How Do Length Errors Propagate Through Different Operations?

[Pixter]

Right having a few labs at the moment and just wanted to get a clarificating conserning errors.

if i have a length error which is 2+-1... if i add two length or subtract two length, what do i do with the error?
the same if i multiply them och divide them...

and then finally if i sqaure,kube or take the sqrt of them..?

it's not that important could probably ask a tutor but have not lectures today so don't have to get the bus to the uni then =).

 

[Hypercase]

for addition and subtraction add the absolute errors.
For division and multiplication add the percentage errors.
for cubes and squares etc. multiply the percentage error by the power.

 

[HallsofIvy]

Hypercase gave very good "rules of thumb". They are not exactly right but very, very accurate for small errors (surely you can do better than "2+-1"!) and far easier than being precise.

If you really want to be precise (of course, your "errors" won't be very precise anyway!) you could do this:
Suppose x is measured at 2+- 0.01, y is measured at 3+- 0.02.
That means that the true value of x is somewhere between 2-0.01= 1.99 and 2+0.01= 2.01. The true value of y is somewhere between 3- 0.02= 2.98 and 3+0.02= 3.02. The largest possible value for x+ y is 2.01+ 3.02= 5.03.
The smallest possible value of x+y is 1.99+ 2.98= 4.97. x+y is somewhere between 4.97 and 5.03: 5+- 0.03. Exactly what Hypercase said: the errors added.
Notice, by the way, that if you subtract the errors do not subtract! They still add. The largest x- y could possibly be is 2.01- 2.98= -0.97, the smallest is 1.99- 3.02= -1.03: x-y is -1+- 0.03- the error is still the sum of the errors.

Now what about xy? The largest xy could be is 2.01*3.02= 6.0702= 6+ 0.0702. The smallest is 1.99*2.98= 5.9302= 6-0.0698. A bit more complicated! To use Hypercases's suggestion, find the percentage error (also called "relative error") in x: 0.01/2 and in y: 0.02/3.
Adding those, the percntage error in xy is 0.03/6+ 0.04/6= 0.07/6 and the absolute error is 6(0.07/6)= 0.07: xy is 6+- 0.07 which, while not exact, is pretty darn accurate.
As Hypercase said, since powers are "repeated multiplication" you do a "repeated addition", i.e. multiplication, of the percentage error.