Homework for the tenth week: Irrationals and decimals

Notes also available as PDF.

Practice is absolutely critical in this class.

Groups are fine, turn in your own work. Homework is due in or before class on Mondays.

- Problem set 6.4:
- 15, 16
- 25-28 (Note: If you write these as {a}^{{1\over 2} }, you can use the rule {(ab)}^{k} = {a}^{k}{b}^{k} and the factorization of each number to simplify the expressions. That summarizes the text’s examples.)
- 41-44 (Again, think of these as fractional exponents. Use the factorizations of each number under the square root, and then use the distributive property to pull out common factors.)
- 49
- 81

- Problem set 6.3:
- 75, 76, 79, 80 (calculators are fine, but correctly denote what repeats)
- 86, 87, 88
- 95, 96

- Problem set 6.5:
- 1-5

- On rounding and floating point arithmetic:
- Round each of the following to the nearest tenth (one place after the
decimal) using round to nearest even, round to zero (truncation),
and round half-up:
- 86.548
- 86.554
- 86.55

- Compute the following quantities with a computer or a calculator.
Write what type of computer/calculator you used and the
software package if it’s a computer. Compute it as shown.
Do not simplify the expression before computing it, and do not
re-enter the intermediate results into the calculator or computer
program. Also compute the expressions that do not include
1{0}^{16}
by hand exactly. There should be a difference between the exact result and
the displayed result in some of these cases. Remember to work from the
innermost parentheses outward.
- (
\overbrace{0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1} 10\text{ times} ) - 1 - (\kern 1.66702pt (
, where 1{0}^{16} often is entered as 1\text{e}16. If the result overflows (signals an error) on various calculators, replace 1{0}^{16} by 1{0}^{8} in this and later portions.\overbrace{0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1} 10\text{ times} )\kern 1.66702pt - 1) × 1{0}^{16} - (\kern 1.66702pt (\kern 1.66702pt (2 ÷ 3)\kern 1.66702pt - 1)\kern 1.66702pt × 3)\kern 1.66702pt + 1
- (\kern 1.66702pt (\kern 1.66702pt (\kern 1.66702pt (2 ÷ 3)\kern 1.66702pt - 1)\kern 1.66702pt × 3)\kern 1.66702pt + 1)\kern 1.66702pt × 1{0}^{16}
- (\kern 1.66702pt (\kern 1.66702pt (6 ÷ 7)\kern 1.66702pt - 1)\kern 1.66702pt × 7)\kern 1.66702pt + 1
- (\kern 1.66702pt (\kern 1.66702pt (\kern 1.66702pt (6 ÷ 7)\kern 1.66702pt - 1)\kern 1.66702pt × 7)\kern 1.66702pt + 1)\kern 1.66702pt × 1{0}^{16}

The object of this first part is to demonstrate round-off error. The first to problems, adding 0.1 repeatedly, may see no error if the device calculates in decimal. The latter four parts should see some error regardless of the base used.

- (
- Now copy down the number displayed by the first
calculation in each of the following. Re-enter it as
x
in the second calculation.
- 1 ÷ 3, then 1 ÷ 3 - x where x is the number displayed.
- If you have a calculator or program with π, π, then π - x where x is the number displayed.

The object here is to see that the number displayed often is not the number the computer or calculator has stored.

- Round each of the following to the nearest tenth (one place after the
decimal) using round to nearest even, round to zero (truncation),
and round half-up:

Note that you may email homework. However, I don’t use Microsoft^{TM} products
(e.g. Word), and software packages are notoriously finicky about translating
mathematics.

If you’re typing it (which I advise just for practice in whatever tools you use), you likely want to turn in a printout. If you do want to email your submission, please produce a PDF or PostScript document.