Why are zeros after a decimal point significant?

[HallsofIvy]

So basically significant zeros are a simple, but somewhat limited way to show measurement accuracy. If I say there are 8.0 people in a room, I am sort of implying that 8.1 might be possible.

No, you are not. Read DrewD's post, above:
"Perhaps, as your headcount point shows, it is more appropriate to state that sigfigs are used for continuum measurements, not discrete measurements."

Significant zeros also show up where you have integer-like limits to accuracy. For example, US currency is limited to 2 significant digits. A penny is .01, but you can't divide that further using physical coins and bits. You can include or omit the sub-dollar amounts, $2 or $2.00, but $2.0 would look strange and make people think some kind of misprint had happened.

I would not consider that a matter of "significant digits", spefically because " A penny is .01, but you can't divide that further using physical coins and bits." It is, again, matter of counting, not measuring.

 

[Borek]

I suspect you have read somewhere that, due to Einstein's "General Theory of Relativity" says that this should be "2 point something".

I believe it was suggested as an explanation when the anomalies of the perihelion precession of Mercury orbit became evident for the first time. These anomalies are now nicely explained by GR and are typically listed between tests showing GR is correct.

 

[voko]

If you are referring to the "2" in r^2, no it is NOT. That is Newton's law of gravity and the "2" is precisely that- we are squaring the r.

If you teach physics axiomatically, perhaps.

 

[pwsnafu]

If you teach physics axiomatically, perhaps.

Huh? Even if it isn't axiomatically taught, that's what Newton's law of universal gravitation says. It's a statement. Whether or not that statement matches observations is irrelevant, as a mathematical expression that 2 is exact.

 

[voko]

Huh? Even if it isn't axiomatically taught, that's what Newton's law of universal gravitation says. It's a statement. Whether or not that statement matches observations is irrelevant, as a mathematical expression that 2 is exact.

Axiomatic physics illustrated.

 

[pwsnafu]

Axiomatic physics illustrated.

In that case you are going to have to explain how non-axiomatic science exists at all. Because in all languages (including all mathematical sciences) you have "what the statement says" and "whether the statement is true". The only way I can interpret you is that the former doesn't exist (i.e. the statement is not saying what the statement is claiming to say).

 

[voko]

In that case you are going to have to explain how non-axiomatic science exists at all. Because in all languages (including all mathematical sciences) you have "what the statement says" and "whether the statement is true".

There is a science called "physics". Anything that science says is a model or a consequence from a model, and no model is ever (well, not ever, but for quite a while now) considered "true". It is just considered (or not) to be in some good agreement with experimental and observational data within some specified or implied limits.

As regards gravitation, there is even a model where it is constant. It works fine where it is applicable. Is it "true" or not? There are other models. Newton's model; Einstein's model (and the post-Newtonian spin-off); Yukawa-modified gravity, just to name a few.

 

[pwsnafu]

There is a science called "physics". Anything that science says is a model or a consequence from a model, and no model is ever (well, not ever, but for quite a while now) considered "true". It is just considered (or not) to be in some good agreement with experimental and observational data within some specified or implied limits.

As regards gravitation, there is even a model where it is constant. It works fine where it is applicable. Is it "true" or not? There are other models. Newton's model; Einstein's model (and the post-Newtonian spin-off); Yukawa-modified gravity, just to name a few.

You are right I shouldn't have said "truth". I did mean "agreement with observations" (is there is a word for that? I guess "accurate"?) when I made my post.

Getting back. You claimed that the 2 in r², was not exactly two (i.e. not an integer 2), and when this was pointed out you claimed that was axiomatic. Newton's statement is that it is an integer 2 and not 2 point something. It just happens to fail in certain situations, and (just as you say) we can use other models.

Am I interpreting you correctly?

If that is so, I fail to see how your second paragraph is not also axiomatic physics. We have what the Newton's model says (the 2 is exact) and then afterwards we have whether it it satisfies the observations. You can't compare it to data unless you understand the model first.

Or maybe I have this "axiomatic vs non-axiomatic" division completely wrong?

 

[voko]

Getting back. You claimed that the 2 in r², was not exactly two (i.e. not an integer 2), and when this was pointed out you claimed that was axiomatic. Newton's statement is that it is an integer 2 and not 2 point something. It just happens to fail in certain situations, and (just as you say) we can use other models.

Am I interpreting you correctly?

I said that the law was a result of measurement. Or observation, if you will. As such, it is bound to have some uncertainty - that's why we have this entire "significant figure" business to begin with. There is no a priori reason why the 2 in the law is on a completely different footing than, for example, the G in that same law.

 

[jbriggs444]

I said that the law was a result of measurement. Or observation, if you will. As such, it is bound to have some uncertainty - that's why we have this entire "significant figure" business to begin with. There is no a priori reason why the 2 in the law is on a completely different footing than, for example, the G in that same law.

You are suggesting, I suppose, that if our ability to measure r accurately over a range of radii were impaired and our ability to measure M and m accurately over a range of test objects were improved and if we treated the problem purely as modelling a set of data points by regression to a function of the form F = GmM/rⁿthen we could plausibly know G to a greater accuracy than n rather than the reverse.

Sure, that seems fair.

 

[D H]

There is no a priori reason why the 2 in the law is on a completely different footing than, for example, the G in that same law.

Yes, there is. Gravitation is instantaneous in Newtonian physics and space is isotropic, three dimensional, and distinct from time. That 2 is exact given those assumptions.

Discard those assumptions and you don't get Newtonian gravity. You get general relativity, which doesn't look like Newtonian gravity. In the limit of small masses, small velocities, and large distances, general relativity does simplify to Newton's law of gravitation -- and the 2 is exact.

 

[CompuChip]

Sorry to interrupt this fascinating discussion, but is this still at all relevant to the original question? Or has the original topic starter left us long ago?