tell me that the radius of the circle is .5 inches plus or minus .01 inches, then I would give my answer as either the interval (3.078761, 3.2044245) or as 3.14159 plus or minus .062832. And that would also be a true statement.

This is quite correct. Generally saying something like "the radius of the circle is .5 inches plus or minus .01 inches" is more correct (or at least more flexible) than the significant figures way of doing it, but the sig figs way is "simpler" so it's usually taught first. And most people, unless they're scientists, won't say that they measures the radius of the circle to be 0.5 +- 0.01 inches.
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Kirby
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### Re: Seemingly ridiculous rules/conventions

NathanielJ wrote:This is quite correct. Generally saying something like "the radius of the circle is .5 inches plus or minus .01 inches" is more correct (or at least more flexible) than the significant figures way of doing it, but the sig figs way is "simpler" so it's usually taught first. And most people, unless they're scientists, won't say that they measures the radius of the circle to be 0.5 +- 0.01 inches.

The trouble with sig figs, is that the decimal accuracy of the input data isn't always equivalent to the true decimal accuracy of the answer.

Continuing with the circle example... suppose it is measured that a circle has a radius of .5 inches. Keeping with sig fig rules (I'm a little rusty, forgive me)... the circumference would be reported as 3.1 inches. Keeping with the implicit assumption of sig figs, one should note that the true circumference could be in the interval (3.05, 3.15).

However, if we realize from the start the radius is .5 +- .05 inches, then we would deduce that the circumference is actually in the interval (2.827, 3.456)!

To be honest, I have no idea how closely sig fig rules are adhered to in the real world - but the problems associated with it (as noted above) are too severe to warrant teaching in a classroom setting. I remember having quizzes in chemistry that amounted to memorizing the rules of significant figures. Instead, I think it would be a much better use of everyone's time if it was just taught that measured data has a margin of error (and no need to get into fancy confidence intervals...), and that that margin of error can be manipulated mathematically just like the original observed data, to find a calculated answer. Would it really be all that complicated?

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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:Continuing with the circle example... suppose it is measured that a circle has a radius of .5 inches. Keeping with sig fig rules (I'm a little rusty, forgive me)... the circumference would be reported as 3.1 inches. Keeping with the implicit assumption of sig figs, one should note that the true circumference could be in the interval (3.05, 3.15).

0.5 has only 1 sig fig (leading zeroes don't count, it could have been written as 5*10-1), but you gave the answer with 2 sig figs. The answer should have been 3.

Nevertheless, I agree with you that sig figs are a rougher than explicitly using an error margin, because sig figs don't really allow for scaling of the error when you multiply or divide. Using a similar example:

circumference = r 2pi = 62.832 +- 0.314, i.e. in the interval [62.518, 63.146]

Using sig figs would give 62.8 implying the interval [62.75, 62.85] which is too short, by a factor of 2pi of course.
The advantage of sig figs is that it is half as many calculations, and unless there is scaling by a much more accurate number (or a mathematically given number such as 2pi), it is still pretty good.

Kirby
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### Re: Seemingly ridiculous rules/conventions

jaap wrote:0.5 has only 1 sig fig (leading zeroes don't count, it could have been written as 5*10-1), but you gave the answer with 2 sig figs. The answer should have been 3.

Nevertheless, I agree with you that sig figs are a rougher than explicitly using an error margin, because sig figs don't really allow for scaling of the error when you multiply or divide. Using a similar example:

circumference = r 2pi = 62.832 +- 0.314, i.e. in the interval [62.518, 63.146]

Using sig figs would give 62.8 implying the interval [62.75, 62.85] which is too short, by a factor of 2pi of course.
The advantage of sig figs is that it is half as many calculations, and unless there is scaling by a much more accurate number (or a mathematically given number such as 2pi), it is still pretty good.

I should have expected that I would botch the sig figs...

Anyways, sig figs would similarly fall apart if you're dealing with formula that are even slightly more complex (involving squares, roots, etc.). And this just begs the question: why use sig figs at all? By using significant figures for calculations, you fail to express the true margin of error of your calculated answer. To students, sig figs seems arbitrary and pointless - it does not capture the range of reasonable values for calculated data that observed data should produce.

Is it laziness? As far as calculations go, in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask? For students, perhaps asking them to give an interval on ALL tests/quizzes and whatnot might be tedious - but teaching the skill once, and asking that students remember how to use that skill seems only logical.

Perhaps we should teach students how to deal with measuring error the right way?

PM 2Ring
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask?

This. You have digital calculators, people. Doing the calculation twice won't kill you.

When I was a lad, we used slide rules & log tables, so sig figs were a natural way to go. But even then I preferred the max & min error approach.

Syrin
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote: And this just begs the question

http://en.wikipedia.org/wiki/Begs_the_question

I figure I'm entitled to some modicum of pedantry on a mathematics forum.

Hix
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### Re: Seemingly ridiculous rules/conventions

Syrin wrote:I figure I'm entitled to some modicum of pedantry on a mathematics forum.

I would have been the pedant if you hadn't beat me to it.

gmalivuk
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### Re: Seemingly ridiculous rules/conventions

Well for better or worse, it now also (and almost exclusively) means "asks" or "raises" the question, and not just "already assumes the answer to" the question.

If you want to talk unambiguously about the fallacy, use the Latin name, petitio principii. After all, if we can talk about "ad hominems" and "non sequiturs", I don't see why we can't use Latin for this one as well.
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Kirby
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### Re: Seemingly ridiculous rules/conventions

Syrin wrote:
Kirby wrote: And this just begs the question

http://en.wikipedia.org/wiki/Begs_the_question

I figure I'm entitled to some modicum of pedantry on a mathematics forum.

Foiled again! My argument is henceforth invalid.

c0rt3z
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### Re: Seemingly ridiculous rules/conventions

[imath]\pi[/imath] is just half of what it should be.

squareroot1
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### Re: Seemingly ridiculous rules/conventions

Yeah, and e should be its reciprocal.

Edit: Not 1/3.14159..., that would just be silly.

Kurushimi
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### Re: Seemingly ridiculous rules/conventions

I think the taylor series for e^x looks cooler like this.

PM 2Ring
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### Re: Seemingly ridiculous rules/conventions

squareroot1 wrote:Yeah, and e should be its reciprocal.

But d/dx(ex) = ex is so pretty! However, I will agree that the Taylor series for e-x & its connection with subfactorials is very cool.

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### Re: Seemingly ridiculous rules/conventions

Knowing about error bounds and whatnot is a good thing, both in case you ever need that detail, or just to know the theory behind the practise... but for the most part, in practise, you only need to know the accuracy of a measurement to an order of magnitude or so anyway... it's just there to get a rough idea of how inaccurate the measurement is. If you need precision about the accuracy of the measurement, you're probably going to be using stats and confidence intervals anyway (I don't think I've ever known a measurement to have a perfectly rectangular probability density, which would let you say "it's in this interval" and nothing more...).

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### Re: Seemingly ridiculous rules/conventions

Studying for the first actuary exam, I find it somewhat annoying that they'll give certain probabilities with only one significant figure, but getting the correct answer requires you to have used 3 or 4, at least at some points in the calculation.
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achan1058
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### Re: Seemingly ridiculous rules/conventions

PM 2Ring wrote:
Kirby wrote:in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask?

This. You have digital calculators, people. Doing the calculation twice won't kill you.
But how do you determine how many digits do you write for your min/max?

Kirby
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### Re: Seemingly ridiculous rules/conventions

achan1058 wrote:
PM 2Ring wrote:
Kirby wrote:in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask?

This. You have digital calculators, people. Doing the calculation twice won't kill you.
But how do you determine how many digits do you write for your min/max?

All of them (Where possible). OR a fixed number (for instance, the AP program decided that 3 is appropriate in all cases). OR however many a standard calculator will give you.

achan1058
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:
achan1058 wrote:
PM 2Ring wrote:
Kirby wrote:in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask?

This. You have digital calculators, people. Doing the calculation twice won't kill you.
But how do you determine how many digits do you write for your min/max?

All of them (Where possible). OR a fixed number (for instance, the AP program decided that 3 is appropriate in all cases). OR however many a standard calculator will give you.
Well, suppose my max/min has 1 digit of accuracy (because of say limitation of instrument), then I multiply by 2pi. I get some numbers out. Now, if you write the full number, you are still giving the impression that your figures are accurate to high positions, and that the very high position digits are not meaningful anyways. To put it into an extreme, suppose instead of using a hand calculator, I use Maple with a setting of 5000 digit precision. If my input max/min has 2 digits of accuracy, should I write the 5000 digit output? On the other hand, if my max/min is accurate up to 10^(-10), should I merely write 3 digits as my answer?

What I mean to say is, using max/min merely converts the error margin imposed by sig figs to the error of error imposed on the error of the max/min. You can never get sig fig away in any meaningful way.

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### Re: Seemingly ridiculous rules/conventions

I get slightly confused when I see sin-1(x)... which is usually interpreted as arcsin(x) but could be csc(x) as well. Ambiguous notations methinks.

Kurushimi
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### Re: Seemingly ridiculous rules/conventions

Actually, its more annoying that [imath](\sin x)^2[/imath] can be written as [imath]\sin^2 x[/imath]. The other one is keeping in tune with the notation for inverse functions. That breaks the notation usually used for exponentiation.

Then again, I think I'd rather have it like that and always use arcsin x or asin x when referring to the inverse sine. But having both [imath]\sin^{-1} x[/imath] and [imath]sin^2 x[/imath] is confusing.

BlackSails
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### Re: Seemingly ridiculous rules/conventions

Sin^-1 (x) makes sense if you consider sin to be an operator. Sin^-1 is then the inverse operator.

Of course, then sin^2 = sin*sin then makes no sense at all, it should be sin(sin())

Kirby
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### Re: Seemingly ridiculous rules/conventions

achan1058 wrote:
Spoiler:
Kirby wrote:
achan1058 wrote:
PM 2Ring wrote:
Kirby wrote:in most cases using a margin of error would amount to running the calculations twice - once with values that result in a minimum reasonable answer, and once with values that result in a maximum. Is that too much to ask?

This. You have digital calculators, people. Doing the calculation twice won't kill you.
But how do you determine how many digits do you write for your min/max?

All of them (Where possible). OR a fixed number (for instance, the AP program decided that 3 is appropriate in all cases). OR however many a standard calculator will give you.
Well, suppose my max/min has 1 digit of accuracy (because of say limitation of instrument), then I multiply by 2pi. I get some numbers out. Now, if you write the full number, you are still giving the impression that your figures are accurate to high positions, and that the very high position digits are not meaningful anyways. To put it into an extreme, suppose instead of using a hand calculator, I use Maple with a setting of 5000 digit precision. If my input max/min has 2 digits of accuracy, should I write the 5000 digit output? On the other hand, if my max/min is accurate up to 10^(-10), should I merely write 3 digits as my answer?

What I mean to say is, using max/min merely converts the error margin imposed by sig figs to the error of error imposed on the error of the max/min. You can never get sig fig away in any meaningful way.

Maybe I wasn't clear: you do min/max to determine an interval. The interval, in this case, is interpreted as (in the circle example), "If the radius of a circle is in the interval (A, B), then the circumference of the circle is in the interval (C, D)." C and D can both be expressed to arbitrary precision because the precision of your answer is really the interval - not the sig figs of C or D individually.

To put that another way, sig figs are used to express error in measurement. The error of the measurement is significant because it tells what range of values the measurement would "agree" with. Suppose I measure a twig to be 2.0 inches, but then someone informs me that the twig is in fact 1.99 inches... I would agree, because the implied interval of my measurement (1.95, 2.05) contains 1.99.

Similarly, if I use measured values to make calculations, it might be important to know what range of values that calculated value could take on. The measurement itself allows an interval, therefore, the calculated value would also be best expressed as an interval. The sig figs of the interval are merely arbitrary; what really matters is the interval itself.

achan1058
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:Maybe I wasn't clear: you do min/max to determine an interval. The interval, in this case, is interpreted as (in the circle example), "If the radius of a circle is in the interval (A, B), then the circumference of the circle is in the interval (C, D)." C and D can both be expressed to arbitrary precision because the precision of your answer is really the interval - not the sig figs of C or D individually.
But how do you determine the accuracy of your max/min? They cannot possibly be arbitrary precise. I mean, it's a coarse upper/lower bound. With your example, it would be say 1.95-2.05, say. It is clearly meaningless to write it out as 1.95000000000000000000000 to 2.05000000000000000000000, even if your calculator gets that much precision. It is also absurd to write 2 to 2 as well. How would you determine what is the amount of digits you write for your interval?

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### Re: Seemingly ridiculous rules/conventions

achan1058 wrote:I mean, it's a coarse upper/lower bound.

Indeed... if I measure something as 2.0cm, accurate to the nearest mm, the range of possible values isn't strictly [1.95,2.05) with absolute precision... if it's a value close to 1.95, on either side of it, I could round it either way out of uncertainty... so there's a feathered range on the edges. 1.94, for instance, might still be possible, if, when measuring something that's 1.94cm, I'll still call it 2.0cm sometimes (even if calling it 1.9cm is more common)... and similarly, 1.96cm I might call 1.9cm sometimes (even if calling it 2.0cm is more common)... so if I say I measure it as 2.0cm, it could be anywhere within something like [1.93,2.07), with values near the boundaries less likely than values near the middle (a flat area in the middle, but tapered off at the edges).

As I said before, it's not just flat intervals with clearly-defined endpoints, it's probability functions and confidence intervals. And even the probability function itself is only known to a certain level of precision, so the confidence interval itself has only a certain level of precision... and you could in theory talk about the confidence interval of the confidence interval. You don't, though, because you very rarely care about how accurate your confidence interval is... you only need it in the first place as a rough guide to how accurate your original value was. And "order of magnitude" is about as rough as they get, while still being useful. If you need more accuracy than an order of magnitude, bring out the stats.

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Kirby
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### Re: Seemingly ridiculous rules/conventions

achan1058 wrote:But how do you determine the accuracy of your max/min? They cannot possibly be arbitrary precise. I mean, it's a coarse upper/lower bound. With your example, it would be say 1.95-2.05, say. It is clearly meaningless to write it out as 1.95000000000000000000000 to 2.05000000000000000000000, even if your calculator gets that much precision. It is also absurd to write 2 to 2 as well. How would you determine what is the amount of digits you write for your interval?

It's a bit of a moot point. The established math/scientific bureaucracy (only the educators?) decided that sig figs should be used to account for error in measurement. In this thread, we further established that keeping your answer sig figs equivalent to your measured sig figs doesn't always result in an appropriate margin of error. The alternative I proposed was an interval to give a better sense of what calculated values the measured value allows. Of course the endpoints of the interval are course - but they're a lot less coarse than the alternative sig figs convention. Furthermore, the number of significant figures in the interval would not be meaningful. The only interpretation of an interval (A, B), would have to be something like "Any real number R such that A <= R <= B is a permissible value."

That interpretation assumes that the interval of the measured value has clearly defined endpoints. As phlip mentioned, this will rarely be the case. But in most circumstances, the exact probability distribution will be unknown. So instead we make the assumption that the interval of the measured value has, in fact, precise endpoints.

Kurushimi
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### Re: Seemingly ridiculous rules/conventions

One thing I don't like is how we were forced to rationalize denominators in Algebra class. What good could that possibly do? Does it make the answer any easier to calculate? To understand? I don't think so.

achan1058
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### Re: Seemingly ridiculous rules/conventions

Kirby wrote:It's a bit of a moot point. The established math/scientific bureaucracy (only the educators?) decided that sig figs should be used to account for error in measurement. In this thread, we further established that keeping your answer sig figs equivalent to your measured sig figs doesn't always result in an appropriate margin of error. The alternative I proposed was an interval to give a better sense of what calculated values the measured value allows. Of course the endpoints of the interval are course - but they're a lot less coarse than the alternative sig figs convention. Furthermore, the number of significant figures in the interval would not be meaningful. The only interpretation of an interval (A, B), would have to be something like "Any real number R such that A <= R <= B is a permissible value."

That interpretation assumes that the interval of the measured value has clearly defined endpoints. As phlip mentioned, this will rarely be the case. But in most circumstances, the exact probability distribution will be unknown. So instead we make the assumption that the interval of the measured value has, in fact, precise endpoints.
The only way I can see it make any sense with precise end points is if you artificially inflate the error margin. So, you will say 1.95 to 2.05 when realistically you can never hit either endpoints (maybe say your error actual min/max is more like 1.97-2.03). Then, whenever you have the need to truncate, you always round down on the min and round up on the max. That will preserve your ideal on the min/max. The only problem of course is that your error margin might be needlessly large if you do enough operations.

phlip
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### Re: Seemingly ridiculous rules/conventions

Kurushimi wrote:One thing I don't like is how we were forced to rationalize denominators in Algebra class. What good could that possibly do? Does it make the answer any easier to calculate? To understand? I don't think so.

Calculate each of these, and tell me which was easier:$\Re\left(\frac{2}{1+3i}\right)$$\Re\left(\frac{1+3i}{2}\right)$
Or, less obnoxiously, it makes it easier to manipulate if the denominator is a single term, and the numerator has the multiterm sum... since [imath]\frac{a+b}{c} = \frac ac + \frac bc[/imath] but [imath]\frac {a}{b+c}[/imath] doesn't break down easily at all.

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joshz
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### Re: Seemingly ridiculous rules/conventions

Right, but I think he was referring to things like:
$\frac{1}{\sqrt{2}}$being simplified to:
$\frac{\sqrt{2}}{2}$
You, sir, name? wrote:If you have over 26 levels of nesting, you've got bigger problems ... than variable naming.
suffer-cait wrote:it might also be interesting to note here that i don't like 5 fingers. they feel too bulky.