If measurement errors in independent quantities are small, a dependent quantity can be expanded to first order. For z=x/y, the relative error is Δz/z = Δx/x + Δy/y; higher-order error terms are neglected.

For r=(1-a)/(1+a), if the measured dimensionless a has small error Δa, find the error Δr.

In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0 s. For |x| << 1, ln (1 + x) = x up to first power in x. The error Δλ, in the determination of the decay constant λ, in s⁻¹, is :-