Decomposing fractions calculator software#

Doing so to and results inĭoing this is relatively simple by hand, but you can also use spreadsheet software to speed things along. We can repeat this to create as many decompositions as we like. This allows for decomposition into two unit fractions, as follows:įor example, for the unit fraction we have which can be factored as. Assume that the unit fraction’s denominator can be factored as, where and are allowed to be 1. Let’s look at how a unit fraction can be decomposed into two different unit fractions. We can generalise our methods for decomposition and composition into five rules. Since we know that we will be dealing with unit fractions, we can save space by omitting the unit numerators, writing fractions like, , as, ,Ĭlearly we can make sums of unit fractions adding up to 1 longer by decomposing each individual unit fraction into a sum of unit fractions. So let’s leave perfect numbers to one side and take another approach. For example 496 is out because two of its divisors have three digits: While we could similarly decompose these numbers as above, the condition that each denominator must be 99 or less limits which ones we can use. As an example, 6 is a perfect number (as you can see above), as is 28 (with divisors 14, 7, 4, 2, and 1):ĭividing both sides by 28 results in a decomposition of 1 into a sum of unit fractions with five terms: Starting with perfect numbersĪ natural number is called a perfect number if the sum of its divisors (including 1, but excluding ) equals itself. On, so the motivated reader may wish to take a stab at it before reading on and then compare our solutions.

$decomposing fractions calculator$

In this article I would like to show how I found my 42-term solution.

$decomposing fractions calculator$

Number of terms, so for all I know a 43-term solution awaits. What I did not accomplish was a proof that 42 was the largest possible Of decomposition lead to different solutions.

Decomposing fractions calculator trial#

I decided to restrict myself to unit fractions with two-digitĭenominators, in other words, natural numbers no greater than 99.īy trial and error I managed to find a sum of 42 terms that met the conditions. It's a question I began thinking about around 20 yearsĪgo when I started work on a problem involving unit fractions.

$decomposing fractions calculator$

How long can we make such a sum without repeating any of the unit

$decomposing fractions calculator$