# What is Significant digits: Definition and 50 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. ### How to correctly round in physics?

I have never been sure how to round it according to the rules. My steps: ##\sqrt{5 \cdot 9.81 \cdot 0.10-\frac{1}{4\cdot \pi \cdot 8.85 \cdot 10^{-12}}\frac{(1.0\cdot 10^{-6})^2}{0.150\cdot 0.10}}## In [ ] is number of significant digits using multiplication/adding etc. rules. ##\sqrt{5 \cdot...
2. ### B How to handle significant digits of exact values?

I learned the basics of sigdigs in school, but that era has more in common with the pyramid of Giza than modern day. I was figuring the angle of a 3d-designed part in my head the other night. It's a cylindrical polyhedron with 32 faces (because 32 seemed like a good number). I didn't expect to...
3. ### 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...
4. ### Significant digits rule when determining radius from diameter

Probably, to satisfy the significant digits rule for division, we should consider ##r = 5.0 \div 2.0##. But I'm unable to come up with a reason why significant digits rule should not apply to ##r= d \div 2##. Also, if we apply significant digits rule to this calculation then we loose accuracy...
5. ### Number Precision When Adding/Subtracting

If you read any introductory physics or chemistry course, in the very first chapter, you'll come across measurement, uncertainty, etc. There, you learn some rules about rounding off numbers, etc. One of the rules discussed there is that if you add/subtract two numbers with different decimal...
6. ### Writing a result from a table and how many digits should the answer have?

Homework Statement [/B] A 0.02 meters long conductor is placed perpendicular in a homogen magnetic field. Current through the conductor is changed. And we have measured the data here: How can we write the magnetic flux density the correct way? How do I know how many digits my answer should...
7. ### MHB Round To Two Significant Digits

What does it mean to round to TWO SIGNIFICANT DIGITS? SAMPLE: Round 542,867.21054031 to two significant digits. Does this means to round to two decimal places or does it mean to round to the tens place?
8. ### 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...
9. A

### Significant Digits for Uncertainty Values

Homework Statement Question on worksheet: What is the random uncertainty in this set of measurements? Use the instrumental uncertainty to limit the number of digits in your random uncertainty result. Data for dropping a pencil from a height of one meter using a stopwatch that measures to the...
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. ### 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...
12. ### B Why we take the least # of significant digits when multiplying?

This question just flew right into my head when I was in the mid of a shower :), and I cannot resist finding out the answer. Question: Why do we take the least number of significant digits when multiplying/ dividing, and the least number of decimal places when adding/subtracting? Thank you!
13. ### 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...
14. ### Error propagation and significant digits

Moderator's note: Thread moved to homework section. Thus no template. I have an exercise in which I have to calculate the Area from the following measurements: L = 22.1 ± 0.1 cm W = 7.3 ± 0.1 cm Of course, A = W * L = 161.33 but since I have a measurement with just 2 significant digits the...
15. ### Inconsistency in Significant Digits

A question posed as an example: converting 1.55 and 0.55 inches to cm. 1.55 * 2.5400000000... = 3.937 which rounds to 3.94 cm per the rules and 0.55 * 2.5400000000... = 1.397 which rounds to 1.4 cm per the rulesIf both inputs are known to the same precision (hundredth of an inch), why are the...
16. ### Condition Number and Significant Digits

Homework Statement IF A is nxn matrix, and k(A)=||A||||A^-1|| then a large k(A) means the solution has more significant digits than is k(A) were small true or false? Homework Equations The Attempt at a Solution Well k(A) is defined as 10^(k) right so the larger the condition...
17. ### Uncertainty significant digits

When dealing with the intermediate value of example: 570 +/- 100, how do you put this into it's final form? If I'm not mistaken, doesn't the uncertainty only contain 1 significant digit? Doesn't the uncertainty digit have to correspond to the final significant digit on the actual value as well...
18. ### What are the rules for adding numbers with uncertainties?

A measured value of 35m is being multiplied by the countable number 3 (say you went 3 times around a track that was 35m long). What's the correct number of sigfigs in the answer? Method 1: 35 m x 3 = 105m = 110m By significant digits rules of multiplication, this would be 110m. (The 35m...
19. ### Why Do Significant Digits Include Uncertain Numbers?

Homework Statement How many significant digits does the number 3568 +/- 42 have? Homework Equations The Attempt at a Solution The true value is ranging between 3526 and 3610. Why is there two significant digits and not one? I can't even find a specific definition of significant...
20. ### Significant Digits Ruler -To what decimal point

Significant Digits Ruler -To what decimal point To what decimal point do I write the number 30 if the uncertainty is 0.0625? The ruler has 1/8 marks. Thank you :)
21. ### Significant digits including only one zero

Homework Statement The problem asks us to find deceleration from 60km/h to 0km/h in 1,2 seconds.Homework Equations The speed formulasThe Attempt at a Solution I know how to apply the formula. However, in the problem, it is written 0km/h. How many significant digits is that? How many significant...
22. ### Do All Numbers Have Unlimited Significant Digits?

Homework Statement How do I know if a number has unlimited significant digits or not? do constants and molar masses and densities all have an unlimited number? does this mean i don't count it when i do my math, etc? Homework Equations The Attempt at a Solution
23. ### Significant digits question

Homework Statement 1) how do I know if the significant digit is supposed to be of " infinite sig-digs" ? If it is a constant value that is universally known, ike density then i understand that. But let's say i were to do ... 35+273.15 Homework Equations addition? The Attempt at a Solution...
24. ### Converting to SI units w/ significant digits & scientific notation

Hi. I'm asked to convert 2 year into the SI Unit seconds: Known: 1 Hr = 3600 s 1 Day = 86400 s So using a calculator... 1 yr = 365*86,400 = 31,536,000 I want to express the answer in scientific notation: Because the lesser number on has 3 significant digits, the scientific...
25. ### A question about significant digits of digits in parentheses at the end

I was looking at the value of Avogadro's number 6. 022 141 29(27)×1023 and was trying to figure out what exactly is meant by the parentheses at the end. I'll take a guess and say that the best determined value has these digits, but that the error is high enough so that only by using the...
26. ### What is the number of significant digits in a integer with trailing 0's ?

It seems that there is a possible ambiguity in the number of significant digits if there are trailing 0's, but no decimal point or overline or underline. Specifically, I am looking at the definition of the second, which the Halliday & Resnick book has listed as 9 192 631 770 So is this 9...
27. ### Different significant digits in uncertainty and data?

Hi, I've just got a question about uncertainties that came up while doing a lab report. I have a set of data for currents required to cancel the deflection of an electron beam caused by the Earth's magnetic field: 0.195 A 0.220 A 0.250 A 0.251 A 0.239 A Because of the difficulty in...
28. ### Question on estimated uncertainty and significant digits

Question on "estimated uncertainty" and significant digits Homework Statement The value is 136.52480, and the "estimated uncertainty" is 2. How many digits should be included as significant? Homework Equations Not sure whether estimated uncertainty refers to a 1σ (68.26%) confidence...
29. ### Significant digits and easy average speed/ velocity questions

Homework Statement Here is a few examples where i am wondering as to how many significant digits to leave 3.5L/100km * 2279km = 79.765 ---> In this case am i supposed to round to 2 significant digits because of 3.5 which has two? 1.2L/100km * 63km = 0.756, i leave it as 0.76 right...
30. ### Sig Figs: Displacement of Car from Origin - 2.8x10^2 km?

How many sig figs would be in the final answer of the following question. A car is driven 215 km west and then 85km southwest. What is the displacement of the car from the point of origin (magnitude and direction)? Now in the book they list the answer as 282km and 12 degrees south of west...
31. ### Significant Digits in Physics: 10,3 km & 576 km/h

I'm reading my physics book and I found something confunsing. In one place my book says that 10,3 km have 5 significant digits (I concluded that is because 10,3 km is 10 300 m ) and in one place it says that 576 km/h has also 3 significant digits. Is it because km/h? How many significant digits...
32. ### Question on significant digits

Hi all. Let's say I have a set of data as follows (the mass of a sample of some chemical measured several times): 23.132 g 24.532 g 21.532 g 22.853 g 23.193 g (I just made that data up, but imagine that a analytical scale put out those numbers, exactly as shown, on its display.)...
33. ### Significant digits in calculation

Hello everyone: I am trying to figure out the how many significant figures (sf), should I keep in the final answer for multiplication and division 1. How many hours and minutes are 20 kiloseconds? 2. 3. for hours: 20*10e3 / 3600 = 5.5555 hrs 20 has only 1...
34. ### How to Determine Significant Figures in Multiplication/Division?

just a quick question... (not a HW question) i know that when you do multiplication/division between numbers you take the lowest amount of sig figs of the two and apply it to your answer... HOWEVER: what would be the answer for a question such as: 0.00420 x 60 = ? the first part has...
35. ### How many significant digits in 0 and 0.0?

Homework Statement I was helping some chemistry student and there was a homework problem on converting Fahrenheit to Celsius. The first two were 16°F and 16.0°F which were easy, but the last one was 0.0°F which had me wondering about 0 and 0.0. Homework Equations C° = 5/9(°F - 32) or...
36. ### Significant Digits: Estimating Next Digit

Homework Statement Suppose you conduct an experiment in which you find a certain measurement to be 0.1mm. We have already discussed in class that this number contains 1 significant digit (the "1" to the right of the decimal), but how many times would you have to conduct the experiment in order...
37. ### Significant Digits in Science: Questions Answered

quick question, does significant digits really apply in the science world because in some years of science we had to use them and in another we didnt so what's the answer? Thanks in advance.
38. ### Solving Significant Digit Problems with Varying Order of Magnitude

I have reasons to believe that my question asks to write the answer with 3 significant digits (i.e. the given numbers are with 3 digits). But there are several figures to be worked out, so I list them (for example): 3.45 3.89 7.90 ... 10.23 or 10.2 (or even 10.20)? The problem is that...
39. ### Significant Digits for Longer Operations

Homework Statement Ok so I am having difficulty doing significant digits for longer operations (i.e. cosine law) where you also have square roots and squares (I know they are simply multiplications) Homework Equations c^2 = a^2 + b^2 - 2ab(cosC) The Attempt at a Solution Here are...
40. ### Solved: Significant Digits - Multi-Equation Answers

[SOLVED] Significant Digits I know how to do significant digits, but was wondering about answers involving multiple equations. If I find acceleration first, then find net force, should I wait to round the answer until the end? a=vf-vi/t and then fnet=ma To reiterate, should I keep...
41. ### Unit conversions and significant digits

Let's us say I was given 100.0 kg of something. That number has 4 significant digits. Now let's say I wanted to convert that to grams. It would be 100,000g. but even if I write that as 100,000. it would be 6 significant digits. Is my only option for this case to write it in scientific...
42. ### How to calculate significant digits

like, major help I have no idea what to do when calculating significant digits and stuff like that those who help I thank very much so:smile:
43. ### Equations of motion and proper Significant Digits.

Homework Statement Solve the following to the proper number of significant figures: a) A loaded truck has mass (2.12 X 10^4 +/- 6 X 10^2) kg and when empty, it has a mass of (1.72 X 10^4 +/- 5 X 10^2) kg. What is the relative uncertainty of the mass of the gravel dumped from the truck...
44. ### Question about significant digits

Homework Statement A 2.6 kg mass hangs at one end of a rope that is attached to a support on a railroad car. When the car accelerates to the right, the rope makes an angle of 8.5° with the vertical, as shown in Figure P4.68. Find the acceleration of the car.Homework Equations (Fnet)x=max...
45. ### Significant Digits: Questions & Answers

I just started physics and have a few questions on significant digits. I have to put this number into scientific notation 2 999 900 My answer is 2.3 * 10 (to the power of 6) Is that right? My other question is 8.83/0.002 = "4415" To write it with sig digits I can only use one...
46. ### Significant digits in intermediate answers

Just wondering if anyone could clarify what qualifies as an intermediate answer. Is this limited to answers within the same calculation, or all the answers from steps leading to the final one? ie. What is the current needed to produce 10.0g of lithium in 1.00h? Step 1: Determine moles of...
47. ### Sig Figs: Debate on # of Trailing 0's?

I apologize in advance if this topic does not belong in the section of the forum -- it was the closest match I could find. I am a grade 10 high school student (16 years old), and I was having a bit of a discussion with my science teacher (who was previously a chemical engineer). I respect...
48. ### Why Significant Digits Matter in Multiplication & Division

A thread here asking about significant digits got me to thinking. I understand why they work the way they do with addition and subtraction, but multiplication and division are so easy that I've never bothered to think about it. So - can someone explain why a three s.d. number times a two...
49. ### Significant Digits: a) 4 or 5 & b) 5 or 7

hi guys... i'm taking an online course and i believe my work was marked wrong... State the number of significant digits. a) 0.0029550 m b) 7.050800 kN can someone please tell me the # of signifcant digits! is a) 4 or 5 and b) 5 or7 thanks guys...
50. ### Sig Figs: Solve Mixed Equations and Get the Correct Answer

Please, if someone helps, I will forever be in your debt! Just Gen Chem I...but its about Sig Figs...know one seems to know the rules about which law to use when you have mixed equations 0.0001235 X 0.0012 + ( 5.48 - 0.004) X 9.1 + (8.2 x10 raised to the neg 6) Give the correct answer...