How do you calculate with uncertainties if one or two of the values in the equation doesn't have any uncertainties?

Answer 1

Lets start out with calculating an area where the measurements of the length and width both have an error involved. A is the length of one side with an error of 'a'. B is the length of the other side with an error of 'b'.

Ordinarily when we talk about errors we always say +/- but I don't have such a symbol on my keyboard and typing +/- all the time is labourious so I'm just going to use + and hope you do the conversions yourself (ok).

When calculating the area we multiply length time width, in our case A*B. When using errors however we must use (A+a)*(B+b). Multiplying this out we get

AB +Ab+Ba+ab.

AB is the area without errors considered. Ab+Ba+ab is the error. Consider it as a collection of areas (which it is in this case but the conceptualization can be applied to other problems of these dimensions) with the A side having a short bit at its terminal end of length a, the B side having a short bit at its terminal end of length b. The short bit at the end of side A makes a thin slice along the width of side B (of area Ba), and the short bit at the end of side B makes a thin slice along the width of side A (of area Ab). Also at the common end of slice Ba and Ab is a little rectangle (of area ab). Drawing a picture at this point may prove helpful.

Now if the error involved in one of these measurements is so close to zero that we can comfortably ignore it then the equation of the area becomes:

AB + Ab if a<<<<0. or AB + Ba if b<<<<0

substitute 0 (for a and then for b) into the equation above and see what you get.

Usually when we deal with errors we only consider those terms that involve the greatest source of error, as the error produced by this term will usually 'include' any and all errors produce by minor errors. Remember errors are +/- factors. If the value of 'a' (the % error involved in the measurement of A) is 90%, and the value of 'b' is 1%. Then (for example if A=B=100) The Ba error would be +/- 90 while the Ab error would be +/-1. Only in two very rare possibilities would these two errors be cumulative +91 or -91. In most cases the +1 error would 'rattle around inside' the boundaries of the +90 error.

Lets consider the formula for a volume: LWD, Length times width times depth, and use the notation (A+a),(B+b),(C+c) for the measurements of the independent dimensions and their associated errors. Multiplying these measurements out using the formula for volume gives:

(AB+Ab+aB+ab) (C+c) -> ABC +ABc + AbC + Abc + aBC + aBc + abC + abc

ABC is the major volume without errors considered.

ABc, aBC, AbC are volumes over three of the surfaces (of areas AB, BC, and AC) with a thin depth of c,a and b respectively.

Abc, aBc, and abC are volumes along the three edges of the (rectangular cube) of areas bc, ac, and ab of lengths A,B and C respectively.

abc is a little rectangual cube at the terminal ends of the above mentioned edge volumes. Again a drawing might be helpful at this point.

Now if only one error is significent (lets say a) then we only consider the error terms in which 'a' is the only error. (inclusion of other errors makes the term insignificant). In this case that would be aBC. So the formula for volume would be

ABC +/- aBC

If for a moment we consider two of the errors to be equally significant (lets say a and b) then the formula and error would be

ABC +/- aBC+AbC

where any term involving c or a and b together are ignored.

If for a moment we consider all errors to be equally significant then the formula and error would be.

ABC +/- aBC+AbC+ABc

If in addition we consider the volume to be a perfect cube then we can substitute 'x' for A,B, and C and e for a,b, and c.

x3 +/- 3x2 e

This error would be the three surfaces areas of the common error depth. The three edge error volumes and the tiny error cube are being ignored as their volumes are dependent on two and three errors combined which are tiny values squared and cube which makes them insignificant indeed.

If you know calculus the error is just the 1st derivative of the employed formula with only the major term employed for the error. The argument and derivation is actually the same as that given above.