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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!

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    $\begingroup$ It would be good if you define what fraction means to you. $\endgroup$
    – user370967
    Jul 6, 2017 at 12:49
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    $\begingroup$ From en.wikipedia.org/wiki/Fraction_(mathematics): 'the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions)' $\endgroup$
    – Shuri2060
    Jul 6, 2017 at 12:51
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    $\begingroup$ it is fraction but not a rational number,as you have mentioned $\endgroup$
    – haqnatural
    Jul 6, 2017 at 12:51
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    $\begingroup$ People, people, let's all calm down. None of this matters because differentiating any constant will result in zero. $\endgroup$
    – user307169
    Jul 6, 2017 at 13:09
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    $\begingroup$ $1/{\sqrt 2}$ is not a fraction over the ring of integers $\,\Bbb Z\,$ but it is a fraction over the ring $\,\Bbb Z[\sqrt 2],\,$ the numbers of form $\,j+k\sqrt 2\,$ for integers $j,k.\ \ $ $\endgroup$ Jul 6, 2017 at 15:51

4 Answers 4

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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.

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    $\begingroup$ +1 This answer implicitly and correctly makes the point that there is no universally accepted meaning for the term - so both the student's teacher and the OP are wrong to insist on a "right definition". . $\endgroup$ Jul 6, 2017 at 13:07
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    $\begingroup$ @EthanBolker - While I agree with your point, don't be too quick to call the teacher wrong. We don't know the context of the teacher's comments, and the teacher has both the need and the right to determine the terminology that will be used in class and to explain the issues with other terminology $\endgroup$ Jul 6, 2017 at 16:25
  • $\begingroup$ @PaulSinclair I agree - if the teacher has really talked about the definition, and understands that this is really a discussion about the definition = what we choose to call a particular mathematical object, not what the object is - i.e. its mathematical properties. Most teachers don't, though in my experience they're open to the idea when I point it out to them. I encountered this recently in a first grade class puzzling about the number of faces of a cone. $\endgroup$ Jul 6, 2017 at 17:44
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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.

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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: $$\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$$

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  • $\begingroup$ Hence $\frac{4}{1}$ is a fraction but $4$ is not. Ok... $\endgroup$
    – leonbloy
    Jul 6, 2017 at 18:23
  • $\begingroup$ @leonbloy Where in there do I claim $\frac 41$ is a fraction? I'm purely talking about things with a square root (radical) in the denominator/numerator... $\endgroup$
    – lioness99a
    Jul 7, 2017 at 7:43
  • $\begingroup$ My point (poorly stated) is that we first need to agree on what a fraction is. Either it's a "kind of number" (as the accepted answer aptly notes) or it's just some notation. In the first case, it coincides with the rationals (and then $1/\sqrt{2}$ is not a fraction). If we opt for the latter, then $\frac41$ is a fraction. $\endgroup$
    – leonbloy
    Jul 7, 2017 at 13:00
  • $\begingroup$ @leonbloy I see where you're coming from now. I didn't consider that as I've always been taught that anything of the form $\dfrac{a}{b+\sqrt{c}}$ was a fraction, without really considering what that meant. $\endgroup$
    – lioness99a
    Jul 7, 2017 at 13:12
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"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.

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    $\begingroup$ Is $\frac{2}{1}$ not a fraction? $\endgroup$
    – Shuri2060
    Jul 6, 2017 at 12:52
  • $\begingroup$ I wouldn't call it a fraction. I'd call it rational. $\endgroup$
    – B. Goddard
    Jul 6, 2017 at 12:53
  • $\begingroup$ Depends on the definition you use I guess. $\endgroup$
    – Shuri2060
    Jul 6, 2017 at 12:54
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    $\begingroup$ @YvesDaoust This should be good. Please tell us what the difference is between the ill-defined "fraction" and the ill-defined "fractional number." Since neither has rigorous definition, it's just plain silly for you to assert that someone might confuse the two terms. $\endgroup$
    – B. Goddard
    Jul 6, 2017 at 14:29
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    $\begingroup$ @YvesDaoust As predicted: You have your own personal definitions and everyone has to bow to your will. Pah. $\endgroup$
    – B. Goddard
    Jul 6, 2017 at 15:08

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