Cordial Maths

One of the trickiest topics in school mathematics is fractions. Why can’t they just behave like familiar, friendly whole numbers? But your Maths Masters are here to help, and we’ve discovered a wonderful new way to add fractions. Here’s an example:

Much easier! And for those who would like to apply the method more generally, the formula is

At this stage you may be suspicious. So, ok, we’ll confess: this excellent method for addition is not really our invention.

The above method of “adding” fractions has been discovered and rediscovered by schoolchildren for centuries. Of course, there’s at least one problem with the method: it usually gives the wrong answer.

So, we’ll all still have to add fractions in the traditional manner, with those annoying common denominators. However, the weird addition above does turn out to have some very remarkable properties.

Here’s an interesting experiment, perfect for a sunny spring day. (So, you may have to leave Victoria). Buy some concentrated raspberry cordial and mix yourself a glass of cordial and water. Of course the more cordial you add, the redder the liquid. We’ll now consider two different mixes:

First Glass: 4 parts cordial and 7 parts water.

Second Glass: 5 parts cordial and 6 parts water.

Now create a third mix by combining the contents of the two glasses:

Third Glass: 4 + 5 parts cordial and 7 + 6 parts of water.

So, the proportions of cordial and water in the third glass are exactly given by the result our weird fraction sum. Intriguing!

Now, since 5/6 is greater than 4/7, the second glass will be redder than the first. What about the third glass? Since we’ve just combined the contents of the first two glasses, the third glass will be redder than the first but not as red as the second. That means we have a cordial-powered proof that

Very neat!

The strange sum we’ve been considering is called a mediant, the name reflecting the in between property that we’ve just observed. Of course it’s misleading to use a plus sign, and so our weird operation should be represented in some other way. Most commonly the symbol ⊕ is used. Then, for example,

However, there is something very strange about the mediant. To illustrate, first note that 5/6 obviously equals 10/12. However, if we calculate the mediant with 10/12 in place of 5/6 we find that

The two fractions 9/13 and 14/19 are definitely not equal. So, the mediant cannot be operating on the actual fractions, the numbers. Rather, the mediant is an operation on particular representations of fractions.

The mediant is definitely a peculiar creature, but it is still of genuine use. Our cordial mixing above is one illustration, but there are much more impressive applications.

Start with any positive whole number: we’ll choose 6 to illustrate. Now write down in order all the fractions with denominators at most 6; the fractions should be in lowest form, and we’ll include 0/1 and 1/1 at the beginning and end. So, starting with 6, our list of fractions is

Lists constructed in this way are called Farey sequences. These sequences have an amazing property: any fraction in a Farey sequence is the mediant of the fractions on either side. So for example, in our list above 3/5 = 1/2 ⊕ 2/3. That is very strange, and very, very cool.

It turns out that Farey sequences are much more than just weird fun. The Riemann hypothesis is perhaps the most famous and most important unsolved problem in mathematics (and is worth $1,000,000). And, the Riemann hypothesis can be expressed as a question about Farey sequences.

Amazing. And all that from a glass or two of cordial.

Puzzle to Ponder: Are there examples where the mediant of two “fractions” is equal to the sum of the two fractions?

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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