When Do You Know to Round Off to Significant Figures

Any digit of a number within its measurement resolution, as opposed to spurious digits

Significant figures (likewise known as the pregnant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to signal the quantity of something.

If a number expressing the result of a measurement (east.1000., length, pressure level, volume, or mass) has more digits than the number of digits allowed past the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so just these can be pregnant 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 one mm, then the get-go 3 digits (ane, i, and 4, showing 114 mm) are sure and so they are meaning figures. Digits which are uncertain simply reliable are also considered meaning figures. In this example, the concluding digit (8, which adds 0.8 mm) is too considered a meaning figure even though there is incertitude in it.^[i]

Another example is a volume measurement of 2.98 Fifty with an dubiety of ± 0.05 Fifty. The actual volume is somewhere between ii.93 L and 3.03 L. Even when some of the digits are not certain, equally long as they are reliable, they are considered significant because they indicate the actual book within the adequate degree of doubt. In this example the bodily volume might be 2.94 L or might instead exist three.02 L. And so all three are meaning figures.^[two]

The following digits are not significant figures.^[3]

• All leading zeros. For example, 013 kg has two significant figures, ane and 3, and the leading nothing is not significant since it is non necessary to signal the mass; 013 kg = 13 kg and so 0 is not necessary. In the example of 0.056 yard there are two insignificant leading zeros since 0.056 m = 56 mm and and so the leading zeros are not necessary to indicate the length.
• Trailing zeros when they are only placeholders. For example, the trailing zeros in 1500 m as a length measurement are not significant if they are but placeholders for ones and tens places as the measurement resolution is 100 chiliad. In this case, 1500 m means the length to measure is shut to 1500 thou rather than saying that the length is exactly 1500 yard.
• 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 near significant is the digit with the highest exponent value (simply the left-about significant figure), and the least significant is the digit with the lowest exponent value (simply the right-about significant figure). For example, in the number "123", the "ane" is the well-nigh significant effigy as it counts hundreds (10²), and "3" is the least significant figure as it counts ones (10⁰).

Significance arithmetic is a set up of approximate rules for roughly maintaining significance throughout a computation. The more than sophisticated scientific rules are known as propagation of uncertainty.

Numbers are often rounded to avoid reporting insignificant figures. For case, it would create false precision to express a measurement every bit 12.34525 kg if the scale was just measured to the nearest gram. In this case, the significant figures are the first 5 digits from the left-nearly digit (1, 2, three, 4, and 5), and the number needs to be rounded to the pregnant figures then that it will be 12.345 kg every bit the reliable value. Numbers tin also be rounded but 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 (base-x, decimal numbers) is causeless in the following.

Identifying significant figures [edit]

Rules to identify significant figures in a number [edit]

Digits in light bluish are significant figures; those in black are not.

Note that identifying the significant figures in a number requires knowing which digits are reliable (eastward.g., by knowing the measurement or reporting resolution with which the number is obtained or candy) since only reliable digits can be meaning; eastward.g., 3 and four in 0.00234 g are not significant if the measurable smallest weight is 0.001 1000.^[4]

• Not-zero digits within the given measurement or reporting resolution are meaning.
  • 91 has two meaning figures (nine and i) if they are measurement-allowed digits.
  • 123.45 has five significant digits (1, ii, 3, 4 and 5) if they are within the measurement resolution. If the resolution is 0.1, and then the last digit 5 is non significant.
• Zeros between two significant non-zero digits are significant (significant trapped zeros) .
  • 101.12003 consists of eight pregnant figures if the resolution is to 0.00001.
  • 125.340006 has seven significant figures if the resolution is to 0.0001: i, 2, 5, three, four, 0, and 0.
• Zeros to the left of the start not-zero digit (leading zeros) are not meaning.
  • If a length measurement gives 0.052 km, then 0.052 km = 52 thou so five and 2 are just pregnant; the leading zeros appear or disappear, depending on which unit is used, so they are not necessary to indicate the measurement scale.
  • 0.00034 has 4 significant zeros if the resolution is 0.001. (three and four are beyond the resolution then are not significant.)
• Zeros to the right of the terminal not-zero digit (abaft zeros) in a number with the decimal point are significant if they are within the measurement or reporting resolution.
  • 1.200 has four significant figures (1, 2, 0, and 0) if they are allowed by the measurement resolution.
  • 0.0980 has three meaning digits (nine, 8, and the last goose egg) if they are within the measurement resolution.
  • 120.000 consists of six significant figures (1, 2, and the 4 subsequent zeroes) except for the last nix If the resolution is to 0.01.
• Trailing zeros in an integer may or may not be significant, depending on the measurement or reporting resolution.
  • 45,600 has 3, four or 5 significant figures depending on how the last zeros are used. For case, if the length of a road is reported as 45600 g without data nigh the reporting or measurement resolution, then it is not clear if the road length is precisely measured as 45600 m or if it is a crude estimate. If information technology is the rough interpretation, then only the first 3 non-zero digits are significant since the abaft zeros are neither reliable nor necessary; 45600 m tin can be expressed as 45.half-dozen km or as 4.56 × 10^four m in scientific notation, and neither expression requires the abaft zeros.
• An exact number has an infinite number of significant figures.
  • If the number of apples in a bag is iv (exact number), then this number is four.0000... (with infinite trailing zeros to the right of the decimal point). As a upshot, 4 does non touch the number of significant figures or digits in the result of calculations with information technology.
• A mathematical or physical constant has significant figures to its known digits.
  • π, as the ratio of the circumference to the diameter of a circle, is 3.14159265358979323... known to 50 trillion digits^[5] calculated as of 2020-01-29, and that calculated 'π' approximation has that many significant digits, while in practical applications far fewer are used (and π itself has infinite significant digits, equally all irrational numbers practice). Often iii.14 is used in numerical calculations, i.e. iii significant decimal digits, with vii right binary digits (while the more than accurate 22/vii is too used, fifty-fifty though information technology also only amounts to the same 3 significant correct decimal digits, it has 10 correct binary digits), which is a good enough approximation for many practical uses. Most calculators, and computer programs, can handle iii.141592653589793, 16 decimal digits, that is commonly used in computers and used by NASA for "JPL's highest accuracy calculations, which are for interplanetary navigation".^[6] For "the largest size there is: the visible universe [..] you would need 39 or 40 decimal places."^[6]
  • The Planck constant is ${\displaystyle h=half-dozen.62607015\times 10^{-34}J\cdot southward}$ and is divers equally an verbal value and then that it is more properly defined as ${\displaystyle h=6.62607015(0)\times 10^{-34}J\cdot s}$ .^[seven]

Ways to announce pregnant figures in an integer with abaft zeros [edit]

The significance of abaft zeros in a number non containing a decimal point can be ambiguous. For example, it may not always be clear if the number 1300 is precise to the nearest unit of measurement (just happens coincidentally to exist an exact multiple of a hundred) or if information technology is only shown to the nearest hundreds due to rounding or uncertainty. Many conventions be to address this issue. Yet, these are not universally used and would simply be effective if the reader is familiar with the convention:

• An overline, sometimes also chosen an overbar, or less accurately, a vinculum, may exist placed over the last significant figure; any trailing zeros following this are insignificant. For example, 1300 has three meaning figures (and hence indicates that the number is precise to the nearest ten).

• Less often, using a closely related convention, the last significant figure of a number may be underlined; for case, "1300" has two significant figures.

• A decimal point may be placed after the number; for example "1300." indicates specifically that trailing zeros are meant to be significant.^[8]

As the conventions in a higher place are not in general use, the post-obit more widely recognized options are available for indicating the significance of number with abaft zeros:

• Eliminate ambiguous or non-significant zeros by changing the unit prefix in a number with a unit of measurement. For example, the precision of measurement specified every bit 1300 1000 is ambiguous, while if stated as 1.thirty kg it is not. Also 0.0123 L can be rewritten as 12.three mL

• Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes one.30×xⁱⁱⁱ . Likewise 0.0123 tin can be rewritten as 1.23×x⁻² . The office of the representation that contains the significant figures (1.30 or one.23) is known as the significand or mantissa. The digits in the base and exponent ( x³ or 10⁻² ) are considered verbal numbers so for these digits, significant figures are irrelevant.

• Explicitly state the number of significant figures (the abbreviation due south.f. is sometimes used): For example "twenty 000 to 2 southward.f." or "xx 000 (2 sf)".

• State the expected variability (precision) explicitly with a plus–minus sign, as in twenty 000 ± one%. This also allows specifying a range of precision in-between powers of 10.

Rounding to significant figures [edit]

Rounding to significant figures is a more general-purpose technique than rounding to n digits, since information technology handles numbers of unlike scales in a uniform way. For instance, the population of a city might only exist known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in fault by hundreds of thousands, but both take two meaning figures (5 and 2). This reflects the fact that the significance of the error is the same in both cases, relative to the size of the quantity being measured.

To round a number to n significant figures:^{[9] [10]}

1. If the n + 1 digit is greater than 5 or is 5 followed past other non-zero digits, add 1 to the n digit. For example, if we want to circular 1.2459 to iii significant figures, and so this step results in 1.25.
2. If the due north + one digit is 5 not followed past other digits or followed by only zeros, then rounding requires a tie-breaking rule. For example, to round one.25 to two significant figures:
   • Circular one-half away from zero (too known equally "5/4")^{[ citation needed ]} rounds upwardly to 1.three. This is the default rounding method unsaid in many disciplines^{[ citation needed ]} if the required rounding method is non specified.
   • Round half to even, which rounds to the nearest even number. With this method, 1.25 is rounded downward to one.2. If this method applies to 1.35, then it is rounded upwardly to 1.4. This is the method preferred by many scientific disciplines, because, for example, it avoids skewing the average value of a long list of values upwardly.
3. For an integer in rounding, replace the digits subsequently the n digit with zeros. For case, if 1254 is rounded to 2 pregnant figures, then 5 and 4 are replaced to 0 so that it will be 1300. For a number with the decimal point in rounding, remove the digits later the n digit. For example, if fourteen.895 is rounded to three pregnant figures, then the digits after eight are removed so that it will be 14.9.

In financial calculations, a number is often rounded to a given number of places. For example, to two places after the decimal separator for many world currencies. This is done because greater precision is immaterial, and usually information technology is non possible to settle a debt of less than the smallest currency unit.

In UK personal taxation returns, income is rounded downward to the nearest pound, whilst tax paid is calculated to the nearest penny.

As an illustration, the decimal quantity 12.345 can be expressed with various numbers of significant figures or decimal places. If insufficient precision is bachelor then the number is rounded in some manner to fit the available precision. The post-obit tabular array shows the results for various total precision at ii rounding means (N/A stands for Non Applicable).

Precision	Rounded to significant figures	Rounded to decimal places
6	12.3450	12.345000
5	12.345	12.34500
4	12.34 or 12.35	12.3450
3	12.3	12.345
2	12	12.34 or 12.35
1	ten	12.3
0	N/A	12

Another example for 0.012345. (Retrieve that the leading zeros are not pregnant.)

Precision	Rounded to significant figures	Rounded to decimal places
vii	0.01234500	0.0123450
6	0.0123450	0.012345
v	0.012345	0.01234 or 0.01235
four	0.01234 or 0.01235	0.0123
3	0.0123	0.012
ii	0.012	0.01
i	0.01	0.0
0	Northward/A	0

The representation of a not-zero number ten to a precision of p significant digits has a numerical value that is given by the formula:^{[ commendation needed ]}

${\displaystyle x^{northward}\cdot \operatorname {round} \left({\frac {x}{10^{n}}}\right)}$

where

${\displaystyle north=\lfloor \log _{10}(|10|)\rfloor +1-p}$

which may need to be written with a specific marking as detailed above to specify the number of pregnant trailing zeros.

Writing incertitude and implied incertitude [edit]

Meaning figures in writing uncertainty [edit]

It is recommended for a measurement result to include the measurement incertitude such equally ${\displaystyle x_{all-time}\pm \sigma _{x}}$ , where x_best and σ_x are the best estimate and uncertainty in the measurement respectively.^[11] 10_best can be the average of measured values and σ_x can be the standard deviation or a multiple of the measurement deviation. The rules to write ${\displaystyle x_{best}\pm \sigma _{x}}$ are:^[12]

• σ_ten has just 1 or two significant figures every bit more precise incertitude has no meaning.
  • ane.79 ± 0.06 (correct), 1.79 ± 0.96 (correct), 1.79 ± 1.96 (incorrect).
• The digit positions of the last meaning figures in 10_best and σ_x are the same, otherwise the consistency is lost. For case, in 1.79 ± 0.067 (incorrect), it does not brand sense to have more accurate dubiousness than the best estimate. 1.79 ± 0.ix (incorrect) also does not make sense since the rounding guideline for improver and subtraction below tells that the edges of the true value range are 2.7 and 0.9, that are less accurate than the best judge.
  • 1.79 ± 0.06 (correct), 1.79 ± 0.96 (right), 1.79 ± 0.067 (wrong), 1.79 ± 0.9 (incorrect).

Implied uncertainty [edit]

In chemistry (and may also be for other scientific branches), uncertainty may be implied by the terminal meaning figure if it is not explicitly expressed.^[two] The unsaid uncertainty is ± the one-half of the minimum calibration at the concluding pregnant figure position. For case, if the volume of h2o in a canteen is reported equally iii.78 L without mentioning uncertainty, and so ± 0.005 L measurement uncertainty may be unsaid. If 2.97 ± 0.07 kg, so the actual weight is somewhere in 2.xc to 3.04 kg, is measured and it is desired to study it with a single number, then 3.0 kg is the best number to report since its implied uncertainty ± 0.05 kg tells the weight range of ii.95 to 3.05 kg that is shut to the measurement range. If 2.97 ± 0.09 kg, and then three.0 kg is even so the best since, if 3 kg is reported then its implied uncertainty ± 0.five tells the range of ii.five to 3.v kg that is likewise wide in comparing with the measurement range.

If there is a need to write the implied uncertainty of a number, and so it can be written equally ${\displaystyle ten\pm \sigma _{x}}$ with stating it as the implied dubiety (to prevent readers from recognizing it as the measurement uncertainty), where 10 and σ₁₀ are the number with an extra zero digit (to follow the rules to write uncertainty higher up) and the unsaid dubiety of it respectively. For instance, 6 kg with the implied dubiousness ± 0.5 kg can be stated as 6.0 ± 0.5 kg.

Arithmetic [edit]

Equally at that place are rules to determine the significant figures in directly measured quantities, at that place are also guidelines (not rules) to decide the significant figures in quantities calculated from these measured quantities.

Significant figures in measured quantities are about important in the decision of significant figures in calculated quantities with them. A mathematical or physical abiding (e.g., π in the formula for the area of a circumvolve with radius r as πr ² ) has no outcome on the determination of the significant figures in the result of a calculation with information technology if its known digits are equal to or more than the significant figures in the measured quantities used in the calculation. An exact number such equally ½ in the formula for the kinetic free energy of a mass yard with velocity v as ½mv ^two has no bearing on the significant figures in the calculated kinetic energy since its number of pregnant figures is space (0.500000...).

The guidelines described beneath are intended to avoid a calculation result more than precise than the measured quantities, only it does non ensure the resulted implied uncertainty close enough to the measured uncertainties. This trouble can be seen in unit conversion. If the guidelines give the implied uncertainty also far from the measured ones, and so it may be needed to decide significant digits that give comparable uncertainty.

Multiplication and division [edit]

For quantities created from measured quantities via multiplication and segmentation, the calculated outcome should have as many meaning figures every bit the least number of significant figures among the measured quantities used in the calculation.^[13] For example,

• i.234 × 2 = 2.468 ≈ 2
• 1.234 × 2.0 = 2.468 ≈ ii.5
• 0.01234 × ii = 0.0ii468 ≈ 0.02

with 1, two, and one significant figures respectively. (2 here is assumed not an exact number.) For the first example, the first multiplication cistron has iv significant figures and the second has one significant figure. The cistron with the fewest or least pregnant figures is the 2d one with only ane, so the concluding calculated effect should also accept i significant figure.

Exception [edit]

For unit conversion, the implied uncertainty of the upshot can be unsatisfactorily higher than that in the previous unit of measurement if this rounding guideline is followed; For example, 8 inch has the implied doubtfulness of ± 0.v inch = ± 1.27 cm. If it is converted to the centimetre scale and the rounding guideline for multiplication and division is followed, then 20.32 cm ≈ 20 cm with the implied dubiety of ± 5 cm. If this unsaid dubiety is considered as also underestimated, and then more proper significant digits in the unit conversion result may be ii0.32 cm ≈ xx. cm with the unsaid uncertainty of ± 0.5 cm.

Another exception of applying the higher up rounding guideline is to multiply a number by an integer, such as 1.234 × ix. If the above guideline is followed, then the result is rounded every bit 1.234 × 9.000.... = 11.106 ≈ 11.11. However, this multiplication is essentially calculation 1.234 to itself 9 times such as 1.234 + 1.234 + ... + 1.234 so the rounding guideline for addition and subtraction described beneath is more than proper rounding arroyo.^[14] As a event, the terminal answer is 1.234 + ane.234 + ... + 1.234 = 11.x6 = 11.106 (one significant digit increase).

Improver and subtraction [edit]

For quantities created from measured quantities via addition and subtraction, the final significant figure position (due east.k., hundreds, tens, ones, tenths, hundredths, and and so along) in the calculated result should be the aforementioned as the leftmost or largest digit position among the final meaning figures of the measured quantities in the adding. For example,

• i.234 + 2 = 3.234 ≈ three
• ane.234 + 2.0 = 3.two34 ≈ 3.two
• 0.01234 + 2 = ii.01234 ≈ 2

with the terminal meaning figures in the ones place, tenths place, and ones place respectively. (two here is assumed not an verbal number.) For the first instance, the start term has its last meaning figure in the thousandths place and the second term has its last significant figure in the ones place. The leftmost or largest digit position among the last meaning figures of these terms is the ones place, then the calculated result should also have its concluding significant effigy in the ones place.

The dominion to calculate meaning figures for multiplication and division are non the aforementioned every bit the dominion for addition and subtraction. For multiplication and sectionalization, only the total number of significant figures in each of the factors in the calculation matters; the digit position of the last significant figure in each factor is irrelevant. For improver and subtraction, only the digit position of the last significant figure in each of the terms in the calculation matters; the total number of significant figures in each term is irrelevant.^{[ citation needed ]} However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.^{[ citation needed ]}

Logarithm and antilogarithm [edit]

The base-10 logarithm of a normalized number (i.e., a × x ^b with 1 ≤ a < ten and b as an integer), is rounded such that its decimal part (called mantissa) has every bit many significant figures as the significant figures in the normalized number.

• log₁₀(3.000 × x⁴) = log₁₀(ten⁴) + log₁₀(iii.000) = four.000000... (exact number so infinite significant digits) + 0.4771212547... = four.4771212547 ≈ 4.4771.

When taking the antilogarithm of a normalized number, the result is rounded to have as many significant figures every bit the significant figures in the decimal part of the number to be antiloged.

• ten^4.4771 = 2999eight.5318119... = 30000 = iii.000 × x⁴.

Transcendental functions [edit]

If a transcendental role ${\displaystyle f(x)}$ (due east.k., the exponential function, the logarithm, and the trigonometric functions) is differentiable at its domain element x, and then its number of significant figures (denoted every bit "significant figures of ${\displaystyle f(ten)}$ ") is approximately related with the number of pregnant figures in x (denoted every bit "significant figures of ten") past the formula

${\displaystyle {\rm {(significant~figures~of~f(x))}}\approx {\rm {(significant~figures~of~x)}}-\log _{ten}\left(\left\vert {{\frac {df(x)}{dx}}{\frac {x}{f(x)}}}\right\vert \right)}$ ,

where ${\displaystyle \left\vert {{\frac {df(x)}{dx}}{\frac {x}{f(ten)}}}\right\vert }$ is the condition number. Run into the significance arithmetic commodity to find its derivation.

Circular only on the final calculation result [edit]

When performing multiple stage calculations, do not round intermediate phase calculation results; continue equally many digits equally is practical (at least 1 more digit than the rounding rule allows per stage) until the end of all the calculations to avoid cumulative rounding errors while tracking or recording the significant figures in each intermediate result. Then, round the final result, for instance, to the fewest number of significant figures (for multiplication or division) or leftmost last significant digit position (for improver or subtraction) among the inputs in the concluding calculation.^[15]

• (2.3494 + 1.345) × 1.2 = 3.69four4 × 1.2 = 4.43328 ≈ 4.4.
• (2.3494 × one.345) + 1.2 = 3.159943 + 1.two = 4.359943 ≈ 4.iv.

[edit]

When using a ruler, initially utilise the smallest mark every bit the first estimated digit. For example, if a ruler's smallest mark is 0.1 cm, and 4.5 cm is read, and so it is iv.5 (±0.ane cm) or 4.4 cm to iv.6 cm as to the smallest marker interval. Withal, in do a measurement can usually be estimated by eye to closer than the interval between the ruler's smallest mark, eastward.grand. in the above case it might exist estimated every bit between 4.51 cm and 4.53 cm.

It is also possible that the overall length of a ruler may not be accurate to the caste of the smallest mark, and the marks may be imperfectly spaced within each unit. Even so assuming a normal good quality ruler, it should be possible to estimate tenths between the nearest ii marks to achieve an extra decimal place of accuracy.^[16] Failing to practise this adds the error in reading the ruler to whatever error in the calibration of the ruler.^[17]

Estimation in statistic [edit]

When estimating the proportion of individuals carrying some particular characteristic in a population, from a random sample of that population, the number of significant figures should not exceed the maximum precision allowed by that sample size.

Relationship to accuracy and precision in measurement [edit]

Traditionally, in various technical fields, "accuracy" refers to the closeness of a given measurement to its true value; "precision" refers to the stability of that measurement when repeated many times. Hoping to reverberate the style in which the term "accurateness" is actually used in the scientific community, in that location is a recent standard, ISO 5725, which keeps the same definition of precision but defines the term "trueness" as the closeness of a given measurement to its true value and uses the term "accuracy" as the combination of trueness and precision. (See the accuracy and precision commodity for a total discussion.) In either case, the number of significant figures roughly corresponds to precision, not to accuracy or the newer concept of trueness.

In computing [edit]

Computer representations of floating-betoken numbers employ a course of rounding to pregnant figures (while usually not keeping track of how many), in full general with binary numbers. The number of correct significant figures is closely related to the notion of relative fault (which has the advantage of beingness a more accurate measure of precision, and is contained of the radix, also known as the base, of the number system used).

See also [edit]

• Benford's law (get-go-digit police force)
• Engineering science notation
• Error bar
• False precision
• IEEE 754 (IEEE floating-point standard)
• Interval arithmetic
• Kahan summation algorithm
• Precision (computer science)
• Round-off error

References [edit]

1. ^ "Pregnant Figures - Writing Numbers to Reverberate Precision". Chemical science - Libretexts. 2019-09-04. {{cite web}}: CS1 maint: url-condition (link)
2. ^ ^{a b} Lower, Stephen (2021-03-31). "Significant Figures and Rounding". Chemistry - LibreTexts. {{cite web}}: CS1 maint: url-condition (link)
3. ^ Chemical science in the Community; Kendall-Chase:Dubuque, IA 1988
4. ^ Giving a precise definition for the number of correct pregnant digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (PDF) (2nd ed.). SIAM. pp. iii–5.
5. ^ Almost accurate value of pi
6. ^ ^{a b} "How Many Decimals of Pi Do We Actually Need? - Edu News". NASA/JPL Edu . Retrieved 2021-ten-25 .
7. ^ "Resolutions of the 26th CGPM" (PDF). BIPM. 2018-11-16. Archived from the original (PDF) on 2018-eleven-nineteen. Retrieved 2018-11-20 .
8. ^ Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry . Austin, Texas: Holt Rinehart Winston. p. 59. ISBN0-03-052002-9.
9. ^ Engelbrecht, Nancy; et al. (1990). "Rounding Decimal Numbers to a Designated Precision" (PDF). Washington, D.C.: U.Due south. Department of Education.
10. ^ Numerical Mathematics and Calculating, past Cheney and Kincaid.
11. ^ Luna, Eduardo. "Uncertainties and Significant Figures" (PDF). DeAnza Higher. {{cite web}}: CS1 maint: url-status (link)
12. ^ "Significant Figures". Purdue Academy - Section of Physics and Astronomy. {{cite web}}: CS1 maint: url-status (link)
13. ^ "Significant Figure Rules". Penn State University.
14. ^ "Doubt in Measurement- Significant Figures". Chemical science - LibreTexts. 2017-06-xvi. {{cite web}}: CS1 maint: url-status (link)
15. ^ de Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Figures (Typhoon)" (PDF). Freshman Physics Laboratory. California Institute of Technology, Physics Mathematics And Astronomy Sectionalization. Archived from the original (PDF) on 2013-06-eighteen.
16. ^ Experimental Electrical Testing. Newark, NJ: Weston Electrical Instruments Co. 1914. p. nine. Retrieved 2019-01-14 . Experimental Electrical Testing..
17. ^ "Measurements". slc.umd.umich.edu. Academy of Michigan. Retrieved 2017-07-03 .

External links [edit]

• Significant Figures Video by Khan university

donovanarmishath.blogspot.com

Source: https://en.wikipedia.org/wiki/Significant_figures

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