Write rule for error in result due to multiplication and division

Explanation

Suppose two physical quantities A and B have measured values `A+-Deltaa,B +-DeltaB` respectively. Where `DeltaA and DeltaB` are their absolute errors.
For a product : Suppose the product of A and B is Z and the absolute error in Z is `DeltaZ`.
`:.Z= AB`
`:.Z+-DeltaZ=(A+-DeltaA)(B+-DeltaB)`
`:.Z+DeltaZ=AB+-DeltaB+-BDeltaA+_(DeltaA DeltaB)`
Dividing the left side by 2 and right side by AB,
`1+-(DeltaZ)/(Z)=1+(DeltaA)/(A)+-(DeltaB)/(B)+-((DeltaA)/(A))((DeltaB)/(B))`
`(DeltaA)/(A) and (DeltaB)/(B)` are very small hence their product will be very small and can be neglected. Neglecting negative sign for maximum error
`(DeltaZ)/(Z)=(DeltaA)/(A)+(DeltaB)/(B)`
For a Quotient : Suppose the quotient of A and B is Z and the absolute error of Z is `DeltaZ`
`:.Z=(A)/(B)`
`:.Z+DeltaZ=(A+-DeltaA)/(B+-DeltaB)`
`Z(1+-(DeltaZ)/(Z))==(A(1+-(DeltaA)/(A)))/(B(1+-(DeltaB)/(B)))`
`(1+(DeltaB))/(B))^(-1)` writing first two terms of binomial expansion,
`(1+-(DeltaB)/(B))^(-1)=(1-+-(DeltaB)/(B))`
From equation (1),
`:.1+-(DeltaZ)/(Z)=(1+(DeltaA)/(A)) (1+-(DeltaB)/(B))`
`:.1+-(DeltaZ)/(Z)=1+(DeltaA)/(A)+-(DeltaB)/(B)+-((DeltaA)/(A))((DeltaB)/(B))`
But `((DeltaA)/(A)) and ((DeltaB)/(B))` are smaller, so their product will be zero by ignoring them,
`+(DeltaZ)/(Z)=+-(DeltaA)/(A)+-(DeltaB)/(B)`
For relaive errros in Z, ignoring, regative signs
`(DeltaZ)/(Z)=(DeltaA)/(A)+(DeltaB)/(B)`
Hence the rule : When two quantities are multiplied or divided, the relative error in the result is the sm of the relative errors in the multipliers.