What do you mean by propagation of errors? Explain the propagation of errors in addition and multiplication.

Explanation

In the final calculation of an experiment a number of measured quantities may be involved. Different types of instruments might have been used for taking readings, which have different degrees of accuracy.
The net error in the final result depends on
(i) The errors occurred in the individual measurements
(ii) On the nature of mathematical operations
performed to get the final result.
Error in the sum of two quantitiees. Let A `DeltaA` and `DeltaB` be the absolute errors in the two quantities A and B respectively. Then,
Measured value of `A= A pm DeltaA`
Measured value of `B=Bpm DeltaB`
Consider the sum, `Z=A+B`
The error `DeltaZ` in Z is then given by
`Zpm DeltaZ=(Apm DeltaA)+(Bpm DeltaB)`
`=(A+B)pm(DeltaA+DeltaB)`
`=Zpm(DeltaA+DeltaB)(or)`
`DeltaZ=DeltaA+DeltaB`
Error in the product of two quantities : Let `DeltaA and DeltaB` be the absolute errors in the two quantities A, and B, respectively. Consider the product `Z=AB`
The error `DeltaZ` in Z is given by
`Z pm DeltaZ=(Apm DeltaA)(B pm DeltaB)`
`=(AB)pm (A DeltaB)pm (B DeltaA)pm (DeltaA. DeltaB)`
Dividing L.H.S by Z and R.H.S. by AB, we get,
`1pm (DeltaZ)/(Z)=1pm (DeltaB)/(B)pm (DeltaA)/(A)pm (DeltaA)/(A).(DeltaB)/(B)`
As `DeltaA//A, DeltaB//B` are both small quantities, their product term `(DeltaA)/(A).(DeltaB)/(B)` can be neglected. The maximum fractionl error in Z is
`(DeltaZ)/(Z)=pm((DeltaA)/(A)+(DeltaB)/(B))`