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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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Calgary25954 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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EPT
