# Do you use mixed fractions?

I’m at the MATRIX conference in Leeds, where I’ve just been talking to Adam Atkinson. He told me that he’s trying to compile a definitive list of countries that don’t use mixed fractions.

Here’s a mixed fraction: $2 \frac{2}{3}$
And here’s a non-mixed fraction: $\frac{8}{3}$
Actually, here’s an interesting fact about that number: $2 \sqrt{ \frac{2}{3} } = \sqrt{ 2 \frac{2}{3} }$
This only makes sense if you believe in mixed fractions (and unicode character U+2062, “invisible times”)

This is going to be one of those wipe-your-bum-standing-up situations: it’s entirely possible that you can be on either side of this divide and not know the other exists. Apparently, in some countries mixed fractions just don’t exist: an integer written next to a fraction is incorrect.

So, to help Adam on his way, I thought I’d start another in our long-running series of Aperiodical Surveys. Please tell us where you live, and if mixed fractions are OK in your book.

• #### Christian Lawson-Perfect

Mathematician, koala fan, Aperiodical editor. Usually found paddling in the North Sea, or fiddling with computers.

### 7 Responses to “Do you use mixed fractions?”

1. Paul Taylor

Are we talking about just when ‘doing maths’? Or are there countries where you couldn’t even say “I’m on leave for the next 3½ days” or whatever?

For the record, some people in non-mixed-number countries might take “3 1/2” to be a product and say it was 1.5. To say “3 1/2 weeks” in Italian you would say “3 settimane e 1/2” so the query as posed doesn’t really constitute a problem. Don’t know about French, Spanish, Portuguese or Romanian. (Belgium and Switzerland also on the list, I was told at Matrix 2016). However, you can say the number “3 e 1/2” in Italian, so your question works if we remove the things there are 3 1/2 of. Maybe people would just write 3.5? This conversation tends to go along similar lines to the one involving learning there are places where “Mary”, “marry” and “merry” are homophones.

We saw some road signs near Leeds indicating towns e.g. 1 1/4 miles or 1 3/4 miles away.

There’s also a _temporal_ element to all of this. I am informed that Italy, Spain, Portugal and Switzerland used to have mixed numbers (within living memory) but no longer do. However, at least in the case of Italy we are talking about a good while ago. You might need to be 80+ now to have done them at school there. One of the Spanish contingent at Matrix said they vanished in Spain in the 60s.

I had imagined that South America might be full of countries without mixed numbers, but if it was colonized before mixed numbers were abolished in Spain and Portugal, then perhaps not.

At this point am also keen to know if France (and Belgium and Romania) _ever_ had mixed numbers. And if anywhere has switched from not having them to having them.

Note: I’ve had a few cases of people saying mixed numbers exist in (place) giving “3+1/2” as an example. I would not count it as a mixed number country if the + is obligatory. Also, some places apparently DO cover them at school as a usage found in foreign media or products, or as a thing that can be used in, say, shoe sizes but not for doing any kind of mathematical operation with. I’d count all of those as being places where mixed numbers did not exist.

2. F.M.

This question will not show the divide clearly, because I felt like answering both. This has to do with the relative size of the digits in 3½: the 1 and the 2 in ½ are much smaller than the 3 standing next to them when using these font sizes. Much less confusing than a real mixed fraction. Your illustration on the right hand side of this very page (https://aperiodical.com/2016/09/do-you-use-mixed-fractions/), the page I’m writing my comment on, shows the problem. All digits are the same size, and when a digit stands next to a fraction, a (continental) European student will most likely always tell you that this is a multiplication, much like when you omit the multiplication sign in $3x$: “$3$ times $x$” = $3\times x$, follows the same logic (and this is good!) as the one used in spoken language : “3 times 1 pencil” = 3 pencils

Therefore, the logic when you read a mixed fraction such as 3½ where all digits are the same size and if there is to be a consistent way of reading all mathematical equations is the same as when reading $3x-2$, where you perform a multiplication between $3$ and $x$. It is, after all, read: “3 1/2” “Three one halves” therefore “one half + one half + one half” = “three halves”. 3 naturally thought of as a quantifier.

The confusion arises perhaps when forcing the student to pronounce a much troubling “and” between $3$ and $\frac{1}{2}$. His/her brain therefore now has two different approaches when analyzing a juxtaposition of two quantities. In a mixed fraction, the $+$ operator is omitted, but the rest of the time, as in the expression 3y(4x-5√2) you remember that it is the $\times$ sign that’s omitted.

I shudder to think what kind of confusions can arise when students compare: $x\frac{1}{2}$ and $\frac{1}{2}x$… If they haven’t learned mixed fractions, there’s not a single doubt in their mind that both are a multiplication and yield the same result. This visually also reminds the reader of the commutativity of factors in a multiplication of two numbers.

I feel the need to add that I teach maths in France and that nowhere in this territory are mixed fractions even heard of. I came across them when a English pupil of mine showed them to me in a notebook. While some may interpret this as me defending my side of the fence, I found the concept at the time useless and the source of much confusion. Especially since a $+$ operator will solve the problem!