Chase Maier
Technical Lead & Senior Software Engineer
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Anomalous Cancellation Calculator
The was a personal project done individually and voluntarily while I was taking my first mathematics course, Honors Calculus I (Math 152), at the University of Montana. As part of this class we had a discussion about the anomalous cancellation of fractions. As we know, when attempting to simplifying a multiplication of fractions, it is possible and mathematically valid to cancel common factors from the numerators and denominators of these fractions. Unfortunately, when learning this method some students do not fully understand the restrictions placed on this process and will attempt to cancel out individual digits which make up a number.
For example, in order to try to simplify the fraction 26⁄65 these students will attempt to cancel common digits. In this case the digit 6 is common and the cancelation would look like 26⁄65 = 2⁄5 = 0.4. When we inform this student that their technique was incorrect and show them that the correct way to solve this problem is to identify 13 as a common factor of both the numerator and denominator and simply, we find that 26⁄65 = 2⁄5 = 0.4 (the same as the result generated by the student).
This technique has been shown on two other fractions shown to the right. If you do the math you can confirm that this technique in fact acheived correct results for both. Can the cancelation of digits be a valid mathematical technique to simply fractions in general? The short answer: No. It is very easy to show that this technique only works on very special cases.
Nevertheless, our instructor wanted to know in what cases it does actually work. Is there a rule or technique we can use to identify these fractions where this anomalous cancellation of digits actually yields a valid simplification? I realized that having a list of such fractions to examine would help to identify patterns and draw conclusion. Shortly thereafter I created this application which, given an integer to serve as an upper bound as input, would generate a complete list of all fractions where anomalous cancellation resulted in a valid simplification such that the numerator and denominator of the fractions was between 1 and the defined upper bound. Several examples of this output can be seen in the media gallery on this page. As this was more of a proof of concept application I did not restrict the output to proper fractions but this would have been a very simple adjustment to the algorithm and, looking back at the project, would have increased the performance of the search drastically.
It turns out that the instructor who posed these questions did not actually expect anyone in our class to be able to generate such a list as a brute force search is the only known way to solve this problem. I was able to share this application with the instructor and the class which allowed for an expanded discussion on the topic.
