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!
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.
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.
From Wikipedia:
A fraction may also contain radicals in the numerator and/or the denominator. If the denominator contains radicals, it can be helpful to rationalize it, especially if further operations, such as adding or comparing that fraction to another, are to be carried out...
In your case: $$\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$$
"Fraction" just means "not a whole number." For instance, irrational numbers can be written as "decimal fractions", like $\pi = 3.14159\ldots.$ If you had said "rational numbers" instead of "fractions", you'd be right.
