Exercises

1. To four decimal places (5SD), `pi` is `3.1416`, and `sqrt(2)` is `1.4142`. If you divide these approximations, how many significant digits of `pi//sqrt(2)` will you get? Don't guess — do the calculation to be sure.
2. 1. Write down all the digits of the pop-up calculator's approximation to `9pi//5`.
   2. Write down a 4SD approximation to `9pi//5`.
   3. What is the correct 3SD approximation? [Hint: Look at part (a), not part (b).]
3. Suppose your bank pays quarterly interest on savings accounts at an annual rate of `2.5`%. That is, the interest added at the end of each quarter is calculated at a rate of `0.025//4 = 0.00625`.

   1. If you deposit `\$160` at the start of a year, how much money will be in your account at the end of the year?
   2. How many significant digits are there in your answer?
4. 1. Experiment with the pop-up calculator to determine the smallest positive number it can recognize and display.
   2. Similarly, determine the largest number it can recognize and display.
   3. Estimate the total number of different numbers your calculator can recognize and display. Is this number finite and moderately large, finite and enormously large, or infinite?
5. The disastrous cancellations in Checkpoint 3 actually are avoidable, but only by transforming the expression to another form. Rewrite the expression `sqrt(x^2+1)-x` by rationalizing the numerator. That is, multiply and divide by `sqrt(x^2+1)+x`, and simplify. Evaluate the resulting expression at each of the numbers in parts (a) and (b) of Checkpoint 3.