# Mixed fractions

Adjacency means different things depending on the context:

• $$2x$$ means $$2 \times x$$.
• $$2 \frac{2}{3}$$ means $$2 + \frac{2}{3}$$.

This results in the coincidence $$\sqrt{2 \frac{2}{3}} = 2 \sqrt{\frac{2}{3}}$$.

Adam Atkinson says that mixed fractions aren't understood everywhere, which prompted Christian Lawson-Perfect to run a survey on whether they exist[1]. At least one respondent asked "Are there really countries where this notation is not used?" and at least one respondent asked "Are there really countries where this notation is used?".

Our best information is that France, Spain, Italy and Portugal are unmixed. Though it is hard to reconcile this with mixed numbers appearing on the syllabus for a particular age range on the Portuguese Ministry of Education web page. Or with their presence in at least some Italian middle school textbooks. Note that even Italian middle school teachers who believe mixed numbers exist may not believe that negative mixed numbers exist or what they might mean if they did.

For the record,

• $$-1 \frac{1}{4}$$ means $$-\frac{5}{4}$$ in mixed countries

So saying "mixed numbers have an invisible plus in them" is not the way to explain them.

Adam Atkinson talks about Italy and mixed numbers here [2] but this only begins to convey how confusing the situation is if you ask multiple people from the same notionally unmixed country about them. He has been told _authoritatively_ that Belgium is and is not mixed and has no idea what to believe.