TITLE: Is $\frac{1}{\sqrt{2}}$ a fraction or not?
QUESTION [7 upvotes]: Is $\frac{1}{\sqrt{2}}$ a fraction or not? 
I just came across a Grade 8 student's class work where the teacher has given the above as an example to differentiate fractions from rational numbers. I think this is wrong fundamentally, as $\sqrt2$ is irrational while all fractions would be rational. Please advise. Thanks!

REPLY [19 votes]: The word "fraction" is used outside of the scope of rational numbers as well. In that case, it's better not to use the term "fraction" to denote a kind of number, but rather to refer to the way of writing or representing it: i.e. as a structure with a numerator and a denominator.
Loosely speaking you could say "$\pi$ is not a fraction" because it cannot be written as a quotient of integers, but it's safer to say "$\pi$ is not a rational number" where you rely on a precise definition of rational number.

Generally, you want to be able to call anything of the form
$$\frac{a}{b}$$
a fraction because then you can refer to its two characterizing components: the numerator $a$ and the denominator $b$ with $a$ and $b$ not necessarily integers but any numbers or even more general expressions. Note that you can write any number $x$, not necessarily integer, as a fraction:
$$x = \frac{x}{1}$$but we generally don't refer to "$x$" as "a fraction". We do want to be able to call something like
$$\frac{1+\sqrt{3}}{1-\sqrt{3}}$$a fraction, although it's not a rational number, because then we can say things such as "multiply numerator and denominator with $1+\sqrt{3}$". The same goes for more complicated expressions (with fractions in fractions!), or even with one or more variables, such as:
$$\frac{e^2 - \frac{\sqrt{5}}{1+\sin\frac{\pi}{7}}}{1-\frac{\sqrt{3}}{2}} \quad ; \quad \frac{e^x-\sin y}{x^2+y}$$

Summarizing:

Is $\frac{1}{\sqrt{2}}$ a fraction or not?

I think we prefer to be able to call this a fraction, with numerator $1$ and denominator $\sqrt{2}$, but then obviously we shouldn't mix "fraction" with "rational number" or use them interchangeably.

REPLY [4 votes]: I think 'fraction' is not really a well-defined mathematical term, but rather, is used to refer to a visual object which represents a number.
I would simply call anything which has a top and bottom a 'fraction', ie. anything in the form $\frac{???}{???}$.
This might include algebraic terms in it, in which case you wouldn't necessarily know if the object is rational or not. But I think most would agree that the object is a 'fraction'.
Whether something is a 'fraction' isn't a well defined property you can place on any number.
For example, you wouldn't call the object '$2\times 0.3$' a fraction, but you would for '$\frac{3}{5}$', although the two expressions are equal.

In the end, I think if the teacher said something along the lines of: a fraction is an object in the form $???\over???$ where the top is called 'the numerator', and the bottom, 'the denominator', then I believe the example shown demonstrates why not all fractions are rationals well enough.