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Basically I need someone to convince me as to why leading zeroes are not counted as significant digits.
I've been listening to this bullshit in school for forever. And I've never had someone that has really explained it to my understanding..
As I see it, significant digits are a way of showing how much accuracy you took in you're measurements. If I weigh something and I get let's say, 10.000405 grams it is considered to have 8 significant digits. Let's say I weigh the same thing, but it loses ten pounds, and is now 0.000405 grams. I used the same tool to obtain this result and am measuring to the same degree of accuracy. But now I only have 3 significant digits. This does not make sense to me whatsoever.
 
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.000405 kilograms = .405 grams
They have the same # of sig figs. How would life work if converting kg to g changed the # of sig figs by 3?
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0.000405 * 10 to the exponent 3 = 0.405
At least I think that's the reason.
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For the same reason that when you go 0121km/hr you've only measured 3 digits... the leading zero's tell you the size of the thing you're measuring... not a degree of accuracy of the the thing you're measuring.
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Can someone maybe explain the flaw in my problem then?
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I don't see why it's a problem that you get a different number of significant digits in the two cases given in the OP.
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On May 20 2010 12:37 Ian Ian Ian wrote:
Basically I need someone to convince me as to why leading zeroes are not counted as significant digits.
I've been listening to this bullshit in school for forever. And I've never had someone that has really explained it to my understanding..
As I see it, significant digits are a way of showing how much accuracy you took in you're measurements. If I weigh something and I get let's say, 10.000405 grams it is considered to have 8 significant digits. Let's say I weigh the same thing, but it loses ten pounds, and is now 0.000405 grams. I used the same tool to obtain this result and am measuring to the same degree of accuracy. But now I only have 3 significant digits. This does not make sense to me whatsoever.
Actually, according to your loss of 10 lbs example, your final answer of 0.000405 g counts as "6 sig figs past the decimal place" since you performed addition/subtraction rather than multiplication/division.
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On May 20 2010 12:41 Ian Ian Ian wrote:
Can someone maybe explain the flaw in my problem then?
On May 20 2010 12:41 meeple wrote:
For the same reason that when you go 0121km/hr you've only measured 3 digits... the leading zero's tell you the size of the thing you're measuring... not a degree of accuracy of the the thing you're measuring.
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motbob
United States12546 Posts
All I can do is explain why your proposed situations are inherently different.
In the first situation, the mass of the object is precise to the one millionths place, and the mass of the object is 10 million times one millionth. In the second situation, the mass of the object is precise to the one millionths place, but the mass of the object is only 400 times one millionth.
In other words, the first example is much more precise compared to the mass of the object. That is what significant digits tell us.
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Is it sad that I instinctively thought of "getting a chick's phone number", because I've used this line several times for that?
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The main flaw is that you've lost 10 pounds but it seems you've only lost 10 grams... heh but I guess you're talking about your problem there...
Well, although it seems unintuitional to "lose" significant digits in a measurement... it's actually perfectly common. There isn't any real problem with it...
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Also... I agree with what motbob said about you lose significant digits because your accuracy in relation to the size of the second number is much less than the accuracy in relation with the size of the first number.
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Still stupid imo 
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extrapolate. Ever use a slide rule and you'll learn fast how you only use 3 digits for everything, frankly it's not as relevant as we have calculators...
significant digits is about precision, not about scientific notation.
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On May 20 2010 12:53 Ian Ian Ian wrote:Still stupid imo
This poses some problems in numerical modeling.
What you noticed is a simple principle of mathematics which eliminates precision by the use of subtraction..... In real situations, you need to be careful, because it can screw up the results greatly.
Lets imagine that your scale for removing the weight could only measure to 3 decimal places. You subtract 10.000 kg from 10.000405 kg. How sure can you actually be that the remaining amount is 0.000405 kg?
10.000405
- 10.000???
------------------
0.000???
Your precision is now down to 3 decimal places, for a total of 3 significant digits. You have no way of knowing the precise amount that is left.
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Forget about the subtraction then..
Like it just seems to me that if you measure something to 4 decimal places or whatever, it should still have the same amount of significant digits, regardless of it's 0.000# or #.000#
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Calgary25998 Posts
lol its not stupid at all. the leading zeroes are just to get to the actual meat of the number because of our writing conventions.
write it out in scientific notation and wow all your zeroes are gone and meaningless.
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Precision and accuracy are two different things. Something can be accurate, but completely imprecise.
For example: if something weighs 1kg, and a scale measures 1kg, it is entirely accurate. Likewise, if it measures it to be 1.00000000kg, that is also equally accurate. However, the second reading is a lot more precise. If something is a smaller mass and the scale is not adjusted accordingly, of course you will have a less precise answer, however that will not affect the accuracy of the measurement.
So to give you an answer, the leading zeros give no added precision to the number, so it should not have any significant figures. However, latter zeros add precision because it still adds precision (you know that the last zero is indeed close to zero).
You have to look at the way you see significant figures. They are not meant to measure accuracy at all. (An extremely precise reading could be entirely inaccurate)
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Just saying, you called them significant digits in your title, and I usually call them Significant Figures. Significant Figures allows them to be abbreviated to "Sig Figs" (flows off the tongue nicely), but calling them Significant Digits forces you to abbreviate the phrase as "Sig Digs" (violently jumps off the tongue with acid-covered cleats).
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# significant figures = # of digits used in exponential form.
So 100 = 1x10^2 has 1 sig fig, but 1.000000x10^2 has 7 sig figs - it's a statement of how much precision you actually measured.
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On May 20 2010 13:12 Ian Ian Ian wrote:
Forget about the subtraction then..
Like it just seems to me that if you measure something to 4 decimal places or whatever, it should still have the same amount of significant digits, regardless of it's 0.000# or #.000#
No. It does not have the same number of significant digits. It has the same level of precision, however.
Imagine I have a scale that can only read to 1 decimal place. I place a penny on it, and find the reading is 4.5 grams. I have 2 significant digits, and 1 decimal place precision.
I then put 10 pennies on the scale. The reading I get is 45.4 grams. I have 3 significant digits, and 1 decimal place precision.
I then put 100 pennies on the scale. The reading I get is 454.3 grams. I have 4 significant digits, and 1 decimal place precision.
I put 1000 pennies on the scale. I get 4543.2 grams. I have 5 significant digits, and 1 decimal place precision.
Because I have a known count of the number of pennies, I can divide up the 4543.2 grams by 1000 pennies to find the weight of the average penny to be 4.5432 grams. This is using a scale which has 1 decimal place precision to find a result with 4 decimal place precision.
Precision and significant digits are two completely different, but related, concepts.
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I just finished a first year physics course and there was not talk at all about significant digits. All there was was error analysis with those cool little plus/minus signs and fair bit of tedious calculation to find exactly what the plus/minus was. Are significant digits ever actually used and for what?
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On May 20 2010 13:46 Kwidowmaker wrote:
I just finished a first year physics course and there was not talk at all about significant digits. All there was was error analysis with those cool little plus/minus signs and fair bit of tedious calculation to find exactly what the plus/minus was. Are significant digits ever actually used and for what?
They're used in Chemistry pretty often. Physics tends to ignore sig figs and units.
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Bill307
Canada9103 Posts
I remember being all confused the first time we were taught significant digits in high school. Then I realized it was just jargon for something I already understood intuitively.
Intuitively, you know that 0001 and 1 are the same thing. Intuitively, you know that 0.001 kg and 1 g are the same thing. So leading zeroes don't change anything.
Intuitively, if I tell you I bought 1 kg of peanuts, you know that I probably didn't buy exactly 1 kg of peanuts: I'm just rounding it off. You also have no idea how precise I was: did I round it to the nearest 10 g and it just happened to come out to 1 kg? Did I round it to the nearest 100 g and the real weight is something like 1.043 kg? You don't know: it's ambiguous.
So when you're working in the field of science, where precision is very important, you know that there has to be a system of telling people how precise your measurements are. One such system is scientific notation.
See, normally (not talking about sig figs or scientific notation here...) there's no reason to add trailing zeroes after the decimal point, e.g. if I write 0.400 the trailing zeroes normally serve no purpose: I'd might as well write 0.4 and it'd be the same thing. So science says, let's use those trailing zeroes for something: let's have them indicate that our measurement is more precise than just 0.4.
Say I've measured out 1.000 kg of calcium chloride with precision to the gram, but we're writing it in grams to be consistent with our other figures, so we write 1000 g. How will other scientists know that those zeroes are significant? How do they know we didn't just round it to the nearest 100 grams or something? That's where scientific notation comes in. If we write it as 1.000 x 10^3 grams, there is no question that the trailing zeroes must be significant, otherwise we would've simply written 1 x 10^3 instead.
And that's all there is to significant digits. It's all very practical, designed to allow scientists to communicate with each other more clearly and with less confusion. Schools just fail to teach it from a practical standpoint, in my experience. =P
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On May 20 2010 12:37 Ian Ian Ian wrote:
Basically I need someone to convince me as to why leading zeroes are not counted as significant digits.
I've been listening to this bullshit in school for forever. And I've never had someone that has really explained it to my understanding..
As I see it, significant digits are a way of showing how much accuracy you took in you're measurements. If I weigh something and I get let's say, 10.000405 grams it is considered to have 8 significant digits. Let's say I weigh the same thing, but it loses ten pounds, and is now 0.000405 grams. I used the same tool to obtain this result and am measuring to the same degree of accuracy. But now I only have 3 significant digits. This does not make sense to me whatsoever.
Well, if you have 1001 grams and take away 1000 you still get 1 grams. You subtracted two quantities with 4 significant digits and got one with 1 significant digit. This problem has nothing to do with leading zeros.
If you want to you can think of significant figures as relative precision. Measuring your mass to kilograms is less precise than measuring the mass of the Moon to kilograms, even though both measurements are in kilograms.
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Your scale probably has a variance of 0.01mg or so. When you measure something relatively large compared to the variance of the scale, you get a lot of significant figures because the scale is pretty sure of those last 10.000405 When you measure something small, and you're hoping to get 0.0004050182, the scale really has no idea if those last 0.0004050182 are even close to correct. You may be able to find a scale that can give you a small enough variance to measure that, but the max capacity will probably be 10mg or so.
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On May 20 2010 14:09 Bill307 wrote:
I remember being all confused the first time we were taught significant digits in high school. Then I realized it was just jargon for something I already understood intuitively.
Intuitively, you know that 0001 and 1 are the same thing. Intuitively, you know that 0.001 kg and 1 g are the same thing. So leading zeroes don't change anything.
Intuitively, if I tell you I bought 1 kg of peanuts, you know that I probably didn't buy exactly 1 kg of peanuts: I'm just rounding it off. You also have no idea how precise I was: did I round it to the nearest 10 g and it just happened to come out to 1 kg? Did I round it to the nearest 100 g and the real weight is something like 1.043 kg? You don't know: it's ambiguous.
So when you're working in the field of science, where precision is very important, you know that there has to be a system of telling people how precise your measurements are. One such system is scientific notation.
See, normally (not talking about sig figs or scientific notation here...) there's no reason to add trailing zeroes after the decimal point, e.g. if I write 0.400 the trailing zeroes normally serve no purpose: I'd might as well write 0.4 and it'd be the same thing. So science says, let's use those trailing zeroes for something: let's have them indicate that our measurement is more precise than just 0.4.
Say I've measured out 1.000 kg of calcium chloride with precision to the gram, but we're writing it in grams to be consistent with our other figures, so we write 1000 g. How will other scientists know that those zeroes are significant? How do they know we didn't just round it to the nearest 100 grams or something? That's where scientific notation comes in. If we write it as 1.000 x 10^3 grams, there is no question that the trailing zeroes must be significant, otherwise we would've simply written 1 x 10^3 instead.
And that's all there is to significant digits. It's all very practical, designed to allow scientists to communicate with each other more clearly and with less confusion. Schools just fail to teach it from a practical standpoint, in my experience. =P
Couldn't have said it better.
On a side note it's nice to know what the expression was in English. (For leading numbers, that is.)
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EPT
