Consider the ratio r = (1-a)/(1+a) to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is Δa (Δa/a << 1), then what is the error Δr in determining r?
2Δa(1+a)²
2aΔa(1-a²)
Δa(1+a)²
2Δa(1-a²)
Solution:
r = (1-a)/(1+a) Δr = Δ(1-a)/(1+a) + Δ(1+a)/(1+a) = -Δa/(1+a) + Δa(1-a)/(1+a)² Since Δa/a <<1, Δa is small, so we can approximate: Δr ≈ -Δa/(1+a) + Δa/(1+a) = 2Δa/(1+a)²