Sig fig question: Adding 3000 + 1.234

[barryj]

When I add 3000 + 1.234 I think the answer should be 3000. A sig fig calculator says it is 3001. Please explain.

My reasoning is that 1.234 is deep in the noise of 3000 and should be ignored. However 3000. + 1.234 should be 3001.

Please correct my thinking.

 

[phinds]

Depends on how you interpret 3000. If you consider it to have only 1 or 2 significant digits, you are correct. BUT ... the normal interpretation is that it has 4 significant digits in which case you are wrong. What would you have thought if it was 3001 + 1.234 ?

 

[Vanadium 50]

The better way to express a 1-significant digit 3000 is 3 x 10³.

 

[barryj]

My understanding is 3000 with no decimal is 1 sig fig. 3000. with a decimal is 4 sigs.
so 3001 has 4 sigs the same as 3001. so 3001 = 3001. + 1.234 would be 3002. I was told to line up the decimals, do the addition, then round.

 

[hutchphd]

My understanding is 3000 with no decimal is 1 sig fig. 3000.

And so 3100 has two sig fig? You were taught wrong in my humble opinion. Zeros have feelings, too
Of course if it matters, use scientific notation which is unambiguous.

 

[barryj]

well, 3100 in scientific notation would be 3.1 X 10^3 so two sig figures, right?
I was taught that zeros after the last digit are not sig unless thee is a decimal point.

 

[vela]

My understanding is 3000 with no decimal is 1 sig fig. 3000. with a decimal is 4 sigs.
so 3001 has 4 sigs the same as 3001. so 3001 = 3001. + 1.234 would be 3002. I was told to line up the decimals, do the addition, then round.

I became aware of this convention a few years ago, but I've rarely seen it used—only once in a textbook. One problem with it is that the decimal point could be confused with a period marking the end of a sentence. And it just looks weird to my eye. As others have noted, scientific notation makes the number of sig figs clear without making room for confusion.

Your understanding is correct. The sig fig calculator probably just doesn't follow the decimal-point convention, so the number of sig figs in 3000 without further context is ambiguous.

 

[barryj]

I tutor high school students in chemistry. Sig figs is an important topic in the class so I best undersand it.

 

[hutchphd]

In chemistry class you should simply require scientific notation. No more problem.

 

[phinds]

I tutor high school students in chemistry. Sig figs is an important topic in the class so I best undersand it.

Your students will most likely use calculators. What do you think a calculator considers 3000 to be?

 

[barryj]

On a calculator, 3000 and 3000. would be the same. However using proper sig figs 1.234 + 1.2 = 2.434
and aftering rounding to two sig figs we would have 2.4

 

[vela]

Depends on how you interpret 3000. If you consider it to have only 1 or 2 significant digits, you are correct. BUT ... the normal interpretation is that it has 4 significant digits in which case you are wrong.

I'll disagree. I don't think 3000 would normally be assumed to have four sig figs. It's ambiguous. At best, you might infer from the context how many digits are significant.

 

[barryj]

Vela, I am saying that "3000" has 1 sig fig, "3000." has 4 sig figs

 

[vela]

I know. I wasn't replying to you.

 

[hutchphd]

I also disagree. If I mean 1 sig fig I will write ~3000. If I write 3000 I mean 3.000X10³. I believe your convention is not conventional.

 

[phinds]

On a calculator, 3000 and 3000. would be the same. However using proper sig figs 1.234 + 1.2 = 2.434
and aftering rounding to two sig figs we would have 2.4

But a calculator will not round that way. It will produce 2.434 because it incorrectly assumes that it should base things on whatever figure has the MOST significant digits, thus giving an answer which is not supported by the input numbers.

You should emphasize that to your students.

 

[jack action]

well, 3100 in scientific notation would be 3.1 X 10^3 so two sig figures, right?
I was taught that zeros after the last digit are not sig unless thee is a decimal point.

3100 could also be 3.100 X 10^3, so four sig figures. It is really you who decide how many sig figures there are.

Say you measure a 3 km road with your car odometer. you start at 46382.4 km and end up at 46385.5 km, then you can say you traveled 3.1 x 10^3 meters because your odometer's precision is 100 meters. But if you measure the same length of road with a (very long) measuring tape - that has a scale with a 1-meter increment - then the same measurement may be 3.100 X 10^3 meters because you know the measurement to be within a 1-meter reading error.

Where your numbers come from matters.

But without context, you should assume the number is exact, i.e. the number is always followed by an infinity of zeros. So 1.234 is really $1.234\bar{0} \times 10^0$, and 3000 is really $3.\bar{0} \times 10^3$.

 

[phinds]

But without context, you should assume the number is exact, i.e. the number is always followed by an infinity of zeros. So 1.234 is really , and 3000 is really .

I disagree and I think that your way is not at all standard. You should assume only the number of significant figures specified and 3000 is ambiguous but USUALLY taken to mean 3000.

 

[hutchphd]

As should be obvious from this palaver, students of science should use scientific notation, not a series of arcane and misunderstood suppositions. Wherever and whenever it matters. If necessary write the number both ways. Why would anyone do otherwise? Why would this not be taught?

 

[vela]

I also disagree. If I mean 1 sig fig I will write ~3000. If I write 3000 I mean 3.000X10³. I believe your convention is not conventional.

What? My convention of saying it's ambiguous?

 

[hutchphd]

I don't even know anymore. It seems like this has gone on for ~4600.00x10^-2 days

 

[jack action]

I disagree and I think that your way is not at all standard. You should assume only the number of significant figures specified and 3000 is ambiguous but USUALLY taken to mean 3000.

An integer is exact by definition, so 3000 is $3.\bar{0} \times 10^3$. And [KEYWORDS]without context[/KEYWORDS], 3000 is an integer.

And - always without context - 1.234 is 1234/1000, an integer divided by another integer, both exact numbers. Thus, $\frac{1.234\bar{0} \times 10^3}{1.\bar{0} \times 10^3} = \frac{1.234\bar{0}}{1.\bar{0}} = 1.234\bar{0}$.

Let's add units to these integers to visualize the concept: You have 3000 apples. You could assume 4 significant figures (why?) and then you have 3000 apples ± 1 apple. Or it could be exactly 3000 apples, an exact number. But if you say you have 3 apples, Is it 3 apples ± 1 apple (because only 1 significant figure) or exactly $3.\bar{0}$ apples?

Without context, I can't imagine assuming the quantity of significant numbers based on the number of digits the integer has.

 

[barryj]

I read that any count of something like 12 apples then 12 has infinite number of sig figs.
Also, a definition has an infinate number of sig figs. 1 mile = 5280 feet the 5280 would actually have an infinite number of sig figs.

 

[vela]

An integer is exact by definition, so 3000 is $3.\bar{0} \times 10^3$. And [KEYWORDS]without context[/KEYWORDS], 3000 is an integer.

In the context of this thread, the initial post implies that the 3000 and 1.234 numbers are the result of some sort of measurement with finite precision.

Without context, I can't imagine assuming the quantity of significant numbers based on the number of digits the integer has.

I agree.

 

[DaveE]

Of course if it matters, use scientific notation which is unambiguous.

Without context, I can't imagine assuming the quantity of significant numbers based on the number of digits the integer has.

You should assume only the number of significant figures specified and 3000 is ambiguous

It's ambiguous. At best, you might infer from the context how many digits are significant.

Yup, this. Your question is meaningless since the number format you are using simply doesn't convey the necessary information.

Some may argue they know the correct interpretation. But you can clearly see that in practice it wouldn't be an accepted standard for this example. The thing about definitions is that people have to know and agree with them, or you will always have to include a discussion of what you mean. It isn't any good to be "right" if people don't agree or understand.

Vela, I am saying that "3000" has 1 sig fig, "3000." has 4 sig figs

It's also good if your conventions are consistent and applicable in more general cases. So, using this "decimal only" notation, how would you denote 3.0 x 10³ and 3.00 x 10³?

If you're going to reinvent the wheel, make it a better wheel than what we already have.

 

[jbriggs444]

There are four places where significant figures may be used. Assuming that we are bound and determined to use significant figures instead of a proper error analysis.

1. In printed data received from an external source without explicit information about an error bound. A number of significant digits is inferred from the number of digits present in the printed numeral.

2. In measured data or data received from an external source with explicit information about an error bound. A number of significant digits is determined based on the error bound and the measured (or printed) value.

In either case, the number of significant figures in such data should then be explicitly tracked separately from the data itself.

3. In an intermediate calculation involving received, measured or previously calculated data. All such input data will have a separately tracked number of significant digits. The calculation should be performed to full machine accuracy and the result recorded as such. The number of significant digits in that result should be assigned based on the rules for significant digits.

4. In a final result reported elsewhere. The result will have full machine accuracy along with a separately tracked number of significant digits. The reported result should be rounded to that number of significant digits and reported in scientific notation.

In the context of this thread, it seems that we are invited to consider two printed results, "3000" and "1.234" received from an external source.

First question: What does the 3000 mean? That question has been belabored enough already. My opinion is that it is ambiguous. One cannot know what the author meant without additional context.

Second question: Are we producing an intermediate result or a final result?

If intermediate, we store 3001.234 and note either one significant digit or four.
If final, we report either $3\times10^3$ or $3.001\times10^3$.

Edit: It may be worth considering the Robustness Principle (aka Postel's law): "be conservative in what you send, be liberal in what you accept". So you can accept that "3000" as input but you should never generate it as output.

 

[DrJohn]

Four people are standing on a very long aircraft runway, on a white line that crosses the runway at 90 degrees, their toes are just touching the leading edge. And they are looking to the far end of the runway. And I am standing nearby.

Person A - how far is it from the edge of this line to the end of the runway?
Person B - 3 km
Person A steps back one metre - And how far is it from where I am now to the end of the runway?
I'd say 3 km.

Person C turns to Person D - how far is it from the edge of this line to the end of the runway? (C and D are both a bit deaf ;) so didn't hear the other two)
Person D - 3000 metres
Person C also steps back one metre - And how far is it from where I am now to the end of the runway?
I'd say 3001 metres.

I think both my answers are correct. Person B answered with little precision, Person D's answer implies, to me, a lot of precision.

Person E arrives, stands on the line and says it looks about 3000 metres to the end of the runway from here.
Person E steps back one metre and asks how far is it to the end of the runway from where I am now?
I'd say still about 3000 metres.

And I think that answer is correct as well.

 

[bob012345]

But a calculator will not round that way. It will produce 2.434 because it incorrectly assumes that it should base things on whatever figure has the MOST significant digits, thus giving an answer which is not supported by the input numbers.

You should emphasize that to your students.

Are you suggesting that if I input 1.234 + 1.2 in my calculator it should not give me 2.434? If it gave me 2.4 I'd throw it out.

 

[phinds]

Are you suggesting that if I input 1.234 + 1.2 in my calculator it should not give me 2.434? If it gave me 2.4 I'd throw it out.

But if you wrote 2.434 as the answer on a quiz where significant digits were supposed to be shown correctly, you would get a zero on the problem.

 

[bob012345]

But if you wrote 2.434 as the answer on a quiz where significant digits were supposed to be shown correctly, you would get a zero on the problem.

I'm just saying that the calculator is not incorrect and should not be interpreting numbers for the user. It just adds raw numbers and leaves the interpretation and rounding to the student.

 

[phinds]

I'm just saying that the calculator is not incorrect and should not be interpreting numbers for the user. It just adds raw numbers and leaves the interpretation and rounding to the student.

Read my post. I did NOT say the calculator was incorrect, I said it incorrectly gives an answer to the specific problem on the hypthetical quiz and that students should be aware of that. I stand by that statement.

 

[bob012345]

Read my post. I did NOT say the calculator was incorrect, I said it incorrectly gives an answer to the specific problem on the hypthetical quiz and that students should be aware of that. I stand by that statement.

Sorry, I understand. I think we are both saying it is the student's responsibility to understand the limitations of using a calculator wrt significant digit quiz problems.

 

[pbuk]

Sorry, I understand. I think we are both saying it is the student's responsibility to understand the limitations of using a calculator wrt significant digit quiz problems.

Yes, but even more than that I think there are great limitations inherent in the idea of a "significant digit calculator", and even a "significant digit quiz", because there can be no answers that are unconditionally correct, they will always be subject to context and interpretation.

 

[phinds]

Yes, but even more than that I think there are great limitations inherent in the idea of a "significant digit calculator", and even a "significant digit quiz", because there can be no answers that are unconditionally correct, they will always be subject to context and interpretation.

I disagree. If a question specifically states something like "using standard rounding, show your result to 3 significant digits", that is unambiguous and not subject to interpretation, is it not?

 

[pbuk]

I disagree. If a question specifically states something like "using standard rounding, show your result to 3 significant digits", that is unambiguous and not subject to interpretation, is it not?

Yes this is true, and in the UK education system at least this is generally the only way in which SF is used. However this is very different from the notion of a "significant figure quiz" where you are presented with ridiculous questions like "what is 3,000 + 1.234" and are supposed to guess or memorize arbitrary rules for inferring what is required.