Problems about significant figures

[Raanikeri]

A question asks me to round 12014 to 3 significant figures.
Why is the answer 12000?
12000 only has 2 significant figures - the 1 and the 2, isn't it?

Also, why is 78000 considered to have 3 significant figures? But then 78800 is considered to have 5 significant figures?

When should we count the trailing zeros in whole numbers as significant, and when should we not?

I've just read the rules about significant figures on here https://ccnmtl.columbia.edu/projects/mmt/frontiers/web/chapter_5/6665.html but it doesn't help me at all.

Please help me ?

 

[blamocur]

12000 by itself has 2 significant figures. But when you are asked to round to 3 significant figures you still get 12000 because you are asked to find the nearest number where all but the 3 leading digits are zeros. The 3 leading digits can have zeros in them, but the rest of the digits are required to be zeros.
I think that 78000 and 78800 have 2 and 3 significant figures respectively -- why do you think it should be 3 and 5 ?

 

[Raanikeri]

Thanks ? I found your explanation really helpful. Oh, as for your last question, it wasn't me that thought those things, but it was listed on a website, and I was struggling to understand why that was the case as well.

 

[Dr.Peterson]

A question asks me to round 12014 to 3 significant figures.
Why is the answer 12000?
12000 only has 2 significant figures - the 1 and the 2, isn't it?

A number like 12000 is somewhat ambiguous. It has at least 2 s.f.s, but can have more in reality, depending on the source of the number. There is no way to write this number to show that it has 3, so this is the best you can do. Do you see an alternative answer?

Also, why is 78000 considered to have 3 significant figures? But then 78800 is considered to have 5 significant figures?

Who told you this? Please show the source.

Again, both of these, lacking a decimal point, are ambiguous, but have at least 2 and 3 s.f.s respectively. I would guess that they were given as answers to questions like your first.

 

[Raanikeri]

A number like 12000 is somewhat ambiguous. It has at least 2 s.f.s, but can have more in reality, depending on the source of the number. There is no way to write this number to show that it has 3, so this is the best you can do. Do you see an alternative answer?


Who told you this? Please show the source.

Again, both of these, lacking a decimal point, are ambiguous, but have at least 2 and 3 s.f.s respectively. I would guess that they were given as answers to questions like your first.

I asked the same question on another Math forum https://www.math-forums.com/threads/rounding-to-significant-figures.441695/#post-1098792 and that was the answer that MathLover1 gave me. To be honest I struggled to make heads or tails from any of the answers I got on that forum, which is why I'm very thankful I got some proper answers from your forum! This issue has been bugging me.

 

[Dr.Peterson]

I asked the same question on another Math forum https://www.math-forums.com/threads/rounding-to-significant-figures.441695/#post-1098792 and that was the answer that MathLover1 gave me. To be honest I struggled to make heads or tails from any of the answers I got on that forum, which is why I'm very thankful I got some proper answers from your forum! This issue has been bugging me.

He seems to be using a non-standard notation, putting a bar over zeroes that are to be taken as significant though they otherwise would not be, to remove the ambiguity. You didn't copy that bar (and we probably wouldn't have been sure what it means if you had).

 

[Raanikeri]

He seems to be using a non-standard notation, putting a bar over zeroes that are to be taken as significant though they otherwise would not be, to remove the ambiguity. You didn't copy that bar (and we probably wouldn't have been sure what it means if you had).

I'm was never taught to use a bar over zeros taken as significant...

 

[Cubist]

I have a slightly different way of looking at this which may be influenced by my previous engineering experience.

If a number "x" is calculated to be 78000 to 3 sig fig => x could be anywhere in the range 77950 ≤ x < 78050
If a number "x" is calculated to be 78000 to 2 sig fig => x could be anywhere in the range 77500 ≤ x < 78500

The 3 sig fig version provides a much tighter estimate for the true value of x. This is why it's good to state the number of significant figures in estimates. Alternatively one could just state the range (as above)

EDIT: Sometimes you might see an accuracy stated like this 78000±50 but in the world of math the plus-minus symbol "±" has a different meaning so I'd avoid using this notation

 

[Dr.Peterson]

This is why it's good to state the number of significant figures in estimates.

That's the essential point: that the way the number is written can't always unambiguously represent the actual number of significant figures,unless you add some extra notation or words. When you round (or measure) to 3 s.f's, what you write means that, but can be read differently, if not separately specified.

I'm was never taught to use a bar over zeros taken as significant...

That's why I called it non-standard. I've never been taught that, either!

 

[Dr.Peterson]

I just searched for that non-standard notation, and found it mentioned in Wikipedia (my bolding):

The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if the number 1300 is precise to the nearest unit (just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundreds due to rounding or uncertainty. Many conventions exist to address this issue. However, these are not universally used and would only be effective if the reader is familiar with the convention:​

• An overline, sometimes also called an overbar, or less accurately, a vinculum, may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).
• Less often, using a closely related convention, the last significant figure of a number may be underlined; for example, "1300" has two significant figures.
• A decimal point may be placed after the number; for example "1300." indicates specifically that trailing zeros are meant to be significant.

As the conventions above are not in general use, the following more widely recognized options are available for indicating the significance of number with trailing zeros:​

• Eliminate ambiguous or non-significant zeros by changing the unit prefix in a number with a unit of measurement. For example, the precision of measurement specified as 1300 g is ambiguous, while if stated as 1.30 kg it is not. Likewise 0.0123 L can be rewritten as 12.3 mL
• Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes 1.30×10³. Likewise 0.0123 can be rewritten as 1.23×10⁻². The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent (10³ or 10⁻²) are considered exact numbers so for these digits, significant figures are irrelevant.
• Explicitly state the number of significant figures (the abbreviation s.f. is sometimes used): For example "20 000 to 2 s.f." or "20 000 (2 sf)".
• State the expected variability (precision) explicitly with a plus–minus sign, as in 20 000 ± 1%. This also allows specifying a range of precision in-between powers of ten.