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Mathematics, numbers, accuracy, precision, etc.

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Mathematics, numbers, accuracy, precision, etc.

Unread postAuthor: boyntonstu » Tue Apr 13, 2010 4:29 pm

I used to work for the National Bureau of Standards in Washington D.C..

Today, it is called, NIST, The National Institute of Standards, and Technology.

Jokingly, we referred to it as "The National Bureau of Substandards".

Creating standards using precision measurements was the main function of the laboratories.


Now for some fun:

Mathematics is a language and not representative of the real world.

There is no such thing as a 1 or a 7.

There are no 2 identical items in the real world.

Therefore the equation 1+1 = 2 has implications far beyond reality.

The two 1's are identical in math.

Is there a pair of anything in the real world that are identical?

There can be 2 classes of things but not 2 identical things.

1 apple does not equal the apple next to it.

Electrons cannot be identical because they both can't be in the same location at the same time.

Makes you think.

When you convert NUMBERS from one system to another system there are NO limits on significant digits.

What is the equivalent of 2 grams? 30.864 716 706

Has a measurement been made?

In the fields of engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to its actual (true) value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. Although the two words can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.
Accuracy indicates proximity of measurement results to the true value, precision to the repeatability or reproducibility of the measurement

A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. Eliminating the systematic error improves accuracy but does not change precision.


Thus, significant digits only come into play in measurements, not in conversions.

we used to measure it with a micrometer, mark it with chalk, and cut it with an axe!
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Unread postAuthor: Bluetooth » Tue Apr 13, 2010 4:59 pm

Ok... Interesting... I wish we didn't need to use conversions. The International System of Units is good with me. Even a thing we call the "gas constant" is just a crazy measurement. The mass of any subatomic particle is so small it can't really be measured, so they all count the same to us (while electrons aren't even counted).
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Unread postAuthor: jimmy101 » Tue Apr 13, 2010 7:10 pm

Bluetooth wrote:Ok... Interesting... I wish we didn't need to use conversions. The International System of Units is good with me. Even a thing we call the "gas constant" is just a crazy measurement. The mass of any subatomic particle is so small it can't really be measured, so they all count the same to us (while electrons aren't even counted).

The mass of subatomic particles can be measured to quite exquisite accuracy. 5 or 6 decimal places without hardly trying. (If you have the equipment.) A $10K (that's cheap) mass spectrometer will get you at least 3 and probably more than 4 sig. figs. on the mass of say CO2. Easy enough to calculate the mass of the nucleons from that. Compare a couple different small molecules, containing different sets of atoms, and you can tease out the difference between the nucleon's masses. Or, just look at the masses for the naturally occurring trace of C^13 that is in any carbon containing sample. C^12O2 vs. C^13O2 gives you the mass of a neutron.
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Re: Mathematics, numbers, accuracy, precision, etc.

Unread postAuthor: Ragnarok » Tue Apr 13, 2010 7:14 pm

boyntonstu wrote:Thus, significant digits only come into play in measurements, not in conversions.

No. Significant figures are in play when you're dealing with any continuous variable with a finite precision. The precision of any number is not magically increased by performing a mathematical calculation on it.

You can't go from the +/- 0.05 gram precision suggested by a value of 2.0 grams to a precision of +/- 0.0000000005 grains that a value of 30.864716706 grains would need - that's that's over nine orders of magnitude of precision that have mystically appeared out of nowhere.
They weren't there in the original measurement, so they shouldn't be there in the final answer.

Sure, if you're defining it as 2 grams with infinite precision (or at least 2.0000000000 gram precision), then you can convert to 30.864716706 validly, but you don't get infinite precision in real life measurements.

... unless they're discrete variables, of course, but most conversions are not done with discrete variables. An exception being currency, where I can have £105.23, and convert that into US dollars, but in the end, the final result has to be rounded due to the fact that US dollars are also a discrete variable.

It's not completely unreasonable to add significant figures if it's plausible that there are more than the notation might suggest, but you certainly can't do it on a whim.

Oh yeah, one last note before someone picks me up on it. You do NOT round off in intervening stages. Only when you get to the final answer, else you'll accumulate rounding errors.
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Unread postAuthor: c11man » Tue Apr 13, 2010 7:24 pm

i wonder what will happen if i mention the fact that using copper for my cannons and feel as though it is an infurior building material.
maybe once this thread gets locked too this "mystery" poster will have a fit and make anouther thread called "when and why to use copper for cannons".

on topic i agree with Rag that precision does not appear out of nowere. unless you are running formulas when the inputs are a specific and infinitly precise integer a conversion SHOULD not give you a number with more sig figs than the original value
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Re: Mathematics, numbers, accuracy, precision, etc.

Unread postAuthor: boyntonstu » Tue Apr 13, 2010 7:44 pm

Ragnarok wrote:
boyntonstu wrote:Thus, significant digits only come into play in measurements, not in conversions.

No. Significant figures are in play when you're dealing with any continuous variable with a finite precision. The precision of any number is not magically increased by performing a mathematical calculation on it.

You can't go from the +/- 0.05 gram precision suggested by a value of 2.0 grams to a precision of +/- 0.0000000005 grains that a value of 30.864716706 grains would need - that's that's over nine orders of magnitude of precision that have mystically appeared out of nowhere.
They weren't there in the original measurement, so they shouldn't be there in the final answer.

Sure, if you're defining it as 2 grams with infinite precision (or at least 2.0000000000 gram precision), then you can convert to 30.864716706 validly, but you don't get infinite precision in real life measurements.




I was using discrete variables when I showed the conversion relationship.

We agree!

"... unless they're discrete variables, of course, but most conversions are not done with discrete variables. An exception being currency, where I can have £105.23, and convert that into US dollars, but in the end, the final result has to be rounded due to the fact that US dollars are also a discrete variable. "

I know of a theft case based on a bank's computer programmer who collected 3rd place interest rounding into his account.

Edited by jrrdw.
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Re: Mathematics, numbers, accuracy, precision, etc.

Unread postAuthor: Ragnarok » Tue Apr 13, 2010 8:26 pm

boyntonstu wrote:I was using discrete variables when I showed the conversion relationship.

No, you weren't. You were using continuous variables with infinite precision.

Discrete variables can only be certain values (generally integers, although non-numerical options exist), usually used for a number of objects.
Continuous variables can be any value, with no limit on decimal precision.

Mass is inherently a continuous variable (as is length, temperature, time, current, etc), although for theoretical purposes, it can be defined with infinite precision.
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Re: Mathematics, numbers, accuracy, precision, etc.

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Unread postAuthor: jimmy101 » Wed Apr 14, 2010 7:59 pm

There are "continuous variables" with infinite precision. For example, 1 inch = 2.54cm. Both the numbers (1 and 2.54) are exact and have an infinite number of sig. figs.

If you measure a distance in inches to an accuracy of a part per ten thousand then convert to cm using 2.54 you don't loose any precision even though 2.54 only has three stated sig. figs. The final answer is still accurate to a part per ten thousand.
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Unread postAuthor: Ragnarok » Wed Apr 14, 2010 8:43 pm

jimmy101 wrote:There are "continuous variables" with infinite precision.

I didn't say there weren't, but it only exists on a theoretical level or when a real world measurement has it by definition.

One example of such a real world measurement is the International Prototype Kilogram which cannot mass anything other than precisely one kilogram, because a kilogram is defined as being exactly its mass.
If you measure it as anything else, then it's your equipment that's wrong.

Anyway, we as spudders are not possessed of the technical ability to make our real world measurements to any particularly notable degree of precision.
We can use conversions which are possessed of infinite precision (such as 1 inch = 2.54 cm, 1 lb = 453.59237 grams or 1 mph = 0.44704 m/s*) and therefore not lose any precision in a conversion, but we cannot add precision beyond our original measurements in any conversion.

*Before anyone asks, yes, I know these from memory.
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Unread postAuthor: rcman50166 » Wed Apr 14, 2010 11:24 pm

Yup know all about this stuff. 4th semester engineering student here. Curious, does NIST hold the international kilogram standard? I understand there isn't even a need for it anymore since it can be precisely measured by other means.
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Unread postAuthor: Ragnarok » Thu Apr 15, 2010 6:43 am

rcman50166 wrote:Curious, does NIST hold the international kilogram standard?

No, the IPK is held at the BIPM in Paris. However, there are a number of replicas, of which the US holds four. Two of which are K4 and K20 from the original forty 1884 replicas.

None of the replicas has a precisely identical mass to the original. K20, for example, is the USA's primary national standard, and is (currently) defined as having a mass of 1 kg - 39 µg.

Where those standards are kept, I do not however know.

I understand there isn't even a need for it any more since it can be precisely measured by other means.

There are attempts to replace the definition of the kilogram with a more refined method, but currently, it is still needed, as it IS the definition.

But if you take the advance of things like the Avogadro project, it looks like we may have a new definition in no great number of years.
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Unread postAuthor: boyntonstu » Thu Apr 15, 2010 7:28 am

I am happy that you guys understand the relationship between Mathematical (Maths) language numbers and real world numbers.

Incidentally, In a book "The Proper Study of Mankind" Harper & Brothers 1948, Stuart Chase, he asks this question: Is Social Science a science equal to Physics, Biology, etc?

He answers; Yes, since Heisenberg's Uncertainty Principle proves that statistically valid data is the limit of our measurements.

Physics, Chemistry, and Social Science are sciences, given statistically valid data.

When I taught Aviation Science to Navy pilot students in Pensacola, Florida, the goal was to create a test to determine whether or not the trainee had the skills necessary for dog fighting an airplane.

The theory was to test the student's ability to do flips on a trampoline.

The trampoline was evaluated for statistical validity and was proven to be a good yardstick.

[Anyone who doesn't take truth seriously in small matters cannot be trusted in large ones either. Albert Einstein ]
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Unread postAuthor: CpTn_lAw » Thu Apr 15, 2010 9:21 am

What is the point in this topic? To prove that a table is a real world object that cannot be entirely flat you give its measurement to many sig.fig. ?

Mathematics : a logical way found by man to explain his surroundings, and expand his logical and abstract comprehension of the universe.

Truth : ?

In a universe counting billions of galaxies, each one counting billions of stars, over distances that only make sense when written in maths language (which is funny because of boyntonstu's first post) ; in a world where man has been using electricity for only 200 years (don't play on words.....) ; who are we to pretend to know anything? We find logic and science to give us milestones in our evolution, to give us pillars to hold on to, because we do not know crap about anything. And we are happy, because it seems to make sense, because every discovery is connected to the basic same mathematical language. Just because we cannot explain the fundamental thing that makes us think , otherwise than with maths (which is funny again, explaining a real-world thing with a language that is not representative of the real world).

We are strapped to a rock the size of a pixel in a billion feet long screen, trying to catch our tail, and we pretend to even begin to understand a thing.

We struggle with our techniques to just render a raytraced scene in real-time, making approximations in every algorithm, where nature doesn't seem to be calculating anything.

So yeah, I must be missing the point in this topic.
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Unread postAuthor: POLAND_SPUD » Thu Apr 15, 2010 9:45 am

boyntonstu claimed (and he probably still thinks that this is true) that grains are better than grams becasue...
....well see for your self if you managed to miss this
Getting back to air rifles, imagine using grams or ounces for .177 or .22 bb's or pellets.

For example:

A Gammo X10 .177 target pellet weighs 7.7 grains.
A Gammo X10 .177 hunting pellet weighs 7.6 grains.

or

A Gammo X10 .177 target pellet weighs 0.498 951 grams.
A Gammo X10 .177 hunting pellet weighs 0.492 471 grams.

That is exactly the reason why grains are used for bb's and pellets.


@boyntonstu
[Anyone who doesn't take truth seriously in small matters cannot be trusted in large ones either. Albert Einstein ]

lol you REALLY shouldn't have posted this :D
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Unread postAuthor: boyntonstu » Thu Apr 15, 2010 9:51 am

POLAND_SPUD wrote:boyntonstu claimed (and he probably still thinks that this is true) that grains are better than grams becasue...
....well see for your self if you managed to miss this
Getting back to air rifles, imagine using grams or ounces for .177 or .22 bb's or pellets.

For example:

A Gammo X10 .177 target pellet weighs 7.7 grains.
A Gammo X10 .177 hunting pellet weighs 7.6 grains.

or

A Gammo X10 .177 target pellet weighs 0.498 951 grams.
A Gammo X10 .177 hunting pellet weighs 0.492 471 grams.

That is exactly the reason why grains are used for bb's and pellets.


@boyntonstu
[Anyone who doesn't take truth seriously in small matters cannot be trusted in large ones either. Albert Einstein ]

lol you REALLY shouldn't have posted this :D


You are so kind!
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