View Full Version : Significant Figures ?

Dejhan_Tulip

02-27-2004, 04:46 PM

Hi everyone !

Well, I have this big issue in my head and I think for the most part I am like overthinking the problem...

My actual problem is the understanding of significant figures IN calculations.

For example...

The correct results WITH the correct sig. figs. for: (angles are in degrees)

1314.459 + 3.01

23.15 * 5.2

29.87777 / 2.0

31.7 * sin(76)

pi * 2.3

( 2.765 + 4.1 ) / ( squrt(3*sin(55)))

I think you get the idea... I get confused whenever I start to mix up angles and many digits from different COMBINATIONS of arithmetic operations.

I really hope someone could clarify this for me b/c its killing me and it's taking a LOT of points from my exams :(

Thanks a lot in advance.

DEJHAN

Yan Wang

02-27-2004, 05:06 PM

You should round your results to the below expressions to the stated sig figs:

1314.459 + 3.01: 3 sig

23.15 * 5.2: 2 sig

29.87777 / 2.0: 2 sig

31.7 * sin(76): 2 sig

pi * 2.3: 2 sig

( 2.765 + 4.1 ) / ( squrt(3*sin(55))): 2 sig

If a number is like 0.0001, that's 5 sig fig, but 0001 is 1 sig fig. If it's 2.32340 that's 6 - the 0 at the end counts if it's there. If it's 3.0, it's two, but 3. is also two - the decimal signifies that (though normally no one writes it like that). A number such as a constant like the speed of light is considered to have infinite sig figs. You should do all problems without rounding. Then at the end, round to the # of sig figs of the value in the problem with the least # of sig figs.

ngreen

02-27-2004, 05:14 PM

I think for number 4 it would be 3 sig figs since the sin(76) = .970295.....

Dejhan_Tulip

02-27-2004, 05:16 PM

Thank you but I still have something unclear; since the first calculation would yield a result of: 1317.469

How am I supposed to report that ? As 1317.47 ?

Could you please type ALL the results for the calculations I posted so I can see that and also that will help me to figure out things a bit quicker.

Thanks a lot !

ngreen

02-27-2004, 05:24 PM

Yeah the addition one are wrong.

Answers:

1. 1317.47 , 6 sig figs

2. 120 , 2 sig figs

3. 1.4 , 2 sig figs

4. 30.8 , 3 sig figs

5. 7.2, 2 sig figs

6. 4.9, 2 sig figs

For addition the sig figs are determined by the least amount of decimal places.

For multiplication they are determined by term with lowest amount of sig figs.

Yan Wang

02-27-2004, 05:26 PM

I think for number 4 it would be 3 sig figs since the sin(76) = .970295.....

No. If a problem gives you 76 as the measure of an angle, then you must round your result to the # of sig figs of that given angle. sin(76) would need to be rounded to 2 sig figs if that's the only calculation - 0.57

Heh, I looked at #1 too fast and thought it was multiplication: "1314.459 + 3.01: 3 sig"

For addition and subtraction, you take the least # of digits after the decimal and make sure your answer has that # of sig figs. 1314.459 + 3.01 would equal 1317.47 (rounding to 2 sig figs after decimal).

If you had to add 0.000000001 to 1.1, you'd get 1.1. Because the limiting sig fig is from 1.1.

blindguyinanorg

02-27-2004, 05:32 PM

1314.459 + 3.01=1317.47

23.15 * 5.2 = 120 (2 sig digs)

29.87777 / 2.0 = 15

31.7 * sin(76)=30.3

pi * 2.3=7.2

( 2.765 + 4.1 ) / ( squrt(3*sin(55)))=3----1 sig dig(the 3 is only one sig dig)

for multiplication and division it is the total number of digits in the smallest number

for addition it is the highest place value (1200 would be 2 where as 1200.1 would be 5)

you do all arithmatic first then either round or truncate to correct digits

i am absolutly sure about this, i was the only one in my class (gifted) who could even understand why this is so.

Dejhan_Tulip

02-27-2004, 05:48 PM

1314.459 + 3.01=1317.47

23.15 * 5.2 = 120 (2 sig digs)

29.87777 / 2.0 = 15

31.7 * sin(76)=30.3

pi * 2.3=7.2

( 2.765 + 4.1 ) / ( squrt(3*sin(55)))=3----1 sig dig(the 3 is only one sig dig)

for multiplication and division it is the total number of digits in the smallest number

for addition it is the highest place value (1200 would be 2 where as 1200.1 would be 5)

you do all arithmatic first then either round or truncate to correct digits

i am absolutly sure about this, i was the only one in my class (gifted) who could even understand why this is so.

You actually meant 4 in your last calculation right ? (1 digit)

And what would be the result for 23.15 * 5.9 ? Would 137 be correct ? (3 sig figs? -- but look that 5.9 has only 2 sig figs.)

Yan Wang

02-27-2004, 06:13 PM

So it's 140

nerdcool64

02-27-2004, 07:36 PM

"23.15 * 5.9 = 140 in sig figs"

That is why I hate sig figs. They are only cause for error in your answer. I have yet to be told by my school's teaching staff what purpose they have or how they are beneficial to our society.

sanddrag

02-27-2004, 07:52 PM

If anyone wants a super detailed very correct way to understand and count significant figures I will copy it out of my Chemistry book for you. Just let me know.

ngreen

02-27-2004, 08:20 PM

"23.15 * 5.9 = 140 in sig figs"

That is why I hate sig figs. They are only cause for error in your answer. I have yet to be told by my school's teaching staff what purpose they have or how they are beneficial to our society.

It shows how accurate you measure to. When ever making a measurement you are suppose to only make the measurements you can. You don't measure to micrograms. You measure to milligrams at best. But it doesn't mean that when you measure to 1 milligram you measure to 1.000000000000000 or exactly one to the infinite significant digits. It is impossible. You may measure to 1.0 and leave off .04756345694351 of it. 1.05 of it is your sig fig. The last digit is understood to be uncertain. If When you make multiplications and additions you get decimals places you shouldn't have or are not accurate to what measurements you took. Sig figs get rid of these innacurracies. It give you the answer for the accurateness of measurement you took.

Oh yeah, and the 3 is understood to be 3.0000000 (infinite zeroes) so there is 2 sig figs on the last one.

Dejhan_Tulip

02-27-2004, 09:41 PM

So, if I do the following:

29.87777 / 2.0 = 15 (2 Sig. fig.)

and

29.87777 / 2 = 10 ( 1 Sig. fig. )

Is that correct ?

ngreen

02-27-2004, 09:58 PM

No, two would be understood as 2.00000000 to infinity so the answer would be 14.93889 to seven sig figs. This at least what my chem book says.

Dejhan_Tulip

02-27-2004, 10:41 PM

Is that true ?? So, not 1 sig. fig. but 7 !?!? (for the division by 2)

And also, for ( 2.765 + 4.1 ) / ( squrt(3*sin(55))) what is the correct result to the correct number of sig. figs. ??

Thanks a lot for the help so far guys, I am really putting some concepts together in my head :D

Dejhan

blindguyinanorg

02-28-2004, 10:53 AM

So, if I do the following:

29.87777 / 2.0 = 15 (2 Sig. fig.)

and

29.87777 / 2 = 10 ( 1 Sig. fig. )

Is that correct ?

there is acually 2 possabilities for 29.87777 / 2. if 2 is a measurment then the answer is 10, if you mean half of the value then its 2.0000000000 like ngreen said. same with the sqrt question i previosly answered as 1 digit. ( 2.765 + 4.1 ) / ( squrt(3*sin(55))) would have 2 sig digs if 3 was just a constant.

deltacoder1020

02-28-2004, 01:24 PM

sig figs. are only really meant to be applied to calculations including a measurement, for the reason that has already been stated here.

Thus, if you just want half of the number, don't count the sig. figs of the 2 you divide it by, because you already know that you want exactly half. On the other hand, if you saw a measurement that was "2", that would mean that someone had put something on a scale that did not provide a readout of tenths, and thus the reading is only accurate to the ones place. Thus, you would have to use only one significant figure in your result, because for all you know, the actual measured quantity could have been 2.4999 or something such as that.

For instance, say were are measuring the volume of a cube, and then calculating the density. We know that the mass is 1.00 kg (i.e. we know it to be within one hundredth of a kilogram of exactly one kilogram). Say we measured the volume with something that only give us a reading to the nearest mL, and we got 100mL for the volume. Now, let's say the actual volume was 100.4mL, but we couldn't measure that last .4 with out equipment.

so the actual density is:

1.00 kg / 100.4 mL = 0.00996... kg/cm^3

with sig. figs. for all measured quantities:

1.00 kg / 100 mL = 0.01 kg/cm^3 (our predicted density, to the nearest hundredth)

without sig. figs for our measurement of volume:

1.00 kg / 100 mL = 0.0100 kg/cm^3 (those two extra zeros would imply that our answer is accurate to the nearest ten-thousandth)

essentially, significant figures simply express how sure one can be of the calculated result. If a measured quantity can vary +- 1, then multiplying it by 42 would result in the answer varying by +- 42. That's why some common calculations with sig. figs. (such as 2*3 = 10) seem strange, but when you think about it, you can only really be sure about the tens digit, because of the variance in the measured values.

Dejhan_Tulip

02-28-2004, 03:14 PM

1. Ok, so as a rule I *have* to keep ALL decimals in all my calculations until the end where I round off and put the correct number of sig. figs right ?

2. These would be the results that I get...

1314.459 + 3.01 = 1317.469 1317.47 (6 SF)

23.15 * 5.2 = 120.38 120 (2 SF)

29.87777 / 2.0 = 14.938885 15 (2 SF)

29.87777 / 2 = 14.938885 10 (1 SF)

31.7 * sin(76) = 30.75837… 30 (2 SF)

1001 * sin(11) = 190.9998 190 (2 SF)

1001 * sin(9) = 156.5909 200 (1 SF)

pi * 2.3 = 7.2256631… 7.2 (2 SF)

( 2.765 + 4.1 ) / ( squrt(3*sin(55))) =

log(2.583) = 0.4121244061… 0.4121 (4 SF)

3. The reason why I left the squrt blank was b/c I am working a problem in a book, and that 3 was gotten from the given measurements in the book. I think that is NOT taken as a constant so the answer should be to 1 sig. fig.

Three points are toched in this little post, please try to comment in the 3 of them. Thank you :)

Dejhan

deltacoder1020

02-28-2004, 05:31 PM

consider addition and subtraction as one "kind" of significant figures, and everything else as another. you should apply the significant figures rounding whenever you switch "kinds". So for instance:

((12.0 + 3.45) + 0.16) * 1.243 + 1.01

evaluates as follows. first, we do the addition in the inner parentheses:

12.0 + 3.45 = 15.45

then we do the second addition:

15.45 + 0.16 = 15.61

now, we are done with one "kind" of operations (the multiplication is next), so we apply sig figs:

15.66 --> 15.6

next, we do the multiplication:

15.6 * 1.243 = 19.3908

we're switching back to addition, so we apply sig figs again:

19.3908 --> 19.4

finally, we do the addition, and apply sig figs to the result:

19.4 + 1.01 = 29.41 --> 29.4

Denman

03-17-2004, 04:46 AM

Hi everyone !

Well, I have this big issue in my head and I think for the most part I am like overthinking the problem...

My actual problem is the understanding of significant figures IN calculations.

in calculations, you should never round or use sf.

eg if you have sin 35/cos 35

you should do that on your calculator as

(sin(35))/(cos(35)) as opposed to

sin 35= 0.57357643635104609610803191282616 OR 0.574 TO 3 SF

cos 35=0.81915204428899178968448838591684 or 0.819 to 3sf

if you devide these you get 0.7008547008547008547008547008547 or 0.701 to 3 sf

whereas if you typed it in normally you would get 0.7002075382 or 0.700 to 3 sf

ok i know thats a bad example but it does show it sort of....

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