3.2.2 Converting Decimals

In the similar manner of how we analyzed an integer $n$ in base-10, we can took at decimals in base-10 as well. For example, let’s look at how we see .125 in base-10

$$.125 = 1 \cdot (.1) + 2 \cdot (.01) + 5 \cdot (.001) = 1 \cdot 10^{-1} + 2 \cdot 10^{-2} + 5 \cdot 10^{-3}$$

You can display this in terms of fractions as well:

$$= \frac{1}{10} + \frac{2}{100} + \frac{5}{1000} = \frac{1}{10} + \frac{1}{50} + \frac{1}{200} = \frac{20 + 4 + 1}{200} = \frac{1}{8}$$

Similar to the previous session, we can replace the powers of ten by the power of any fraction. Let’s look at converting $.321_6$ to a base-10 fraction:

$$.321_6 = \frac{3}{6} + \frac{2}{36} + \frac{1}{216} = \frac{108 + 12 + 1}{216} = \frac{121}{216}$$

Because of the complexity and calculations involved, going in the reverse direction is seldom (if ever) used on a number sense test. In addition, the test usually asks for a base-10 fraction representation (be sure to reduce!). Here are some practice problems to help you familiarize yourself with this process:

Problem Set 3.2.2