Two quantities, whose true values are X and Y, are known to have possible errors of ? _x and ? _y. If the two quantities are to be added, what will be the error in the sum? If X and Y are the true values, then the measured values will be X + ? _x and Y + ? _y, bearing in mind that the errors could be positive or negative.

The conclusion drawn from this is that when quantities with known errors are added or subtracted, the actual errors are added. Note that errors are always added, never subtracted, even if the quantities to which the errors belong are subtracted.

If the two quantities X and Y are to be multiplied, then

Since ? _x and ? _y are hopefully small compared with X and Y, for the purposes of this calculation, their product ? _x ? _y can be ignored, Then,

The two factors within the brackets are the fractional errors of X and Y Their sum is the fractional error in the product XY. The sum of the fractional errors multiplied by the product XY, as shown in the expression, will be the actual potential error in the product XY, in whatever units X and Y are measured.

The development for the quotient of X over Y produces the same result. Accordingly, the rule is: The potential fractional error of the product...