However, I feel like it’s something you may perhaps find interesting enough to read this post. So, I’m going to teach you! Take a seat.
Prerequisite information
There are two concepts in math called “Sequences” and “Series.” Without going into so much detail as to bore you, I’ll give you a quick working definition just to get the point across.
Sequence: A Sequence is a list of things (usually numbers) that are in order.
Example:
5,10,15,20,25,30,35,40,45,50, . . . 5*(n-1), 5*(n), 5*(n+1) . . .
Essentially, it’s a pattern that continues on to ‘infinity’. The previous sequence can be defined as 5*n with n being the term of the sequence. The first term, as you can see, is 5*1 = 5. The second term is 5*2 = 10. The 100th term, then, would be 5*100 = 500.
Series: A Series is the summation of the first infinite terms in a Sequence. (if we take every term in the sequence to infinity and add them together, what would it equal?)
Example:
If we were to write this out it would look something like this:
The real question is, "Infinity is impossible to reach, so how in the world do we add all of these terms together?"
Fortunately for us, this is a well-studied and commonly used series known as a Geometric Series. In fact, it's exactly what we're going to use to turn decimals into fractions!
Geometric Series:
Where:
a is the first term (what would the value of the Sequence be if we plugged in n=1?)
r is the "common ratio" (how is the value changing between terms?)
The idea is that we can calculate the sum of this special series with the equation a/(1-r) where a and r are the previously defined terms. Here is a quick example:
Application of information
Whew, now that we got the dirty work out of the way!
Ex 1) Represent 0.19191919191919. . . as a fraction
First of all, how can we strategically represent this number? How about:
It should be obvious, then, that the first term is what? (you guessed it!) 19/100
And the common ratio (how the term is changing) is what? (of course!) 1/100
And, for the sake of verification:
https://www.google.com/search?q=19%2F99 ... 2&ie=UTF-8
Yay!
Ex 2) Represent 0.153215321532. . . as a fraction
Based off of the previous example, we have a really good idea of where to start for this one
Similarly, we can see that the first term is (1532/10000) and the common ratio is (1/10000).
Verification
Your turn!
By now, perhaps you've noticed a pattern? How about you try doing the following:
Represent 9.720572057205. . . as a fraction (proper or improper)
I hadn't tried experimenting with other ways to get the same results. Interesting! Mathgasm!
