Significant Figures and zeros

[Leoragon]

Why are zeroes following a non-zero number in the decimal area considered significant?

2.34000 <-- Like this, why are the three zeroes after the four considered significant?

 

[member 392791]

Significant figures are usually needed for lab measurements.

Suppose you have a weight balance that can measure grams to 5 decimal places. The amount of decimals that your weight balance can accurately measure are all considered significant.

So if you had 2.34001g, that would be significant. But since your weight balance can measure that accurately, if the balance says 2.34000g, then all of those digits are significant because your balance can measure it.

 

[HallsofIvy]

The final "0"s do not add anything to saying what the value of the number is- mathematically, 2.34= 2.340= 2.3400= 2.34000. So the only purpose the "0"s can have is to tell us the accuracy- they are "significant figures".

 

[Leoragon]

What do you mean tell us the accuracy? Isn't 2.34 already accurate enough?

If I'm getting this right. The extra zeroes just tells me the capabilities of the instrument.

 

[symbolipoint]

What do you mean tell us the accuracy? Isn't 2.34 already accurate enough?

If I'm getting this right. The extra zeroes just tells me the capabilities of the instrument.

The instrument indicates the capability (the accuracy(?)) of the measurement and resulting calculations. The significant figures stems from how well, or how precisely, a number on an instrument can be read.

 

[Leoragon]

I still don't get it.

 

[Stephen Tashi]

I still don't get it.

A guy who reports a measurement of 2.31 leaves open the possibility that the exact value might have been 2.3104.

A guy who reports a measurement of 2.3100 does not leave that possibility open.

 

[symbolipoint]

Here's another example.
A person steps on a scale and it shows he weighs 154 pounds. He does NOT know if he really weighs 153.98 pounds, or 153.610 pounds, or 154.2 pounds, or 154.1 pounds, or 154.43 pounds or 154.0009 pounds. This scale only reads to the nearest whole pound. The scale or dial does not indicate to the nearest fraction of a pound; the dial or scale cannot be reliably read between whole pound values and no markings are shown between whole pound values. This scale gives reading, if over 99 pound value, to THREE SIGNIFICANT FIGURES.

 

[ImaLooser]

What do you mean tell us the accuracy? Isn't 2.34 already accurate enough?

If I'm getting this right. The extra zeroes just tells me the capabilities of the instrument.

Well, it depends on what you are doing. 2.34 might be accurate enough. Then again, maybe not. Maybe you need 2.34000. 2.34001 would be too much and 2.33999 too little.

 

[Borek]

I think you are just confused by the fact these digits are zeros.

You feel like 2.34 and 2.3400 mean the same... OK, now is there a difference between 1.06 and 1.0614? Sure there is.

Trick is, it is exactly the same thing. 2.34 (2.3400) is a weight in pounds, 1.06 (1.0614) is a weight of the same object given in kg. It is just obvious 14 means something.

 

[symbolipoint]

Why are zeroes following a non-zero number in the decimal area considered significant?

2.34000 <-- Like this, why are the three zeroes after the four considered significant?

Why do you want to assume that those zeros are significant? Either they are or they are not. Where is the number? From what is it taken or found? 2.34000 is more accurate than 2.34. Does that help? If we add 0.00001 to 2.34000 then we obtain 2.34001.
If we have 2.34 and add to it 0.0001, then we still have 2.34. Does that also help?

 

[Leoragon]

I think you are just confused by the fact these digits are zeros.

You feel like 2.34 and 2.3400 mean the same... OK, now is there a difference between 1.06 and 1.0614? Sure there is.

Trick is, it is exactly the same thing. 2.34 (2.3400) is a weight in pounds, 1.06 (1.0614) is a weight of the same object given in kg. It is just obvious 14 means something.

Why do you want to assume that those zeros are significant? Either they are or they are not. Where is the number? From what is it taken or found? 2.34000 is more accurate than 2.34. Does that help? If we add 0.00001 to 2.34000 then we obtain 2.34001.
If we have 2.34 and add to it 0.0001, then we still have 2.34. Does that also help?

First guy, I just picked a random number. Not something in pounds or kilograms or anything. It's just random.

Second guy, I read in a book that the zeroes are significant. But what I keep getting is that the extra zeroes after the decimal are significant to show that those digits are indeed zero, not some other number; it removes the possibility of a different number. If that is true, then there can be an infinite number of significant numbers that are zero?

 

[HallsofIvy]

First guy, I just picked a random number. Not something in pounds or kilograms or anything. It's just random.

The "significant figures" are meaningless here. "Significant figures" are not a "math" topic, they are a "science" topic. You only use them to say how accurate a measurement (or quantity calculated from measurements are. If I say that a measurement is "3.5 m" I am saying that I am measuring to the "nearest tent meter" and the actual value could be anywhere from 3.45 m to 3.55 m. If, instead, I say "3.5000" meters, I am saying that I am measuring to the nearest tenth millimeter and the actual value could be anywhere from 3.49995 m to 3.50005 m.

Second guy, I read in a book that the zeroes are significant. But what I keep getting is that the extra zeroes after the decimal are significant to show that those digits are indeed zero, not some other number; it removes the possibility of a different number. If that is true, then there can be an infinite number of significant numbers that are zero?

No. As I said before "significant digits" have nothing to do with "numbers" per se. They only have to do with accuracy of measurements. Although I refuse to use the word "infinite", it is certainly true that there are an arbitrary number of different accuracies I could use in lengths and it is certainly possible that the distance I am measuring is smaller than the specific accuracy to which I am measuring. In that case I would say that I to "0" to the accuracy I was using.