Hacker News Reader

Comment by SavantIdiot | Hacker News Reader

"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.

View on HN

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.”

[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

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.

When it comes to measurement, 1.7320 is not the same as 1.732

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).

Yes, I think all we've learned from this article is that the author never learned about significant digits.

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 the catch is that you aren't keeping the number normalized. Significant digits somewhat assume scientific notation. So that last number is 9.9e-1. Two digits.

You didn't lose a digit, you wrote it that way. If your instrument has 3 digits, you should have written 0.990v

You can't make your meter add an extra digit (unless you can turn the dial to a different scale). The digit is lost.

But of course it isn't a big deal to lose that dgiti because a trailing digit is 0.1 as much error fraction when the leading digit is 9 vs 1.

I'm not sure what you mean here, probably because I worded it poorly. Of course your instrument will have different scales, and you should use the appropriate one.

But the measurement was done with a number of significant figures, this is independent of how you write it. Nothing is "lost", if measurement had 2 sig figs it is 0.99, if it had 3 it was 0.990. Or 0.991 or whatever.

Am I missing something?

You skipped the example.

They said if your device measures 1.00, it is three significant digits. If it measures 0.01, it is one significant digit. From the same measuring device.

So it is counter intuitive that the measurement just going down below 1 somehow causes it to "lose" significant digits. It didn't lost precision, though. Just the number of digits that are significant.

0 is also a digit

I'm not sure what you are saying. 0.01 has just one significant digit.

Consider you have a 10 meter stick with centimeter meetings. This can give 0.01 as the smallest measure. A number with 1 significant digit. It can also give up to 10.00, a number with 4 significant digits. In both, the precision of the measurement is to a centimeter.

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.

No, what is weird is that the original author seems to think you can look at a number and know how many significant digits it has.

This is almost completely wrong: the number of significant digits is something given by metadata, you cannot look at a number and know that it has been properly rounded to the correct number of significant digits. You always need to be told this special circumstance because most numbers you encounter have in fact not been properly rounded.

In the world of general chemistry, the significant figures rule is assumed. It's not a great rule because in this case:

    .000300   you have 3 significant figures.

But in this case:

    300000    you could have 1 to 6 significant figures.

The convention to deal with this is always to use scientific notation:

    3.00x10^5  has three significant figures.

It's assumed that you shouldn't write down insignificant digits.

Not sure how you're supposed to do floating point math while following that rule... you'll have better luck with writing C code with all semicolons in place.

You lost me on this one. :(

There is the odd case of trailing zeroes in non decimal numbers, but otherwise if a number is written, when is that not assumed to be significant?

That is, there are fairly standard rules for conveying significant digits, I thought?

Whenever you've not personally done the error propagation and rounding yourself you shouldn't assume someone else has done it for you, and you should check if they have before trusting it.

For example, the number you are viewing could have been produced with finite precision floating point math and then just rounded to an arbitrary chosen (small) number of digits.

I mean... I don't disagree with this statement. But there are norms on how you are supposed to write numbers with significant digits.

Sure, but there is no typographical convention for how to indicate that a number is following such a norm so you can never tell from the number alone.

There's also the thing where the 1 in 0.0491 has the exact same significance as the 2 in 0.0982.

IOW, the final significant figure has continuously variable significance (but discretely intervalled numerically) depending on the absolute value of what is it supposed to be significant to.

This is a feature of granularity.

And rounding error has a variable influence on the intervals and how "discreet" or blatant any error may appear.

Then you've got error propagation and pretty soon you're going to need something like Gustafson's UNUM's:

https://en.wikipedia.org/wiki/Unum_(number_format)

> "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.

I think it was more a matter of attitude. Never a great way to actually explain something.

Reply on news.ycombinator.com