Obtaining the Fractional Form of a Repeating Decimal

I was out for a run recently, and I noticed I had forgotten how to convert a repeating decimal into a fraction; as a reminder, a repeating decimal is a decimal which has a set of digits that repeat infinitely. So I began by attempting to solve the simple repeating decimal of 0.333..., since I know what I'm working towards (1/3).

Typically, such problems are presented in an academic environment where people are told exactly how to obtain it without experimenting. In daily problems we encounter, we typically do not have guidance as in a classroom, so we must experiment and observe while striving towards an answer. This is a common method of problem solving in the real world: We take a guess, try it, see if it works, and repeat if necessary until we arrive at an answer (exact or of varying precision). This is similar to a numerical methods approach, as when attempting to find the zeros of a function. It was this guess and test approach that I used to obtain a solution.

I first began by introducing something new, a variable, x, and letting x equal the repeating decimal. So I had one unknown with one equation, but I needed to work towards the fractional form.

With x=.3, I decided that perhaps manipulating both sides would be helpful. The first thing that came to mind was logarithms; I had been reading something concerning logarithms, so that was probably why I thought of it. Specifically, I thought of the natural logarithm, ln (from the Latin logaritmus naturalis); that is, the logarithm with base e, an irrational number. But seeing that I didn't get anything useful out of it, I tried the common logarithm with base 10. For a moment I thought I could get somewhere since I could write the equation as x = 3*(1/10 + 1/100 + 1/000 + ...), but I realized I was making a mistake thinking I could use the logarithm to move towards an answer.

I finally thought after a few days (I was only thinking of this during running sessions) that I should try multiplying both sides by a number. A number that came to mind was 10, after which I immediately realized this provided a path towards a solution.

When you multiply both sides of x = 0.333... by 10, the result is 10x = 3.333 (we'll call this previous equation 1, and the latter equation 2). We can think of these as two separate equations. Since equation 1 states that x is equal to 0.333..., I can subtract both sides of equation 2 by either term. Choosing to keep like terms together, I obtain 10x-x = 3.333... - 0.333.... This simplifies to 9x=3; because equations 1 and 2 have an infinite number of digits for both equations, we can say with certainty that the subtraction will eliminate an infinite number of digits. This leaves us with simple algebra that results in x=1/3.

QED