Old fractions
I was reading the book `The Man of Numbers` by Keith Devlin which is about the life of Fibonacci and how he introduced/popularised the Hindu-Arabic Numerals to Europe from Arabian/North African traders. The book isn't amazing but one interesting part was a section on fractions in Leonardo of Pisa's time. They would've written fractions like so:
If we translate that to our modern way of writing fractions this is equal to . This seems a bizarre way of doing it but let's say you were writing down a length using the imperial units with these conversion units.
- 12 inches = 1 foot
- 3 feet = 1 yard
- 22 yards = 1 chain
- 10 chains = 1 furlong
- 8 furlongs = 1 mile
Our length might be 2 miles, 3 furlongs, 4 chains, 15 yards, 1 foot and 5 inches. We could write that using the fractions as the following in miles: . Note that it goes from right to left, something the book posits might be a consequence of coming from Arabic which is written in that way. Or we could write a length of time like so in terms of days instead of writing 2 days, 5 hours, 36 minutes and 23 seconds. This would've been exceptionally useful for when measurements were much less standardised across the world and even coins/money weren't decimalised; using 12 denaris to 1 solidus, 20 solidi to 1 libra in medieval Pisa.
Even our system can be written in this way; so we could write the square root of 2: . Note that another way of doing this is using a recursive fraction. So we could write it like . This is different from a continued fraction which has the recursion on the denominator but we can write it like that in this way: Note that it goes on a while but since I'm only using an approximation being more precise wouldn't be meaningful. The algorithm for calculating this can be found on the Wikipedia page.
It is useful to note how much the notation we use to represent a thing tends to drive the way we think about the thing itself. Numbers don't come pre-ordained with labels saying what they should be; those labels are assigned by us and we can use whichever ones we choose to. The standard ones currently used across the globe have been constructed and refined over millenia. Europe used Roman Numerals for centuries until the rise of trade across the Mediterranean made the advantages of Hindu-Arabic numerals for calculations and storage especially for larger numbers irresistible. It also had the advantage of making the language of mathematics more abstract which makes it more generic and more powerful, enabling future advancements.
To experiment with this fraction system and thinking about it; we could construct the following infinite fraction with infinitely many halves: . This would just be which is one of the first series used to demonstrate converging infinite series; it is equal to 2. An interesting way to show this is using the recursive fraction from earlier, we could wonder what is the value equal to the following: . If we expand this out, we can get the following: , and if we continue we see this is equal to the infinite sum from before. Then we can get the value of the sum by the following: