## Tuesday, September 22, 2015

### For the Monday-Wednesday class: Notes for September 21 and 23Also the notes for September 29 for the Tuesday Thursday class

Scales based on other powers of 10 besides 100 (percent) and murder rates per 100,000.

Fractions, decimals and percents.

Changing repeating decimals to fractions in lowest terms.

Approximating and simplifying square roots.

Rounding decimals.

Let's consider 3/7. If you type 3/7 into the calculator, you'll get 0.428571429. If we wanted to write the fraction with a bar over the repeating part, the answer would be
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.428571

Usually, fractions like this are written as decimals rounded to the nearest tenth, hundredth or thousandth.  Let's do all three versions here.

3/7 rounded to the nearest tenth.
Step 1: Erase everything after the first digit, which gives us .4
Step 2: Because the left most digit we erased is a 2, we leave the .4 alone.
3/7 ~= .4 rounded to the nearest tenth.

3/7 rounded to the nearest hundredth.
Step 1: Erase everything after the second digit, which gives us .42
Step 2: Because the left most digit we erased is an 8, we change .42 to .43, rounding up.
3/7 ~= .43 rounded to the nearest hundredth.

3/7 rounded to the nearest thousandth.
Step 1: Erase everything after the third digit, which gives us .428
Step 2: Because the left most digit we erased is a 5, we increase .428 to .429.
3/7 ~= .429 rounded to the nearest thousandth.

Rounding error.

If we multiply 7 × 3/7, we get 3 exactly.  If we multiply 7 by a rounded version of 3/7, we won't get exactly 3. If we rounded the approximation off, when we multiply we will be below the target. If we round the approximation up, we will be above the target.

7 × .4 = 2.8, so we are 0.2 below 3.

7 × .43 = 3.01, so we are 0.01 above 3.

7 × .429 = 3.003, so we are 0.003 above 3.