This blog post serves as an exercise and solution to the following question:
In plain English: is the fraction a repeated decimal (0.142857142857142857…)?
How can we tackle such a problem? First note that we have to deal with a recurring pattern: a pattern that refers to itself. One useful concept to model such problems is the concept of sequences. We can model the repeated decimal by the following sequence:
Are you interested in learning more Python? Order our new "Mastering Pandas" course now on Data Blogger Courses for only
- Learn to visualize data using Pandas
- Learn how to load and store data effectively
- Learn advanced data operations
How does that work? How does this sequence approximate the repeated fraction? First, take a look at . . Now, take a look at the following item in the sequence: . So, it concatenates “142857” and the recurring pattern constructed so far. By that property we know that the sequence equals the repeated decimal if approaches .
We would like to proof that . This is equal to proving that . This equals and this is equal to proving that .
Lets introduce another sequence:
Notice that the statement we need to proof is the same as the following:
Now notice that:
And thus we know that:
Thus (for ):
From this, we get that:
And from this we can deduce the earlier statement, thus:
Thus, is indeed equal to the repeated fraction ! If you have any questions or suggestions, feel free to post them below!
Help building the Data Blogger CommunityHelp to grow our community to spread AI and Data Science education around the globe.
Every penny counts.