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

Explanation

A number of measured quantities may be involved in the final calculation of an experiment. Different types of instruments might have been used for taking readings. Then we may have to look at the errors in measuring various quantities, collectively.
The error in the final result depends on
(i) The errors in the individual measurements.
(ii) On the nature of mathematicla operations performed to get the final result. So we should know the rules to combine the errors.
These type of combination of errors are known as Propagation of errors.
(i) Error in the sum of two quantities:
Let `DeltaA` and `DeltaB` be the absolute errors in the two quantities A and B respectively. Then
Measured value of `A=+-DeltaA`
Measured value of `B=B+-DeltaB`
Consider the sum `Z=A+B`
The error `DeltaZ` in Z is then given by
`Z+-DeltaZ=(A+-DeltaA)+(B+-DeltaB)`
`=(A+B)+-(DeltaA+DeltaB)`
`=Z+-(DeltaA+DeltaB)`
(or) `DeltaZ=DeltaA+DeltaB`
The maximum possible error in the sum of two quantities is equal to the sum of the absolute errors in the individual quantities.
(iii) Error in the difference of two quantities:
Let `DeltaA` and `DeltaB` bet eh absolute errors in the two quantities A and B, respectively. Then,
Measured value of `A=A+-DeltaA`
Measured value of `B=B+-DeltaB`
Consider the difference `Z=A-B`
The error `DeltaZ` in Z is then given by
`Z+-DeltaZ=(A+-DeltaA)-(B+-DeltaB)`
`=(A-B)+-DeltaA+DeltaB`
`=Z+-DeltaA+DeltaB`
(or) `DeltaZ=DeltaA+DeltaB`
The maximum error in difference of two quantities is equal to the sum of the absolute errors in the individual quantities.
(iii) Error in teh 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`
`=(AB)+-(ADletaB)+-(BDeltaA)+-(DeltaA.Deltab)`
Dividing L.H.S by Z And R.H.S by AB we get
`1+-(DeltaZ)/Z=1+-(DeltaB)/B+-(DeltaA)/A+-(DeltaA)/A . (DeltaB)/B`
As `Delta//A,DeltaB//B` are both small quantities, their product term `(DeltaA)/A . (DeltaB)/B` can be neglected. The maximum fractional error in Z is
`(DeltaZ)/Z=+-((DeltaA)/A+(DeltaB)/B)`
The maximum fractional error in the product of two quantities is equa to the sum of the fractional errors in the indvidual quantities.