why do we represent rational number by p/q?

[aamenis]

why do we represent rational number by p/q, why can't you not use m/n, x/y, c/y, a/c, etc. is there any meaning by (p/q)?

 

[Dr.Peterson]

You can always use any letters you want; there is no special significance to the fact that p and q are often used. That's just tradition, and I doubt that it is even close to universal, even in merely stating the definition of a rational number, much less in proofs.

As to why someone starting using p and q, and other authors copied it, possibly "q" represents "quotient", and "p" just fits with that.

 

[HallsofIvy]

You do have to say that the numerator, p, (or x, or a), must be an integer and the denominator, q, (or y or b) must be a positive integer.

 

[JeffM]

You do have to say that the numerator, p, (or x, or a), must be an integer and the denominator, q, (or y or b) must be a positive integer.

Positive integer? Integer other than zero maybe?

 

[HallsofIvy]

While it is not required, it is fairly common to associate the sign of a rational number with the numerator so the denominator is always positive.

 

[JeffM]

While it is not required, it is fairly common to associate the sign of a rational number with the numerator so the denominator is always positive.

Learn something every day.

 

[Dr.Peterson]

It depends very much on context. For many purposes it is sufficient just to say that a rational number is any real number that can be represented as a/b, where a and b are integers. This implies that b can't be zero, since then the fraction would not be a real number; and there is no need to demand that b is positive. The latter would be explicitly stated if you were not merely defining a rational number, but (within a theorem statement or a proof, for example) working with a particular representation, perhaps requiring b to be positive and the fraction to be in lowest terms. The fact that b is non-zero would have to be stated explicitly in making a more formal definition, such as equating the rationals with equivalence classes of the set of ordered pairs (a, b) of integers, where b is non-zero. Here, nothing implies the latter condition.

The important issue for the sake of the question is that whatever variable names we use, they are defined only locally, with no inherent meaning outside of a specific discussion within which they have been defined, so that, for instance, one would never talk in general of "the q of a rational number".

 

[Otis]

When I've been free to do so, I've defined Rationals as \(\frac{Integer}{Natural}\), too.

?

 

[JeffM]

Upon reflection, I have no problem with accepting that the conventional way to represent rational numbers is

[MATH]\dfrac{\text {integer}}{\text {positive integer}}.[/MATH]
Actually I usually say [MATH]- \ \dfrac{1}{2}[/MATH] rather than [MATH]\dfrac{-\ 1}{2}.[/MATH]
I am having great difficulty, however, accepting that it is correct that

[MATH]p,\ q \in \mathbb Z,\ \text { and } q < 0 \implies \dfrac{p}{q} \not \in \mathbb Q,[/MATH]
which leads to the rather odd result that

[MATH]\dfrac{-\ p}{-\ q} \in \mathbb Q \text { and } \dfrac{-\ p}{-\ q} \equiv \dfrac{p}{q} \not \in \mathbb Q.[/MATH]
To be consistent then we should say

[MATH]\dfrac{1}{\sqrt{3}} \not \in \mathbb R[/MATH]
because the normal way to write this value is [MATH]\dfrac{\sqrt{3}}{3}.[/MATH]

 

[Dr.Peterson]

I am having great difficulty, however, accepting that it is correct that

[MATH]p,\ q \in \mathbb Z,\ \text { and } q < 0 \implies \dfrac{p}{q} \not \in \mathbb Q,[/MATH]
which leads to the rather odd result that

[MATH]\dfrac{-\ p}{-\ q} \in \mathbb Q \text { and } \dfrac{-\ p}{-\ q} \equiv \dfrac{p}{q} \not \in \mathbb Q.[/MATH]

This is why I suggested a distinction between contexts, and specifically between definitions and other uses.

A key word in any definition, I think, is "can": "A rational number is any number that can be represented by p/q, where ...". There, we have no need to specify a preferred representation, much less a unique representation (which requires mentioning lowest terms). In particular, if a fraction is written with a negative denominator like 2/-3, it still can be written with a positive denominator, -2/3, so it is still a rational number even if you were to call for positive denominators. But even so, I don't think I've seen a definition that says q>0. That would be said in other contexts.

A common defect in informal "definitions" is a failure to clearly distinguish a number from its representation. We should not make it sound like a rational number is a particular representation p/q. But the following (I can't make myself link to less respectable sources!) are all good, mostly:

Wikipedia: In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. -- good, though "non-zero" could in my opinion be left unsaid​

​

MathWorld: A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q!=0. A rational number p/q is said to have numerator p and denominator q. -- accidentally equates the number with a representation, as if a number has one specific denominator.​

​

Math Open Reference: A rational number is one that can be written as the ratio of two integers. -- good​

Dictionary.com: a number that can be expressed exactly by a ratio of two integers. -- good​

​

Math Is Fun: A number that can be made by dividing two integers (an integer is a number with no fractional part). -- good​

None of these mentions being positive; most don't mention non-zero. All are fine as definitions.

 

[JeffM]

I can accept the immediately preceding post without cavil.

"A rational number is a number that can be expressed as an integer divided by a whole number" works for me.

I may have misread or misconstrued what I thought some were implying, perhaps a unintentionally, namely that an integer divided by a negative integer is not a rational number.

 

[HallsofIvy]

Exactly. An integer divided by a negative integer is a rational number. But the standard way to represent a rational number is "an integer divided by a positive integer". That avoids confusion of "-3/5" and "3/(-5)". Another important difference is that it avoids the difficulty with each of

Math Open Reference: A rational number is one that can be written as the ratio of two integers.

Dictionary.com: a number that can be expressed exactly by a ratio of two integers.

Math Is Fun: A number that can be made by dividing two integers (an integer is a number with no fractional part).

which are NOT good because each would allow something like "5/0" as a rational number!

 

[Dr.Peterson]

Well, no.

I wouldn't say that 5/0 is "a number that can be written as the ratio of two integers" -- it isn't a number at all! That was my point. If they said "any integer divided by an integer is a rational number", that would be wrong. But they don't.

Keep in mind that I'm not saying these would be the definition used in an analysis text; I'm saying these are good enough to use in a place where a simple definition is needed, because they don't lead to wrong conclusions (when read reasonably).

So, no, these are not ideal, but I think they are good enough for their purpose.

 

[Otis]

This may be off-topic, as I'm not referring to definitions, but I regularly remind students that we're free to move a negative sign in a ratio to whatever position is helpful (numerator, denominator or out in front). Here's an example where it's helpful to move a negative sign in the numerator to the denominator.

\(\displaystyle \frac{-5}{3 - x} + \frac{4}{x - 3}\)

\(\displaystyle \frac{5}{-(3 - x)} + \frac{4}{x - 3}\)

\(\displaystyle \frac{5}{x - 3} + \frac{4}{x - 3}\)

?