Chapter 6.4, Problem 86E

Challenge Yourself

Do the calculation of $\phi$ in Example 2 (b) for 20 iterations. What is your result?

Example 2 Approximating $\mathit{\pi },\mathit{ }\mathit{\varnothing },\mathit{ }\mathbf{a}\mathbf{n}\mathbf{d}\mathit{ }\mathit{e}$

Although $\pi$ = 3.141592654…, $\varnothing$ = 1.618033989 …, and e = 2.7182818284 … are irrational numbers and therefore have nonrepeating decimal expansions, there are methods for computing good approximations to these numbers.

b. Use your calculator to approximate $\varnothing$ as follows:

Step 1: Enter 1

Step 2: Press the 1/x or x^-1 key.

Step 3: Add 1 to the result in step 2.

Step 4: and beyond: Keep repeating steps 2 and 3—that is, invert your answer and add 1 repeatedly—until you get the first five decimals after the decimal point in the decimal expansion of $\mathit{\varnothing }.$