Significant numbers used in chemistry confusion

[SparkErosion]

I was reading in my chemistry book about significant numbers. Also about measured and exact numbers. From what I understand, a measured quantity can be different than an exact number. The book said when comparing something, like 1 lb = 16 oz, it is an exact number because the two are the same (or exact). If I have 5 apples, it is an exact number. But it says if I have a measured quantity like temperature, or weight, it can vary according to measurement sometimes.

It says for example if I measure a number with a ruler and it goes by units of 1, between 1 and 2, like 1.5. Both are significant numbers, yet according to the book (and since 5 is a non zero I think). The .5 in 1.5 can vary depending on measurements. Why is it then a significant number? If the .5 in the number is uncertain?

From my book it says: Significant figures are all the digits including the estimated digit.

Does this make any sense? Maybe I'm confusing significant numbers with measured or exact numbers, but, I really want to know what makes numbers significant.Not just that they are. I thought significant numbers had to do with uncertanity measureing them, versus an exact number.

Thanks much for the help.

 

[Ygggdrasil]

In chemistry, you will encounter two different classes of numbers: measurements and exact numbers. Measurements are numbers that are not known exactly (such as number obtained from an experiment) and have some degree of error associated with them. Exact numbers (such as numbers that come from definitions) do not have any error associated with them (for example, one meter is exactly one hundred centimeters, not 99 nor 101 centimeters).

Because measurements have an associated error, we need some way of expressing this error. The best way to do this is to simply write out the amount of uncertainty in the measurement, such as 2±1 kg or 2±0.1 kg. However, instead we often use the number of significant figures in order to express this uncertainty. If your measurement is 2±1 kg, you would report the number with one significant figure as 2 kg. A measurement with a slightly higher precision, such as 2±0.1 kg would be written with two significant figures as 2.0 kg. Because of the precision of the measurement, you know that the first digit in the number is correct and are uncertain only about the tenths digit. Reading a number with three significant figures such as 2.00 kg tells you that whoever obtained this measurement was uncertain only about the value of the hundredths digit.

Because exact numbers have no error, the concept of significant figures does not apply to them. For example, if we say 1 lb = 16 oz, even though the number 16 might look like it has two significant figures, it really should have an infinite number of significant figures (16.000...) because there is no uncertainty associated with the number.

 

[SparkErosion]

Thank you! This helps a lot.
Thinking more about the number 1.5...

5 is a non zero, so its significant, even if it's the uncertain number. But, it also has an infinite number of significant values. 1.5 could be 1.50... to infinity. So in that case, 5 wouldn't be the significant number, it would be the zero. And after each zero, it would be the next (last) digit to be estimated. All the numbers would be significant though.

Thank you for helping me on this.
Now I think I understand why the .5 in 1.5 is significant, even though its the estimated number. Because as a whole there is no uncertanity associated with the number. I was confused because when I have a number that I know is exact (like 5.00) vs (.5, which I thought was measured) , and thus I was confused why they both were significant.Thanks again!

 

[SparkErosion]

One more question...how come there are 3 significant figures in .005 but in .00041 there's only 2? That seems to defy logic to me...both numbers are similar..the 5 is significant, zeros are too in the first number.makes sense.but in the second number for some reason the zeros after the decimal don't count? Is this an error in the book? I understand your post well and thought I understood this subject now... Thanks again!

 

[technician]

It is not correct to say that 0.005 is 3 significant figures! (0.105 would be 3 sig figs.)
The best way to deal with these numbers with 0 after the decimal point is to write them in standard form.
0.005 = 5 x 10^-3 (1 significant figure !)
0.00041 = 4.1 x 10^-4 (2 significant figures)

A good way to appreciate the meaning of significant figures is to look at what they are NOT telling you!
When you write 41 one way to look at this is to say that the number is not 42 or 40 so you could say that 41 is 41 +/-1
Similarly 4.1 means 4.1 +/- 0.1
As far as accuracy is concerned quoting a number to 3 sig figs implies an accuracy of ≈1%
eg 105 means NOT 104 and NOT 106 ... out by +/- 1 in about 100 ...1%...and so on