Is there a term for a number $\in (0, 1)$?

Question

ELU's answers aren't sufficiently specific, as decimals/(decimal) fractions (e.g. 222/100) can $\not\in (0, 1)$

Answer 1

A fraction with absolute value less than $1$ (that is, less than $1$ and more than $-1$) is called a proper fraction.

So, the numbers you mentioned would be called positive proper fractions.

Answer 2

In mathematics, the term "fraction" is wrong, as you said, because the fraction can be $\ge 1$ (or $\le 0$ for that matter).

In English language, in colloquial use, "fraction" is often meant to be a synonym with "a small part of", implicitly in $(0,1)$. Have a look at: http://www.dictionary.com/browse/fraction, any meaning apart from the scientific ones.

As the linked question is about English language, I would argue that the answer "fraction" is right, unless that word is used in mathematical context. If you say "Only a fraction of people I invited actually turned up", it means a small proportion, it cannot mean, for example, $\frac{100}{100}=100\%$ of people turned up. However, in a mathematical text this would be a gross imprecision, and, indeed, as the other answers say, you would need to say "positive proper fraction".