# What is Significant figures: Definition and 153 Discussions

Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and absolutely necessary to indicate the quantity of something. If a number expressing the result of measurement of something (e.g., length, pressure, volume, or mass) has more digits than the digits allowed by the measurement resolution, only the digits allowed by the measurement resolution are reliable so only these can be significant figures. For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, and these show 114 mm) are only reliable so can be significant figures. Among these digits, there is uncertainty in the last digit (8, to add 0.8 mm) but it is also considered as a significant figure since digits that are uncertain but reliable are considered significant figures. Another example is a volume measurement of 2.98 L with the uncertainty of ± 0.05 L. The actual volume is somewhere between 2.93 L and 3.03 L. Even if all three digits are not certain (e.g., the actual volume can be 2.94 L but also can be 3.02 L.) but reliable as these indicate to the actual volume with the acceptable uncertainty. So, these are significant figures.The following digits are not significant figures.
All leading zeros. For example, 013 kg has two significant figures, 1 and 3, and the leading zero is not significant since it is not necessary to indicate the mass; 013 kg = 13 kg so 0 is not necessary. 0.056 m has two insignificant leading zeros since 0.056 m = 56 mm so the leading zeros are not absolutely necessary to indicate the length.
Trailing zeros when they are merely placeholders. For example, the trailing zeros in 1500 m as a length measurement are not significant if they are just placeholders for ones and tens places as the measurement resolution is 100 m. In this case, 1500 m means the length to measure is close to 1500 m rather than saying that the length is exactly 1500 m.
Spurious digits, introduced by calculations resulting in a number with a greater precision than the precision of the used data in the calculations, or in a measurement reported to a greater precision than the measurement resolution.Of the significant figures in a number, the most significant is the digit with the highest exponent value (simply the left-most significant figure), and the least significant is the digit with the lowest exponent value (simply the right-most significant figure). For example, in the number "123", the "1" is the most significant figure as it counts hundreds (102), and "3" is the least significant figure as it counts ones (100).
Significance arithmetic is a set of approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty.
Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34525 kg if the scale was only measured to the nearest gram. In this case, the significant figures are the first 5 digits from the left-most digit (1, 2, 3, 4, and 5), and the number needs to be rounded to the significant figures so that it will be 12.345 kg as the reliable value. Numbers can also be rounded merely for simplicity rather than to indicate a precision of measurement, for example, in order to make the numbers faster to pronounce in news broadcasts.
Radix 10 is assumed in the following.

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1. ### Uncertainty Error and Significant Figures in this Force Tables Lab

TL;DR Summary: I have a Force Table Lab and am not sure if the "percent error" is the correct way to express errors. I also need some help on sig figs for my results. Hello, I am working with the usual undergrad force tables apparatus. I would like to know how to express the results of...
2. ### Consideration of number of significant figures in experimental quantities

I am not sure how to approach this problem. I know that there really is no use taking time values accurate up to the sixth decimal place if my length values are accurate only to the first decimal place, after all errors should be comparable. So I wanted to know how I should quote my time values...
3. ### Significant figures -- wrong answer to Thermodynamics question?

For this problem, The solution is, ## Q = 54500 J ## , however, since the calculation for heat uses the first law which involves ##5 \times 10^{-3) m^3 ## value for the initial volume, should the finial answer not be to 1 sig fig as well? This would give ## 60000 J ##. Many thanks!
4. ### Confused about significant figures in official solutions

I keep running into the issue of getting a different number of significant figures in my solutions than the official answers at the back of the book, so I'm wondering if I'm missing something or if the book is occasionally sloppy about significant figures. In Unit 1, the book states (p.8.)...
5. ### Significant Figures Verification

Good morning, I've completed the problem provided above and have verified my answers are correct, but I'm running into a strange situation when it comes to the solution's answer for the kinetic energy portion. For the kinetic energy imparted to the planet, we're taking 20,000 kg * 9.8 m/s² *...
6. ### Rounding to significant figures

This is the solution from the book. But I only get 0,037 A. What am I doing wrong?
7. ### Finding number of significant figures

I am using these rules to determine number of significant digits. (1) All trailing zeros following decimal are significant (2) All trailing zeros in a number without decimal point are not significant (3) All non-zero digits are significant It seems rule 2 will not apply here and therefore number...
8. ### Tracking significant figures through a complicated problem

In general I think I follow significant figures. On one and two step problems I don't really have such a problem but I seem to constantly miss points because I get lost in complicated ones. I'm hoping I can understand more as getting into science means getting friendly with these. Let's suppose...
9. ### Significant Figures of 225.0 & Molar Volume at STP

What are the number of significant figures of 225.0? I think it is 4, but the solution says it's 3. Also, is the significant figures of molar volume at STP (22.7 L/mol) considered? Thanks.
10. ### Chemistry Uncertainty and Significant Figures

Hi, Would it be reasonable to have uncertainty in your independent variable (e.g. uncertainty of mass measured by electronic balance)? For processing raw data: does each processed answer just have the same number of sigfigs as each raw datum it is calculated from, or do I need to keep track of...
11. ### B Implicit error margins based on significant figures

WARNING: Topic is very pedantic. I have used a set of different physics books over the years, and they have all had a focus on the topic of significant figures, error margins and measurement. I have never quite understood these concepts fully and the relationships between them. One aspect I...
12. ### B Significant figures for special functions (square roots)

I am using square roots, however, I am confused over how many significant figures (s.f.) to keep. Suppose I have ##\sqrt{3.0}##, which has 2 s.f. From three different sources, I'll put a summary in brackets: https://www.kpu.ca/sites/default/files/downloads/signfig.pdf (if 2 s.f. in the data...
13. ### How to Write #s w/ Uncertainty: Rules & Examples

I am having a little trouble to write in the right way a number: I did a calc in which the uncertainty was 150, while the value with physics meaning was about to 1427. I am trying to figure out what is the right way to write it. Now, since the uncertainty can not have more than two significant...
14. ### A bit confused about significant figures

Hi. I understand that the number 20, written like this, has 1 significant figure Suppose I took a pile of measures of length: 18, 19, 20, 21, 22, 23, 24, 25 giving a mean of 172/8 = 21.5 on my calculator If I have to give the mean to the lowest number of significant figures then that would...
15. ### Significant Figures with Experimental Values

I was thinking of choosing 0.740, because it looks the most consistent with the other numbers because they all have a trailing zero. But then, in accordance with sig figs, 0.74 is the right answer. Which one should I choose? Thank you so much!
16. ### I How to Convert a Binary Number to 6 Significant figures

1203.201 which is 0100|1011|0011.0011|0011 How to round the binary representation to 6 significant figures and 12 significant figures? And what is the rounding rule for base 2? If it was rounded to 6 significant figure, such as 0100|10 {2}, then it changes its initial value, doesn't it? So...
17. ### Comp Sci Calculating something correct upto ##m## significant figures

I have solved questions where I have been asked to find something correct to ##m## places of decimal using some series. See this thread. The logic was, the program would terminate at the ##k##th term if the ##k+1##th term is ##<10^{-m}##. But how do I terminate the series here? Say I calculate...
18. ### Lost significant figures through division by exact numbers

I came across a significant figures problem today that I need information on. The problem is this: "What volume of water can a cylindrical container hold of it is 13.0 cm tall and 12.0 cm in diameter? Show your work and express the answer in scientific notation using significant figures." Of...
19. ### I What is the proof of the rules of significant figures?

Please prove the rules of significant figures. I do not know why when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them.
20. ### Uncertainty of measurements & significant figures

Homework Statement This conversion factor 1m = 39.3701 inches was used, when measuring a tape that had inches on it to a full meter. The measuring tape increased by 0.0625 inch increments, so the 1 meter (39.3701) was estimated to be in between the 39.3125 and 39.3750 on the actual measuring...
21. ### Uncertainty - Reynolds number and the friction factor

Homework Statement Correctly present the table of information. The values in the table are deliberately in a wrong format. The calculated Re values have been analysed to have an uncertainty of ± 0.4% and the calculated f values an uncertainty of ± 0.1%. Homework EquationsThe Attempt at a...
22. ### I Tipler/Mosca significant figures

In tipler and mosca it says that the number of significant figures in the result of addition or subtraction is no greater than the least number of significant figures beyond the decimal place of any of the numbers. They give the example of 1.040+0.21342. Clearly, 1.040 has three significant...
23. ### Calculate Avg of 2 Values with Significant Figures: 1.56324 N & 1.13405 N

Homework Statement We measured 2 values and were asked to calculate their average using sig fig The two values were: 1.56324 N and 1.13405 N. ( 1.56324 N + 1.13405 N ) / 2 = 1.348645 N Would the answer be 1.34865 N or 1 N? (Someone told me that we only use the sig figs of measured values...
24. ### SI Units + sig figures

Homework Statement If I am asked to give my answer in acceptable SI units and to 3 significant figures, how would I express my answer? Homework Equations Answer: 589883.4263 J The Attempt at a Solution My instinct would be to put this in KJ, but I don't know if that's an "acceptable SI...
25. ### Significant figures in practical investigation report

Homework Statement I am writing a prac report regarding Hooke's Law, and am uncertain how many significant figures I should write my results in in my results table. I have heard that in my prac report, I must keep my significant figures consistent, that is, all data measured by the same device...
26. ### Adjustment of Significant Figures

Homework Statement The diameter and length of solid cylinder measured with a vernier calipers of least count 0.01 cm are 1.22 cm and 5.35 cm respectively. Calculate the volume of the cylinder and the uncertainty involved within it. Homework Equations V= 1/4 πd2 l The Attempt at a Solution...
27. ### Significant figures problem

Homework Statement A regulation soccer field for international play is a rectangle with a length between 100 m and 110 m and a width between 64 m and 75 m. When calculating the areas what is the appropriate number of significant figures? Homework Equations multiplying/dividing: The least...
28. ### Significant figures and rounding

Homework Statement This isn't a specific HW problem, its just something I keep tripping over for some reason so I guess it qualifies as HW. OK so, for example the number 1.5 has 2 sig fig . But if you compare that to another number with 3 sigfigs, like 1.58, then arent they actually equal...
29. ### I Significant figures and uncertainty in measurements

Hello, I was recently pondering on significant figures and uncertainty reminding myself that there is no perfect measurement: every measurement involves an error caused by the instrument and/or the operator. A measurement should be executed as many times as possible and not just once. The...
30. ### B Greatest Possible Uncertainty and Sig Figs

There is something I seriously don't understand about uncertainty. Suppose there is an electric balance that reads 5.67g The limit of reading is 0.01g The greatest possible error is half of the limit of reading and is thus 0.005g By this logic, and assuming the very best possible situation, I...
31. ### Rounding off to the correct number significant figures

1. The problem statement, all variables, and given/known data If we add 3.46 and 2.3 we get 5.76. we can round it to two significant numbers 5.8 because there is the least two significant digit 2.3 but If we add 2.3 X 10^4 + 1.6 X 10^3 = 2.3 X 10^4 + .16 X 10^4 = 2.46 X 10^4. Here can we round...
32. ### Significant Figures: Counting on 5.400 and the Role of Zeroes

Homework Statement Number of significant figures on 5.400 are? Homework Equations Does the 0s after 4 count as significant figures? The Attempt at a Solution 4 ?
33. B

### I Uncertainties and significant figures

If I have a result that can only be quoted to 2s.f., let's say 20, and I calculate an uncertainty of 0.2, then how do I quote this?
34. L

### Quick Significant Figures Problem

Homework Statement )(9.66x10^-1) + (5.1x10^2) -8.77+2.8 ) / ((8.333x10^-2)(3.001)) Homework Equations NoneThe Attempt at a Solution my answer does not match the answer key which is 2.0 x 10^3.[/B] When i did the problem , i ended up get 3 SF answer. The numerator simplies to 504.996 where...
35. ### Significant figures in chemistry?

Homework Statement I'm having difficulties figuring out how many significant figures to report for several caluclations in my lab report. As it is a report, there are no specific problem statements. Homework Equations No equations, but there is a rule I was told: when dividing/multiplying I...
36. ### How many significant figures in the average area?

Homework Statement I've been given a list of lengths of the side of a small metal sheet in the shape of a square. The length of the side of this square sheet varies with temperature. I have to calculate areas for each length and then calculate the average area. What should be the uncertainity...
37. ### Significant figures in a results table

Homework Statement Basically there is a results table for the time taken for 20 oscillations. Three examples are 9.90, 11.16 and 12.68. I need to work out the time period to the correct number of significant figures. Homework EquationsThe Attempt at a Solution I divide by 20 to get the time...
38. P

### Significant Figures with Kinetic Energy Formula?

Homework Statement An object of mass m = 2.3±0.1 kg is moving at a speed of v = 1.25±0.03 m/s. Calculate the kinetic energy (K = 1 /2mv2 ) of the object. What is the uncertainty in K? I am not exactly sure if I used the error equation correctly when I start using Δ(v2). Could someone verify my...
39. ### Number of significant figures

Homework Statement I don't understand this question . The author want the at least 2 significant correct . why 0.5x10^(2-m) is used ? It's not clear that the author want the answer to be less than 5 ?50? 0.5 ? 0.05 ?or 0.005? Can someone explain it ? Homework EquationsThe Attempt at a...
40. ### Solve Significant Figures Homework

Homework Statement I am given some results of certain experiments with errors and I need to rewrite them correctly in term of significant digits. Homework Equations The professor explained to us that, for example, 1.12345 ± 0.5231 is not correct (or at least not the right way) because you...
41. ### Confusion regarding significant figures in answers

Homework Statement I'm working through some exercises in which I'm supposed to convert various units of speed from SI units to other units Example: 0.4m/s into km/h Homework Equations The Attempt at a Solution Having checked, I know my answer to be (to 3 significant figures) correct. My...
42. ### B Significant Figures: Why Leading Zeros Don't Count

I measure some quantity with an instrument that it is precise to two decimal points. So maybe I get 8.84 V. Then I do some changes in my parameters and get 0.01 V. The two measured values are precise to two decimal points. But the first one has three significant figures, while the second one...
43. ### What is the uncertainty in 5.00mm measured by analog device?

Homework Statement What is the uncertainty in 5.00mm measured by an analog device? Homework Equations No equations, Uncertainty and error analysis question The Attempt at a Solution I know that the uncertainty is half the smallest division but i can't figure out what is the uncertainty and...
44. ### Calculating Standard Error of Mean with Significant Figures

Homework Statement In a physics lab, Logger Pro software generated statistical estimators such as the standard deviation σ = 0.04021 of a sample of size n = 29. Among other things, I must calculate the standard error of the mean σmean. My question is: Must σmean have four sig figs or two...
45. ### Adding and Subtracting Significant Figures

Homework Statement Hi, To add or subtract sig figs, it is to my understanding that the answer is reported so that it reflects the least precise number. For the example 43.4 - 22 - 3, would I report my answer as 18 or 20? Thanks. Any help is very much appreciated! Homework EquationsThe...
46. ### Other Do theoretical physicists use significant figures?

I know the phrase ''theoretical physicist'' may be a general term, do they or ''mathematical physicists'' use significant figures? I know it's kind of a silly question, but it stems from what I learned in chemistry class. Our chemistry teacher talked about the importance of such things and I...
47. ### Significant Figures & Error Analysis

Hi, I've just had my first lab in physics and I'm having a bit of trouble understanding how to determine the significant figures of my final answers and transforming them in scientific notation. For example: Homework Statement [/B] I had to measure three sides of a parallelepiped with a...
48. ### Error and significant figures

Homework Statement We measure the period of oscillation of a simple pendulum .In successive measurements ,the readings turn out to be 2.63 s , 2.56s , 2.42s , 2.71s and 2.80s . What is the period of oscillation taking into account appropriate significant figures ? Homework EquationsThe...
49. ### I Number of significant figures in 5*5.364

How many significant figures should be present in the answer of 5*5.364? 1) Addition rule: The result cannot have more digits to the right of the decimal point than either of the original numbers. 2) Multiplication rule: The result must be reported with no more significant figures as there are...
50. ### Significant Figures Homework: Converting Time from Min to Hrs

Homework Statement I am starting to confuse myself with the proper use of SF. I am to convert time from minutes to hours, keeping in mind proper SF Homework Equations conversion factor: 1 min = 1/60 hr The Attempt at a Solution The timing error is +/- 0.2 (1SF) = 0.003hr (1SF) 20.0...