Take a look at an example of rewriting a repeating decimal as a fraction and see how multiplying by a power of 10 helps.
See solution.
Practice makes perfect
We are asked why we multiply by a power of 10 when writing a repeating decimal as a rational number. Let's rewrite an example repeating decimal as a fraction. We will start with 0.2. First, we will assign a variable x to our repeating decimal.
x =0.2
Next, we will multiply each side of the equation by 10^d, where d is the number of repeating digits in the repeating decimal. Since d=1 in our example, we will multiply both sides of our equation by 10.
10x =2.2
Now we will subtract equivalent expressions of the variable and the repeating decimal from each side of the equation. From the left-hand side we will subtract x and from the right-hand side we will subtract 0.2.
10x - x =2.2 -0.2
Notice that this step lets us subtract all the repeating digits from the number 2.2. This means that we now have a non-repeating decimal on the right-hand side of the equation!
9x =2 ⇔ x = 2/9
As we can see, multiplying by a power of 10 lets us have a number that is greater than our repeating decimal on the right-hand side of the equation. Then, after subtracting the repeating decimal from both sides, we are left with a non-repeating decimal on the right-hand side of the equation.
