Exercise 16 Page 12

A line over 3 indicates that the number 3 repeats. Similarly, a line over 123 indicates that these three numbers repeat. Note that we could also rewrite them with more digits under the line. &0. 333333 ... = 0.3 = 0.333 &0. 1 2 3 1 2 3 1 2 3 ... = 0. 1 2 3 = 0.1 2 3 1 2 3 &0.2 52 52 52 5 ... = 0.2 5 = 0.2 5 However, we could not rewrite the second and third decimal as follows. &0. 1 2 3 1 2 3 1 2 3 ... ≠ 0.1 2 3 1 2 &0.2 52 52 52 5 ... ≠ 0.2 52 Let's take the first repeating decimal. We know that two options for writing this are 0.3 and 0.333. Let's rewrite each of the two forms as a fraction. Because we know they represent the same decimal, we can assign variable x to both of them.

Now, we will multiply each side of each equation by 10^d, where d is the number of repeating digits in the repeating decimal. In our case, we will multiply the first equation by 10^1 and the second one by 10^3.

Next, we will subtract equivalent expressions of the variable and the repeating decimal from each side of each equation. From the left-hand sides we will subtract x. From the right-hand side of the first equation, we will subtract 0.3 and from the right-hand side of the second equation we will subtract 0.333. Then, we will solve each equation for x.

Note that if we divide the numerator and the denominator of the solution of the second equation by 111, we will get 39. x = 333/999 = 3/9 We can see that we ended up with the same fraction that represents x. The first time we used one repeating digit, and the second time three repeating digits. Therefore, the number of repeating decimals doesn't matter, because we will always end up with the same fraction.