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. A

    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. F

    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. C

    Significant figures -- wrong answer to Thermodynamics question?

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  4. Argonaut

    Confused about significant figures in official solutions

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  5. T

    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. Skalvig

    Rounding to significant figures

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  7. V

    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. A

    Tracking significant figures through a complicated problem

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  9. I

    Significant Figures of 225.0 & Molar Volume at STP

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  10. I

    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. J

    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. yucheng

    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. LCSphysicist

    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. maxelcat

    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. I

    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. Dada

    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. Wrichik Basu

    Comp Sci Calculating something correct upto ##m## significant figures

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  18. P

    Lost significant figures through division by exact numbers

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  19. F

    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. E

    Uncertainty of measurements & significant figures

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  21. J

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  22. W

    I Tipler/Mosca significant figures

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  23. J

    Calculate Avg of 2 Values with Significant Figures: 1.56324 N & 1.13405 N

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  24. jybe

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  25. W

    Significant figures in practical investigation report

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  26. Zahid Iftikhar

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  27. S

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  28. S

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  29. F

    I Significant figures and uncertainty in measurements

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  30. Cardinalmont

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  31. I

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  32. Muhammad Danish

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  33. B

    I Uncertainties and significant figures

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  34. L

    Quick Significant Figures Problem

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  35. alphaj

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  36. Anshul23

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  37. Alexander350

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  38. P

    Significant Figures with Kinetic Energy Formula?

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  39. F

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  40. S

    Solve Significant Figures Homework

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  41. F

    Confusion regarding significant figures in answers

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  42. R

    B Significant Figures: Why Leading Zeros Don't Count

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  43. M

    What is the uncertainty in 5.00mm measured by analog device?

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  44. E

    Calculating Standard Error of Mean with Significant Figures

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  45. C

    Adding and Subtracting Significant Figures

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  46. Mikaelochi

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  47. G

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  48. V

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  49. Elena14

    I Number of significant figures in 5*5.364

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  50. J

    Significant Figures Homework: Converting Time from Min to Hrs

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