Explain error of a sum or a difference.

Let two physical quantities $\mathrm{A}$ and $\mathrm{B}$ has measured value $\mathrm{A} \pm \Delta \mathrm{A}$ and $\mathrm{B} \pm \Delta \mathrm{B}$ where,  $(i)$ For addition :

Let $Z$ is quantity obtained by addition of $A$ and $B$.

$\therefore \mathrm{Z}=\mathrm{A}+\mathrm{B}$

Let error is $Z$ be $\Delta Z$

$Z \pm \Delta Z=(A \pm \Delta A)+(B \pm \Delta B)$

$A+B=Z$

$\therefore \pm \Delta Z=\pm \Delta A \pm \Delta B$

For maximum absolute error,

$\Delta Z=\Delta A+\Delta B$

$(ii)$ For Subtraction :

Let difference of $A$ and $B$ is $Z$

$\therefore Z=A -B[\text { Let } A>B]$

$Z \pm \Delta Z =(A \pm \Delta A)-(B \pm \Delta B)$

$=(A-B)-(\pm \Delta A \pm \Delta B)$

$A-B =Z$

$\pm \Delta Z =\pm \Delta A \pm \Delta B$

$For maximum error in $Z$,$

$\Delta Z=\Delta A+\Delta B$