2.1.8 π and e Approximations

Using the standard approximations of: π≈3.1\pi \approx 3.1, e≈2.7e \approx 2.7, and e2≈7.4e^2 \approx 7.4 lead to the beneficial results of:

π2≈10,e3≈20,π⋅e≈8.5\pi^2 \approx 10, \quad e^3 \approx 20, \quad \pi \cdot e \approx 8.5

Knowing these values, we can approximate various powers of ee and π\pi relatively simple and within the required margin of error of ±5%. The following is an example where these approximations are useful:

(e×π)4=e4×π4=e⋅e3⋅(π2)2≈e⋅20⋅100≈e⋅2000≈5400(e \times \pi)^4 = e^4 \times \pi^4 = e \cdot e^3 \cdot (\pi^2)^2 \approx e \cdot 20 \cdot 100 \approx e \cdot 2000 \approx 5400

Problem Set 2.1.8

2π42\pi^4 *
e2×π4e^2 \times \pi^4 *
e4e^4 *
π5\pi^5 *
(e×π)4(e \times \pi)^4 *
π5+e4\pi^5 + e^4 *
π3×e4\pi^3 \times e^4 *
(3π)4(3\pi)^4 *
(e+1.3)5(e + 1.3)^5 *
[(π−.2)(e+.3)]3[(\pi - .2)(e + .3)]^3 *
(π+1.9)3(e+2.3)3\frac{(\pi + 1.9)^3}{(e + 2.3)^3} *
(4e)3(4e)^3 *
e4π4e^4\pi^4 *
πe×eπ\pi^e \times e^\pi *
(3π+2e)4(3\pi + 2e)^4 *
ππe×e\pi^{\pi e} \times e *