Significant figures Definition and 19 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. Argonaut

    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.)...
  2. yucheng

    B Significant figures for special functions (square roots)

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  3. 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!
  4. J

    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 Equations The Attempt at...
  5. jybe

    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...
  6. Cardinalmont

    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...
  7. Muhammad Danish

    Significant figures

    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 ?
  8. alphaj

    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...
  9. Anshul23

    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...
  10. 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...
  11. M

    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...
  12. E

    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...
  13. G

    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...
  14. L

    Significant Figures Question

    Homework Statement Solve for g in your area with the correct number of significant figures. Theta = latitude = 50.7 degrees h = height = 518. m Homework Equations g = 9.80616 - 0.025928 cos2(theta) + .00068 cos^2(2*theta) - 0.000003h The Attempt at a Solution Well, I just plugged in the data...
  15. I

    Need clarification on sig-figs and propagation of error.

    Hello, I have a question asking me to find the volume of a rectangular prism. The dimensions are as follows: x = 20 ± 0.2 cm, y = 30 ± 0.2 cm, z = 70 ± 0.4 cm I am asked to report the answer with the correct number of significant figures and include the error. What I have so far: V = xyz =...
  16. T

    A question about significant figures and rounding

    Homework Statement What is the answer to (72.4meters)*(cos58)? How many significant figures does it have, and why? Homework Equations "The least number of significant figures in any number of the problem determines the number of significant figures in the answer." cos58=0.529919264233... The...
  17. G

    Significant figures vs precision (I think)

    Homework Statement I've got a doppler shift question about calculating the frequency heard by a receiver moving 5.0m/s towards a sound source that is emitting 3000Hz sound. I assumed speed of sound is 343 m/s for an air temperature of 20C. Homework Equations Freceiver = Fsender((V +...
  18. learn.steadfast

    Logarithmic significant figures

    In most areas of science, being able to solve and report problems to the correct number of significant figures is necessary; but I'm having trouble finding a complete set of rules for significant figures when working with non-ideal power equations. eg: I'm talking about equations requiring the...
  19. J

    Significant Figures in Conversion Factors

    So from what I had initially understood, in a conversion problem, you go by whatever number has the smallest SF and round your answer to that number, even if that number is one. In the examples on my homework it seems like my teacher chooses a random number of significant figures for each...