integer counts (e.g., the number of oranges in a bag) definitions of one unit in terms of another (e.g. a minute is 60 seconds) actual prices asked or offered, and quantities given in requirement specifications legally defined conversions, such as international currency exchange scalar operations, such as "tripling" or "halving" mathematical constants, such as π and e Physical constants such as Avogadro's number, however, have a limited number of significant digits, because these constants are known to us only by measurement. On the other hand, c (speed of light) is exactly 299,792,458 m/s by definition. Contents 1 Multiplication and division using significance arithmetic
2 Addition and subtraction using significance arithmetic
3 Transcendental functions
4
Multiplication and division using significance arithmetic[edit] When multiplying or dividing numbers, the result is rounded to the number of significant figures in the factor with the least significant figures. Here, the quantity of significant figures in each of the factors is important—not the position of the significant figures. For instance, using significance arithmetic rules: 8 × 8 ≈ 6 × 101 8 × 8.0 ≈ 6 × 101 8.0 × 8.0 ≈ 64 8.02 × 8.02 ≈ 64.3 8 / 2.0 ≈ 4 8.6 /2.0012 ≈ 4.3 2 × 0.8 ≈ 2 If, in the above, the numbers are assumed to be measurements (and therefore probably inexact) then "8" above represents an inexact measurement with only one significant digit. Therefore, the result of "8 × 8" is rounded to a result with only one significant digit, i.e., "6 × 101" instead of the unrounded "64" that one might expect. In many cases, the rounded result is less accurate than the non-rounded result; a measurement of "8" has an actual underlying quantity between 7.5 and 8.5. The true square would be in the range between 56.25 and 72.25. So 6 × 101 is the best one can give, as other possible answers give a false sense of accuracy. Further, the 6 × 101 is itself confusing (as it might be considered to imply 60 ±5, which is over-optimistic; more accurate would be 64 ±8). Addition and subtraction using significance arithmetic[edit] When adding or subtracting using significant figures rules, results are rounded to the position of the least significant digit in the most uncertain of the numbers being summed (or subtracted). That is, the result is rounded to the last digit that is significant in each of the numbers being summed. Here the position of the significant figures is important, but the quantity of significant figures is irrelevant. Some examples using these rules: 1 + 1.1 2 1 is significant to the ones place, 1.1 is significant to the tenths place. Of the two, the least precise is the ones place. The answer cannot have any significant figures past the ones place. 1.0 + 1.1 2.1 1.0 and 1.1 are significant to the tenths place, so the answer will also have a number in the tenths place. 100 + 110 ≈ 200 We see the answer is 200, given the significance to the hundredths place of the 100. The answer maintains a single digits of significance in the hundreds place, just like the first term in the arithmetic. 100. + 110. = 210. 100. and 110. are both significant to the ones place (as indicated by the decimal), so the answer is also significant to the ones place. 1×102 + 1.1×102 ≈ 2×102 100 is significant up to the hundreds place, while 110 is up to the tens place. Of the two, the least accurate is the hundreds place. The answer should not have significant digits past the hundreds place. 1.0×102 + 111 = 2.1×102 1.0×102 is significant up to the tens place while 111 has numbers up until the ones place. The answer will have no significant figures past the tens place. 123.25 + 46.0 + 86.26 ≈ 255.5 123.25 and 86.26 are significant until the hundredths place while 46.0 is only significant until the tenths place. The answer will be significant up until the tenths place. 100 - 1 ≈ 100 We see the answer is 100, given the significance to the hundredths place of the 100. It may seem counter-intuitive, but giving the nature of significant digits dictating precision, we can see how this follows from the standard rules. Transcendental functions[edit] Transcendental functions have a complicated method for determining the significance of the result. These include the logarithm function, the exponential function and the trigonometric functions. The significance of the result depends on the condition number. In general, the number of significant figures for the result is equal to the number of significant figures for the input minus the order of magnitude of the condition number. The condition number of a differentiable function f at a point x is
x f ′ ( x ) / f ( x )
; displaystyle leftxf'(x)/f(x)right; see Condition number: One variable for details. Note that if a function has a zero at a point, its condition number at the point is infinite, as infinitesimal changes in the input can change the output from zero to non-zero, yielding a ratio with zero in the denominator, hence an infinite relative change. The condition number of the mostly used functions are as follows;[1] these can be used to compute significant figures for all elementary functions:
e x displaystyle e^ x :
x
displaystyle x Natural logarithm function ln ( x ) displaystyle ln(x) : 1
ln ( x )
displaystyle frac 1 ln(x) Sine function sin ( x ) displaystyle sin(x) :
x cot ( x )
displaystyle xcot(x) Cosine function cos ( x ) displaystyle cos(x) :
x tan ( x )
displaystyle xtan(x) Tangent function tan ( x ) displaystyle tan(x) :
x ( tan ( x ) + cot ( x ) )
displaystyle x(tan(x)+cot(x)) Inverse sine function arcsin ( x ) displaystyle arcsin(x) :
x 1 − x 2 arcsin ( x )
displaystyle left frac x sqrt 1-x^ 2 arcsin(x) right Inverse cosine function arccos ( x ) displaystyle arccos(x) :
x 1 − x 2 arccos ( x )
displaystyle left frac x sqrt 1-x^ 2 arccos(x) right Inverse tangent function arctan ( x ) displaystyle arctan(x) :
x ( 1 + x 2 ) arctan ( x )
displaystyle left frac x (1+x^ 2 )arctan(x) right The fact that the number of significant figures for the result is equal to the number of significant figures for the input minus the logarithm of the condition number can be easily derived from first principles. Let x ^ displaystyle hat x and f ( x ^ ) displaystyle f( hat x ) be the true values and let x displaystyle x and f ( x ) displaystyle f(x) be approximate values with errors δ x displaystyle delta x and δ f displaystyle delta f respectively. Then we have x ^ = x ± δ x displaystyle hat x =xpm delta x , f ( x ^ ) = f ( x ) ± δ f displaystyle f( hat x )=f(x)pm delta f , and ± δ f = f ( x ^ ) − f ( x ) = f ( x ± δ x ) − f ( x ) = f ( x ± δ x ) − f ( x ) ± δ x ⋅ ( ± δ x ) ≈ ± d f ( x ) d x δ x displaystyle pm delta f=f( hat x )-f(x)=f(xpm delta x)-f(x)= frac f(xpm delta x)-f(x) pm delta x cdot (pm delta x)approx pm frac df(x) dx delta x The significant figures of a number is related to the uncertain error of the number by δ x ≈ x ⋅ 10 1 − ( s i g n i f i c a n t f i g u r e s o f x ) displaystyle delta xapprox xcdot 10^ 1- rm (significant~figures~of~x) . Substituting this into the above equation gives: f ( x ) ⋅ 10 1 − ( s i g n i f i c a n t f i g u r e s o f f ( x ) ) ≈ d f ( x ) d x x ⋅ 10 1 − ( s i g n i f i c a n t f i g u r e s o f x ) displaystyle f(x)cdot 10^ 1- rm (significant~figures~of~f(x)) approx frac df(x) dx xcdot 10^ 1- rm (significant~figures~of~x) 1 − ( s i g n i f i c a n t f i g u r e s o f f ( x ) ) ≈ log 10 ( d f ( x ) d x x f ( x ) ⋅ 10 1 − ( s i g n i f i c a n t f i g u r e s o f x ) ) = 1 − ( s i g n i f i c a n t f i g u r e s o f x ) + log 10 ( d f ( x ) d x x f ( x ) ) displaystyle 1- rm (significant~figures~of~f(x)) approx log _ 10 left( frac df(x) dx frac x f(x) cdot 10^ 1- rm (significant~figures~of~x) right)=1- rm (significant~figures~of~x) +log _ 10 left( frac df(x) dx frac x f(x) right) ( s i g n i f i c a n t f i g u r e s o f f ( x ) ) ≈ ( s i g n i f i c a n t f i g u r e s o f x ) − log 10 ( d f ( x ) d x x f ( x ) ) displaystyle rm (significant~figures~of~f(x)) approx rm (significant~figures~of~x) -log _ 10 left( frac df(x) dx frac x f(x) right)
Uncertainty is not the same as a mistake. If the outcome of a particular experiment is reported as 1.234±0.056 it does not mean the observer made a mistake; it may be that the outcome is inherently statistical, and is best described by the expression indicating a value showing only those digits that are significant, ie the known digits plus one uncertain digit, in this case 1.23±0.06. To describe that outcome as 1.234 would be incorrect under these circumstances, even though it expresses less uncertainty. Uncertainty is not the same as insignificance, and vice versa. An uncertain number may be highly significant (example: signal averaging). Conversely, a completely certain number may be insignificant. Significance is not the same as significant digits. Digit-counting is not as rigorous a way to represent significance as specifying the uncertainty separately and explicitly (such as 1.234±0.056). Manual, algebraic propagation of uncertainty—the nominal topic of this article—is possible, but challenging. Alternative methods include the crank three times method and the Monte Carlo method. Another option is interval arithmetic, which can provide a strict upper bound on the uncertainty, but generally it is not a tight upper bound (i.e. it does not provide a best estimate of the uncertainty). For most purposes, Monte Carlo is more useful than interval arithmetic[citation needed]. Kahan considers significance arithmetic to be unreliable as a form of automated error analysis.[2] In order to explicitly express the uncertainty in any uncertain result, the uncertainty should be given separately, with an uncertainty interval, and a confidence interval. The expression 1.23 U95 = 0.06 implies that the true (unknowable) value of the variable is expected to lie in the interval from 1.17 to 1.29 with at least 95% confidence. If the confidence interval is not specified it has traditionally been assumed to be 95% corresponding to two standard deviations from the mean. Confidence intervals at one standard deviation (68%) and three standard deviations (99%) are also commonly used. See also[edit] Rounding Propagation of uncertainty Significant figures Accuracy and precision MANIAC III Loss of significance References[edit] ^ http://www.cl.cam.ac.uk/~jrh13/papers/transcendentals.pdf[full
citation needed]
^
Further reading[edit] Delury, D. B. (1958). "Computations with approximate numbers". The
Mathematics Teacher. 51 (7): 521–30. JSTOR 27955748.
Bond, E. A. (1931). "Significant Digits in Computation with
Approximate Numbers". The Mathematics Teacher. 24 (4): 208–12.
JSTOR 27951340.
External links[edit] The Decimal Arithmetic FAQ — Is the decimal arithmetic ‘significance’ arithmetic? Advanced methods for handling uncertainty and some explanations of the shortcomings of significance arithmetic and significant figures. Significant Figures Calculator – Displays a number with the desired number of significant digits. Measurements and Uncertainties versus Significant Digits or Significant Figures – Proper methods for expressing uncertainty, including a detailed discussion of the problems with any notion of s |