Chapter 1, Problem 10P

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 e_i is related to X_i and x_i as x_i = X_i + e_i

(a) Show that the error in a sum X₁ + X₂ is

$(x_1\text{ + }{x}_2)\text{ - (}{X}_1\text{ + }{X}_2\text{) = }{e}_1\text{ + }{e}_2$

(b) Show that the error in a difference X₁ − X₂ is

$(x_1\text{ + }{x}_2)\text{ - (}{X}_1\text{ + }{X}_2\text{) = }{e}_1\text{ - }{e}_2$

(c) Show that the error in a product X₁X₂ is

$x_1{x}_2\text{ - }{X}_1{X}_2\approx X_1{X}_2\left(\frac{e_1}{X_1}\text{ + }\frac{e_2}{X_2}\right)$

(d) Show that in a quotient X₁/X₂ the error is

$\frac{x_1}{x_2}\text{ - }\frac{X_1}{X_2}\approx \frac{X_1}{X_2}\left(\frac{e_1}{X_1}\text{ + }\frac{e_2}{X_2}\right)$