When one knows the true values

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When one knows the true values x1 and x2 and has approximations X1 and X2 at hand, one can see where errors may arise. By viewing error as something to be added to an approximation to attain a true value, it follows that the error ei, is related to Xi, and xi as xi = Xi + ei.
(a) Show that the error in a sum X1 + X2 is
(x1 + x2) − (X1 + X2) = e1 + e2
(b) Show that the error in a difference X1 − X2 is
(x1 − x2) − (X1 − X2) = e1 − e2
(c) Show that the error in a product X1 X2 is
x1 x2 − X1X2 = X1X2 (e1/X1 + e2/X2)
(d) Show that in a quotient X1/X2 the error is
x1/x2 − X1/X2 = X1/X2 (e1/X1 – e2/X2)

 
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