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Question:

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