"It is not transparent to me why this definition is sensible."
Because if I have a six digit meter and I measure 5.0000 volts I know that it is 5.0000 volts. I don't see why that is confusing at all.
"Here, yyy has correct one and three significant digits relative to xxx, but incorrect 2 significant digits, since the truncation at x2x_2x2 and y2y_2y2 do not agree even to the first significant digit. "
This doesn't make sense. Y has the correct one?
I think you might be confused about the purpose of significant digits. SD are intended to be paired with a tolerance. If there is no tolerence quote with the number, it is meaningless to talk about significant digits. You arbitrarily defined the tolerance as +/0.5 (half a unit). That is not always true, almost never true. Consider test equipment, not every multimeter is going to be +/0.5mV, some might be +/-.3uV. It varies.
I agree that it’s better to state the uncertainty [1] together with the measurement, but when this isn’t done, there is a standard convention for significant digits, explained in ISO 80000-1:2009 [2], subclause 7.3.4:
When a number is given without any further information,
it is generally interpreted so that the last digit is
rounded with a rounding range equal to 1 in the last digit
(see Annex B). Thus, for example, the number 401 008 is
generally assumed to represent a value between 401 007,5
and 401 008,5. In this case, the maximum magnitude of the
error in the number 401 008 is 0,5.
This is consistent with the author’s “less than half a unit.”
Exactly this. It may be useful for OP to consider what the hypothetical opposite of significant digits would be: insignificant digits, ie. digits that tell you nothing. A leading zero can be omitted from a number without any loss of information. A trailing zero, when correctly used in a position that is not below the tolerance, gives you significant information. On the other hand, if you start combine numbers with different tolerances, but do not truncate the result, you will have a bunch of trailing digits that are meaningless in practice because they are below the combined tolerance.
I'm not entirely clear on the author's thinking. But consider the case of watching a real world voltmeter read 1.02v, 1.01v, 1.00v, 0.99v. Did we really lose a digit of precision there? The leading digit transition between 1 and 9 is a boundary condition that messes up that rule.
I think you missed what they found weird. The subject for that was that 0.0491 only has three significant digits. It is not obvious why, if you don't normalize the number.
> "Here, y has correct one and three significant digits relative to x, but incorrect 2 significant digits, since the truncation at x_2 and y_2 do not agree even to the first significant digit. "
> This doesn't make sense. Y has the correct one?
I think that's a language issue, like "that x̂ agress to x upto p" in the next section. I think what they're trying to say is the following. If you round x = 0.9949 to 1, 2, or 3 significant digits, you get 1, 0.99, or 0.995 respectively. If you round y = 0.9951 to 1, 2, or 3 significant digits, you get 1, 1.0, or 0.995 respectively. So x and y look equal if you round to 1 or 3 significant digits, but not if you round to 2 significant digits.
This is such a great example of how people that know math(s) really struggle to explain things to people who don't, because they struggle to realise how much lingo they are using.
The first point explains something as basic as absolute and relative errors. Great.
1 page down (s)he's casually thrown in function notation with R representing the real numbers with no explanation. Anyone who isn't familiar with this notation is immediately intimidated. But I'd wager almost anyone who is familiar with it didn't need to read the page before it. But also, I think it's just unnecessary to be this specific in the first place, if you're trying to quickly explain concepts.
It's actually a huge problem in maths and even Wikipedia (I feel) is pretty guilty of this.
If you take a (relatively) key mathematical concept, such as the Taylor series, then the opening gambit is completely incomprehensible to anyone who isn't already knowledgeable in maths: https://en.wikipedia.org/wiki/Taylor_series
This, to me, seems in huge contrast with other scientific fields. Medicine wikipedia, for example, caters towards the "average reader" first, and doctors later. This is entirely appropriate for an encyclopedia. For example: https://en.wikipedia.org/wiki/AV_nodal_reentrant_tachycardia...
One comment regarding Wikipedia: you may be aware of the “Simple English” language option, which can help with making some pages more accessible. It’s in the language menu, or you can just change the language prefix in the URL, such that your example above becomes https://simple.m.wikipedia.org/wiki/Taylor_series. Granted, it’s not immediately obvious that it’s there, but perhaps it’s one reason that the main article can be more precise off the bat rather than explanatory. Furthermore, I wonder if Wikipedia math editors are less concerned with the notion of an encyclopedia as learning device and treat an article as a canonical overview of a topic, with all of the baggage that comes with that.
On the other hand, as someone who knows a bit of math, I love those Wikipedia articles that get straight to the point, and hate those where I first have to dig through a pile of (to me) useless "intuition".
I'm not sure if it is really possible to learn/teach math (like actual math, not just some loose intuitions, because those don't really help you with anything) without going through notation and prior knowledge. Notation exists for a reason.
I run into that same problem a lot while trying to find tutorials that target beginners. Absolute beginners, the kind showing up to my organization's Google Summer of Code chat rooms. It's rare to find tutorials that just keep it simple enough.
I wanted a regex one recently. First off, they all presumed mastery of the concept of a string, but more seriously, none of them had a really good definition of a regex! I still can't think of one myself, but I know it's probably not "pattern to match specific strings", because it uses an uncommon meaning of "pattern" and a jargon meaning of "match". And of course they all immediately used terms like "formal languages".
The perldoc page 'perlretut' did pretty well with "a template that is used to determine if a string has certain characteristics" - ok, a bit abstract for my taste, but let's keep going - then a bit further down they use "modifier" (flag) with no definition. But I really had to look hard for flaws in that one. I guess the only way to actually nail this is to supervise successive groups of students as they try to learn using your tutorial-in-progress...
Is encountering function notation and the symbol for the set of real numbers all that different from encountering unfamiliar vocabulary? It is slightly harder to Google for notation, but otherwise I do not see a major difference.
In the Wikipedia page you linked, I immediately encounter several terms that I do not know the meaning of. I must either click through or search separately to fill in my own gaps.
Your criticism is not totally disagreeable, but we have to wonder where to draw the line. Is a "Hacker's Guide to Numerical Analysis" obligated to hold your hand all the way through basic notation? The notation that you mention in your comment are the "Hello world" of mathematics.
I sort of agree. But there's no right answer here. We can't expect every Wikipedia article to take you through the basics of the subject from scratch. A certain level of knowledge just has to be assumed, and it's hard to know where to draw that line.
Are there any plans for a physical book? My attention span seems to be calibrated to focus for a prolonged period on paper but less so for information on screen.
Anytime you're doing a lot of floating-point math, the imprecision can bite you in non-intuitive ways. But you can mitigate that by fixing the order of operations to minimize error.
I suppose one other specific example of a similar concept would be that when training deep learning models models you're doing a ton of matrix multiplication and tweaking the weights constantly. Especially when using lower-precision numbers, you can hit issues where your coefficient gets too close to 0, and the next thing you know it IS 0, and everything it gets multiplied by also comes out to 0, which messes everything up. So you have to avoid getting too close to 0: https://pytorch.org/docs/stable/amp.html#gradient-scaling.
My rule of thumb is that anything title "...for Hackers" is crap.This link was not an exception.
> Since the relative error is invariant under scaling (x↦αx)(x \mapsto \alpha x)(x↦αx), we will mostly be interested in relative error.
No, No, No, No. This is very much dependent on what you are measuring/modelling, what is the tolerance you are willing to accept, the consequences and so on.
A 1% of error in the change the cashier gave you back maybe is not a big deal. That same error in the national budget is equivalent to 2X the NASA's budget.
Digression: for all authors that want minimalistic website, please don’t push it too hard, as author of this website did. Color scheme is nice, as is font selection, but it lacks navigation links. Putting simple link to website home would do so much for explorers of site.
Since I dumped on the author in another comment, let me defend him here: going up one level in the URL is really not a problem. It’s even easier if you use something like Vimperator, where you just need to type “gu”. And there is a nice table of contents.
[+] [-] SavantIdiot|5 years ago|reply
Because if I have a six digit meter and I measure 5.0000 volts I know that it is 5.0000 volts. I don't see why that is confusing at all.
"Here, yyy has correct one and three significant digits relative to xxx, but incorrect 2 significant digits, since the truncation at x2x_2x2 and y2y_2y2 do not agree even to the first significant digit. "
This doesn't make sense. Y has the correct one?
I think you might be confused about the purpose of significant digits. SD are intended to be paired with a tolerance. If there is no tolerence quote with the number, it is meaningless to talk about significant digits. You arbitrarily defined the tolerance as +/0.5 (half a unit). That is not always true, almost never true. Consider test equipment, not every multimeter is going to be +/0.5mV, some might be +/-.3uV. It varies.
[+] [-] wsh|5 years ago|reply
[1] JCGM 100:2008, Evaluation of measurement data — Guide to the expression of uncertainty in measurement, https://www.iso.org/sites/JCGM/GUM-JCGM100.htm
[2] ISO 80000-1:2009, Quantities and units — Part 1: General
[+] [-] PhaseLockk|5 years ago|reply
[+] [-] yudlejoza|5 years ago|reply
The trailing zero was part of the measurement and, hence, is included in the significant digit count.
That also means you can't just measure 1.7320, and report 1.73200 instead (or 1.732). You have to report only, and exactly, what you measured.
(I'm ignoring rounding in what I said).
[+] [-] leephillips|5 years ago|reply
[+] [-] ticklemyelmo|5 years ago|reply
[+] [-] taeric|5 years ago|reply
[+] [-] mkl|5 years ago|reply
> This doesn't make sense. Y has the correct one?
I think that's a language issue, like "that x̂ agress to x upto p" in the next section. I think what they're trying to say is the following. If you round x = 0.9949 to 1, 2, or 3 significant digits, you get 1, 0.99, or 0.995 respectively. If you round y = 0.9951 to 1, 2, or 3 significant digits, you get 1, 1.0, or 0.995 respectively. So x and y look equal if you round to 1 or 3 significant digits, but not if you round to 2 significant digits.
[+] [-] sengstrom|5 years ago|reply
[+] [-] jphoward|5 years ago|reply
The first point explains something as basic as absolute and relative errors. Great.
1 page down (s)he's casually thrown in function notation with R representing the real numbers with no explanation. Anyone who isn't familiar with this notation is immediately intimidated. But I'd wager almost anyone who is familiar with it didn't need to read the page before it. But also, I think it's just unnecessary to be this specific in the first place, if you're trying to quickly explain concepts.
It's actually a huge problem in maths and even Wikipedia (I feel) is pretty guilty of this.
If you take a (relatively) key mathematical concept, such as the Taylor series, then the opening gambit is completely incomprehensible to anyone who isn't already knowledgeable in maths: https://en.wikipedia.org/wiki/Taylor_series
This, to me, seems in huge contrast with other scientific fields. Medicine wikipedia, for example, caters towards the "average reader" first, and doctors later. This is entirely appropriate for an encyclopedia. For example: https://en.wikipedia.org/wiki/AV_nodal_reentrant_tachycardia...
[+] [-] 1ceaham|5 years ago|reply
[+] [-] _hl_|5 years ago|reply
I'm not sure if it is really possible to learn/teach math (like actual math, not just some loose intuitions, because those don't really help you with anything) without going through notation and prior knowledge. Notation exists for a reason.
[+] [-] howenterprisey|5 years ago|reply
I wanted a regex one recently. First off, they all presumed mastery of the concept of a string, but more seriously, none of them had a really good definition of a regex! I still can't think of one myself, but I know it's probably not "pattern to match specific strings", because it uses an uncommon meaning of "pattern" and a jargon meaning of "match". And of course they all immediately used terms like "formal languages".
The perldoc page 'perlretut' did pretty well with "a template that is used to determine if a string has certain characteristics" - ok, a bit abstract for my taste, but let's keep going - then a bit further down they use "modifier" (flag) with no definition. But I really had to look hard for flaws in that one. I guess the only way to actually nail this is to supervise successive groups of students as they try to learn using your tutorial-in-progress...
[+] [-] epsilonclose|5 years ago|reply
In the Wikipedia page you linked, I immediately encounter several terms that I do not know the meaning of. I must either click through or search separately to fill in my own gaps.
Your criticism is not totally disagreeable, but we have to wonder where to draw the line. Is a "Hacker's Guide to Numerical Analysis" obligated to hold your hand all the way through basic notation? The notation that you mention in your comment are the "Hello world" of mathematics.
[+] [-] chrisjharris|5 years ago|reply
[+] [-] marzullo|5 years ago|reply
[+] [-] domnomnom|5 years ago|reply
[+] [-] singhrac|5 years ago|reply
[+] [-] bloaf|5 years ago|reply
[+] [-] rarefied_tomato|5 years ago|reply
[+] [-] bollu|5 years ago|reply
[+] [-] smlckz|5 years ago|reply
[+] [-] dennis714|5 years ago|reply
[+] [-] chrisjharris|5 years ago|reply
[+] [-] omginternets|5 years ago|reply
[+] [-] macksd|5 years ago|reply
I suppose one other specific example of a similar concept would be that when training deep learning models models you're doing a ton of matrix multiplication and tweaking the weights constantly. Especially when using lower-precision numbers, you can hit issues where your coefficient gets too close to 0, and the next thing you know it IS 0, and everything it gets multiplied by also comes out to 0, which messes everything up. So you have to avoid getting too close to 0: https://pytorch.org/docs/stable/amp.html#gradient-scaling.
[+] [-] apbytes|5 years ago|reply
[+] [-] sampo|5 years ago|reply
[+] [-] unknown|5 years ago|reply
[deleted]
[+] [-] cambalache|5 years ago|reply
> Since the relative error is invariant under scaling (x↦αx)(x \mapsto \alpha x)(x↦αx), we will mostly be interested in relative error.
No, No, No, No. This is very much dependent on what you are measuring/modelling, what is the tolerance you are willing to accept, the consequences and so on.
A 1% of error in the change the cashier gave you back maybe is not a big deal. That same error in the national budget is equivalent to 2X the NASA's budget.
[+] [-] vulcan01|5 years ago|reply
> My rule of thumb is that anything title "...for Hackers" is crap
Remind me what site you're on again?
[+] [-] enchiridion|5 years ago|reply
[+] [-] domnomnom|5 years ago|reply
Well then you're missing the whole point completely.
[+] [-] ubavic|5 years ago|reply
[+] [-] leephillips|5 years ago|reply