Never add several negligible quantities to a dominant addend.
(2007-08-15) Any nonzero number is equal to a multiple of a power of 10.
We may write any nonzero number in a unique way as a product into a power of 10 of a signedcoefficient with a magnitude at least equal to 1 and strictlyless than 10. The aforementioned power of 10 is called the[decimal] order of magnitude and it canbe omitted when it's equal to unity (1 is 10 to thepower of zero). For example, thespeed of light expressed inscientific notation is:
c = 2.99792458 10 8 m/s
A very important feature of scientific notation is thattrailing zeros can only occur in the coefficient after the decimal point (since there's onlyone nonzero digit before that decimal point). Therefore, trailing zeros are always significant digits in scientific notation, as discussed in thenext article (which, incidentally, presents an example where a result can only be given in scientific notation if the implied precision is to be statedcorrectly).
Nikki (Yahoo!2007-08-14) On the precision implied by givingjust so manysignificant figures.
What's the precision of the factors in the following product? How precise is the result? How would that result be best stated?
2.9 3.5 10.0
If nothing else is said, we can only assume that each factor is knownwithin half a unit of the "least significant digit" given. Trailing zeroes are significant only if they occur after the decimal point (as is the case for the thirdfactor above).
For example, 10.0 denotes a quantity which occurs anywherebetween 9.95 and 10.05 with uniform probability. The above product is thus:
between 2.85 3.45 9.95= 97.83 and 2.95 3.55 10.05= 105.25
Therefore, it's best expressed inscientific notationas 1.010 2. Indeed, this denotes a quantity between 95 and 105. Close enoughand nottoo precise.
You can't merely give the result as 101.5 because that wouldbe a gross misrepresentation of the precision involved. Instead, you should state:
2.9 3.5 10.0 = 1.0 10 2
With the standardized notation which specifies precisionvia a standard deviation expressed in units of the least significantdigit (seecomputation) as presentedin thenext article, we could write:
2.9 3.5 10.0 = 101.5(13)
On 2007-09-20, Barry wrote: [edited summary]
How would you report 100.0200 = 1.047128548...with the proper number of significant figures?
What about -log(0.001178) ?
Well, if the value 0.0200 comes from rounding,it's actually between 0.01995 and 0.02005. So the result is between 1.047008 and 1.047249. Stating the result as 1.0471 gives the impression that the true value is between 1.04705 and 1.04715. This is slightly too precise (by a factor of 2) but that's not grossly misleading (so, it's OK in my book). The alternative would be to state the result as 1.047, which is too coarse (by a factor of 5).
If you only rely on significant figures to express the precision of your results,you're always faced with a similar choice between two different levels of precisionthat differ from each other by a factor of 10. Just choose the lesser of two evils, knowing that you will occasionally have tomisrepresent the precision of your result by a factor of 3 (or a bit more).
Such unsatisfying limitations can't be circumvented within the"significant figures" scheme. When the precision of a result has to be stated more rigorously,it's best to give either its upper and lower bounds (at a 99% confidence level) or to indicatean estimate of the standard deviation (as a two-digit number between parentheses afterthe least significant digit, as discussed in thenext article).
In the second example, -log(0.001178) denotes some value
between -log(0.0011785) = 2.92867 and -log(0.0011775) = 2.92904
That's best reported as 2.929 (which says "between 2.9285 and 2.9295").
Interestingly, logarithms are the quintessential example of a case where the number ofsignificant figures in the result is not directly related to the number of significant figuresof the input data. In the following pathological example,the input has only 3 significant figures but the result does have 9 significant figures:
log ( 7.89 10 123456) = 123456.897
(2015-06-21) Use strict inequalities to indicate the rounded value is a true bound.
Strict inequalities are easy: x < 1.5 and x < 3/2 are equivalent.
When non-strict inequalities are used with rounded numbers, they acquire completely differentmeanings, similar to the meaning acquired by equalities in that case... What such an equality states is that a strict inequality is true forthe tightest different bound expressible at the same level of precision. For example, x ≤ 1.5 means that x < 1.6.
The former is more intuitive than the latter, as it gives the best acceptablevalue at the relevant precision. This is familiar to old-school engineers butothers may struggle when confronted with this, especially intables.
(2007-08-14) Standard deviation expresses the uncertainty or precision of a result.
In many cases, theabove rules concerning significant digits are too coarse to convey a good indication of the claimed precision.
Professionals state the accuracy of their figures by giving theuncertainty expressed in units of the last figure between parentheses(seeexamples).
Technically, this uncertainty is expressedeither as the relevant standard deviation or as1/3 of the "firm" bounds you may have on either side of the mean (both definition are equivalent if we identify "firm bounds"with the 99.73% confidence level in a normal Gaussian distribution).
Straight rounding errors are not at all "normally distributed" along a Gaussiancurve. Instead, the error is uniformly distributed over an interval whose width is equalto one unit of the least significant digit retained. This entails a standard deviation of 1/12 = 0.29 in terms of that unit.
In ourprevious example of a product of threerounded value, what we have to determine is the standard deviation of thefollowing random variable:
( 2.9 + 0.1 X ) ( 3.5 + 0.1 Y ) ( 10.0 + 0.1 Z)
Where X, Y and Z are independent random variables, each uniformly distributedbetween -½ and +½. The average (mathematical expectation) of that random variable is 101.5 and its standard deviation is 1.3444711... (:this involves averaging the square of the above inside a cube of unit volume).
Thus, our product can be expressed with standardized precisionas
101.50(134) or 101.5(13)
The latter form is the more common one, since standardized precision is mostoften expressed with 2 significant digits (3 digits is an overkill).
(2007-08-15) Stating a nonzero number as a multiple of a power of 1000.
Engineering notation is superficially similartoscientific notation, except that the exponent of 10 is restricted to a multipleof 3 (thus, the relevant power of 10 is actually a power of 1000). For this to be possible in all cases, the coefficient is allowedto go from 1 (included) to 1000 (excluded).
Because there may be trailing zeros before the decimalpoint in engineering notation, the number of significant digits is notalways clear. This is the main reason why the systematic useof the engineering notation is strongly discouraged in print, unless accuracyis stated with theabove convention.
By extension, we also call engineering notation any systemresembling scientific notation where the absolute magnitude of thecoefficient is not restricted to the 1-10 range (it could,occasionally, be more than 1000 or less than 1). List of results spanning several orders of magnitudeare sometimes more readable this way, since we can merely comparecoefficients as the order of magnitude (a power of 10) remains constant.
(2007-08-07, 2021-07-05) Alternative approaches for robust solutions to quadratic equations.
The quadratic formula was first published early in the ninth century AD (c. 810) by Al Kwarizmi. It's a mainstay of middle-school algebra, giving the two roots of the polynomial a x2 + b x + c when b2 4ac is positive:
x =
b
b2 4ac
2a
WLG, we may consider only monic polynomials Up to a change of sign, their coefficients are justthe symmetric functions of the roots mu and m+u; namely the sum 2m (twice themean m) and the product p:
[ x (m+u) ] ×[ x (mu) ] = x2 2m x + m2u2 = x2 2m x + p
This implies that u2 = m2 p. If that quantity is positive, the square root functioncan be used to obtain a simplified version of the quadratic formula, currently advocated by Po-Shen Loh (coach for the US IMO team).
x =
m
m2 p
There are two problems with all formulas of this type :
They're not robust, as a subtraction of nearly equal quantities may be required, with an unacceptable loss of floating-point accuracy.
The square-root function isn't well-defined (although the two-valued locution"" is). Especially for equations with complex coefficients.
To solve the first problem, we may first compute whichever root is trouble-free because it doesn'tinvolve the subtraction of nearly equal quantity (thus choosing the "+" sign ifm is positive and the "" sign if it's negative). The other root is then computed by dividing p into that root, which doesn't cause any undue loss of floating-point accuracy (a multiplication or a division never does). Other similar techniques can be found in thefollowing section,including the robust expression for a difference of square roots (of which the above can be construed as a special case).
We run into the aforementioned second problem when trying to discuss preciselywhich root is trouble-free in the case of complex coefficients.
Let's consider the following form of quadratic equationwhich uses the hyperbolic sine function (sh). Any normalized quadratic equationwith a negative constant term can be recast into that form:
x 2 + 2a x sh a 2 = 0
Its two real solutions, are then given by the following robust formulas:
-a exp and a exp -
Using the reverse hyperbolic function Argsh to obtain , if need be, will never entail any loss of floating-point precision...
(2007-08-07) How to avoid subtracting nearly equal quantities.
In what follows, the number x need not be small,but it may well be...
In each of the examples below, the floating-point computation on the left-hand sidewill lead to an unacceptable loss of precision when x is small. The given substitute should be used, which is mathematically identical but won'tlead to potentially nonsensical results with floating-point arithmetic.
Inverse of Hyperbolic Tangent(cf. relativisticrapidity) :
Argth (a+x) - Argth (a) = Argth (x / [1-a(a+x)] )
(2019-02-10) 0.0 represents a small result which is not known to be exactly zero.
As floating-point quantities always represent an approximation, there is no reason why identical floating-point numbers should denote the same number.
1.0 1.0 = 0.0
Calculator designers should resist the temptation to equate 0 and 0.0 :
00 = 1 and 0.00 = 1 ( x0.0 is undefined unless x is an exact positive integer.)
More generally, one should distinguish floating-point and exact values. Besides explicit rounding to exact values (under the user's responsibility) an operation on floating-point numbers never yields an exact valueand should never be misrepresented as such.
