Periodic numbers

An infinitely repeating number: $(1.2\dot{3}4\dot{5})=(1.2\bar{3}4\bar{5})=1.2345345\dots$ Some numbers can be written in two ways: $(1.2\dot{6}{)}_7=(1.3{)}_7$ or $(0.\dot{9}{)}_{10}=(1.0{)}_{10}$.

Converting Periodic Numbers into a Fraction §

Converting a periodic (or repeating) decimal into a fraction involves a series of steps. Here’s a step-by-step guide:

1. Define the Number: Let’s say you have a repeating decimal like $0.\dot{6}$ (which means 0.6666…). Let this number be represented by $x$.

   So, $x=0.\dot{6}$

2. Multiply by a Power of 10: Multiply both sides of the equation by a power of 10 that shifts the decimal point just past the repeating part. In this case, we’ll multiply by 10.

   $10x=6.\dot{6}$

3. Set Up an Equation: Subtract the original number from the number you got in step 2. This will eliminate the repeating part.

   $10x-x=6.\dot{6}-0.\dot{6}$

   $9x=6$

4. Solve for x: Now, solve for $x$.

   $x=\frac{6}{9}$

   Simplify the fraction:

   $x=\frac{2}{3}$

So, $0.\dot{6}$ can be represented as the fraction $\frac{2}{3}$.

Another Example: For a number like $0.8\dot{3}$ (which means 0.8333…):

1. Let $x=0.8\dot{3}$

2. Multiply by a power of 10 that shifts the decimal point past the repeating part. Since only the 3 is repeating, we’ll multiply by 100:

   $100x=83.\dot{3}$

3. Subtract:

   $100x-x=83.\dot{3}-0.8\dot{3}$

   $99x=82.5$

4. Solve for $x$:

   $x=\frac{82.5}{99}$

   Simplify the fraction:

   $x=\frac{165}{198}$ $x=\frac{55}{66}$

So, $0.8\dot{3}$ can be represented as the fraction $\frac{55}{66}$.

This method can be applied to any repeating decimal, even if the repeating part is more than one digit long. The key is to choose the right power of 10 to multiply by in step 2. If two digits are repeating, multiply by 100. If three digits are repeating, multiply by 1000, and so on.

This can also be for any base periodic number, where you’d replace 10 with the base, $b$.