Question

How errors propogate in difference when x= a- b
∆x=(∆a+∆b)
∆x=(∆a-∆b)

Answer 1

Suppose a result x is obtained by subtraction of two quantities say a and b  

i.e.  x = a – b

Let Δ a and Δ b are absolute errors in the measurement of a and b and Δ x be the corresponding absolute error in x.

∴ x  ± Δ x  = ( a ± Δ a) – ( b ± Δ b)

∴ x  ± Δ x  = ( a – b ) ±  Δ a  – + Δ b

∴ x  ± Δ x  = x   ±  ( Δ a  + Δ b)

∴ ± Δ x  = ±  ( Δ a  + Δ b)

∴ Δ x   = Δ a  + Δ b

Thus the maximum absolute error in x  = maximum absolute error in a + maximum absolute error in b.

Thus, when a result involves the difference of two observed quantities, the absolute error in the result is equal to the sum of the absolute error in the observed quantities.

Propagation of Errors in Product:

Suppose a result x is obtained by the product of two quantities say a and b

i.e.  x = a × b    ………..  (1)

Let Δ a and Δ b are absolute errors in the measurement of a and b and Δ x be the corresponding absolute error in x.

∴ x  ± Δ x  = ( a ± Δ a) x ( b ± Δ b)

∴ x  ± Δ x  =  ab  ±  a Δ b ±  b Δ a ± Δ aΔ b

∴ x  ± Δ x  =  x  ±  a Δ b ±  b Δ a ± Δ aΔ b

∴ ± Δ x  =  ±  a Δ b ±  b Δ a ± Δ aΔ b ……  (2)

Dividing equation (2) by (1) we have



The quantities Δa/a,  Δb/b and  Δx/x are called relative errors in the values of a, b and x respectively. The product of relative errors in a and b i.e. Δa × Δb is very small hence is neglected.



Hence maximum relative  error in x = maximum relative error in a + maximum relative error in b

Thus maximum %  error in x = maximum % error in a + maximum % error in b

Thus, when a result involves the product of two observed quantities, the relative error in the result is equal to the sum of the relative error in the observed quantities.

Propagation of Errors in Quotient:

Suppose a result x is obtained by the quotient of two quantities say a and b.

i.e.   x = a / b    ………..  (1)

Let Δ a and Δ b are absolute errors in the measurement of a and b and Δ x be the corresponding absolute error in x.



The values of higher power of Δ b/b are very small and hence can be neglected.



Now the quantity  (Δ aΔ b / ab)is very small. hence can be neglected.