Why are zeros after a decimal point significant?

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In summary: But to put it in a way that allows us to apply the rules for significant figures to the problem, say you're adding ##8\pm1## to ##8\pm.1##. The significant digits rules tell us that the answer is ##16\pm1##.(If you're adding ##8\pm1## to ##8\pm.0001##, you can't use significant digits rules, but the answer is still ##16\pm1##.)Or, we can say that "8.000", while it's more than we need to represent the value accurately, is still the best we've got. That is, we can't say that the true
  • #36
voko said:
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).
 
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  • #37
pwsnafu said:
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.
 
  • #38
voko said:
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?
 
  • #39
pwsnafu said:
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.
 
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  • #40
voko said:
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.
 
  • #41
voko said:
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.
 
  • #42
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?
 
<h2>1. Why do we use zeros after a decimal point?</h2><p>Zeros after a decimal point are used to represent numbers that are less than one. They help to show the precision or accuracy of a measurement or calculation.</p><h2>2. Are zeros after a decimal point always significant?</h2><p>No, zeros after a decimal point are not always significant. It depends on the context and the number itself. In some cases, they may be used as placeholders or trailing zeros and do not add any additional information.</p><h2>3. How do zeros after a decimal point affect significant figures?</h2><p>Zeros after a decimal point are considered significant figures, and they are included when counting the total number of significant figures in a number. However, trailing zeros may or may not be significant depending on the measurement or calculation.</p><h2>4. Can zeros after a decimal point change the value of a number?</h2><p>Yes, zeros after a decimal point can change the value of a number. For example, 0.01 and 0.010 are two different numbers with different values. The additional zero after the decimal point in the second number makes it more precise and changes its value.</p><h2>5. Why is it important to understand the significance of zeros after a decimal point in science?</h2><p>In science, accuracy and precision are crucial. Zeros after a decimal point can affect the precision of a measurement or calculation, which can, in turn, affect the accuracy of the results. Therefore, understanding the significance of zeros after a decimal point is important in ensuring the reliability of scientific data and conclusions.</p>

1. Why do we use zeros after a decimal point?

Zeros after a decimal point are used to represent numbers that are less than one. They help to show the precision or accuracy of a measurement or calculation.

2. Are zeros after a decimal point always significant?

No, zeros after a decimal point are not always significant. It depends on the context and the number itself. In some cases, they may be used as placeholders or trailing zeros and do not add any additional information.

3. How do zeros after a decimal point affect significant figures?

Zeros after a decimal point are considered significant figures, and they are included when counting the total number of significant figures in a number. However, trailing zeros may or may not be significant depending on the measurement or calculation.

4. Can zeros after a decimal point change the value of a number?

Yes, zeros after a decimal point can change the value of a number. For example, 0.01 and 0.010 are two different numbers with different values. The additional zero after the decimal point in the second number makes it more precise and changes its value.

5. Why is it important to understand the significance of zeros after a decimal point in science?

In science, accuracy and precision are crucial. Zeros after a decimal point can affect the precision of a measurement or calculation, which can, in turn, affect the accuracy of the results. Therefore, understanding the significance of zeros after a decimal point is important in ensuring the reliability of scientific data and conclusions.

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