Why are negative numbers introduced before quotients in the real number subsets?

Question

This is a question regarding why the order of the real number subsets commonly used in the mathematics community is such:

$$ \mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R} $$

Here the concept of negative numbers is introduced before the concept of quotients. Theoretically, it is possible to introduce the concept of quotients before the concept of negative numbers.

If we defined the second subset (after the natural numbers) as equivalence classes of pairs of natural numbers (p, q) with q ≠ 0, using the equivalence relation defined as follows: $$\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2}) \Longleftrightarrow p_{1}q_{2}=p_{2}q_{1}$$ then the third as consisting of the second subset, 0 and the negatives of the second subset.

The establishment of mathematical convention should have much less to do with the student's competence and much more to do with providing a logical sequence of concept introductions to promote a sound understanding of mathematics.

I believe that the most straightforward approach to sequencing the introductions of mathematical concepts is not always the most logically sound approach.

Why would I advocate for introducing quotients before negative numbers, in the real number subset order? If we assume ~1500 years between the first use of division around 2000 B.C.E. and the Greek counting table at Salamis around 300 B.C.E., this leads me to believe the natural development of mathematics prioritised quotients over negative numbers.

A university course on the history of mathematics is primarily about understanding the stories behind the mathematics we use today. A chronological account would tell many other stories of the changes in societal mentality throughout history regarding mathematics which do not apply to the mathematics we use today.

Mathematics can be a fantastic subject to study with great societal value but we should always aim to focus the logic and reasoning behind our ideas on the truth. The bounds of mathematics should not be restricted to the current thought of today.

If such a thing as the "natural order of events" exists and it prioritised introducing quotients before negative numbers and the factors around the conception of the Renaissance era were consistent with said order of events (thus denouncing any feeling of historical exceptionalism), there is reason to factor this consideration when deciding the fundamentals and their effect on our advancements within mathematics.

I believe the truth is more important than our current comprehension of mathematics. If there is a strong historical case to be made that humans advanced using quotients before negative numbers, this should be a consideration when deciding the future of mathematics. In particular the conventions of which our future advancements will be made with.

If you think the current system has merits that supercede these I would appreciate an explanation as to why.

Answer 1

Note that the OP appears to have slightly tangled up two distinct issues of: (1) ordering of subsets, and (2) order of what concepts are introduced in school. That is, it would be simply false to try and say, $ \mathbb{Q}\subseteq\mathbb{Z}$. But it's true that we do have a choice about what order we introduce things in a curriculum; and even that historically quotients were used before negatives.

However, given that we do have negatives in our toolkit today, it seems clear that understanding negatives is a lot simpler than arbitrary fractions. Handling negatives is really just an extra bit of information -- a binary decision of whether you have a positive or negative sign in front of your whole number; and you can clearly depict this on a number line by extending it both left and right with discrete marks. On the other hand, quotients are more complicated in that they add a denominator with an infinite number of possible integer values, there are issues of equating fractions in different forms, reducing to simplest terms, factoring and canceling, etc.; and there is no way to clearly show all rationals present in a given slice of the number line.

Ultimately, we observe the following:

• Integers are essentially the result of all possible subtractions of two natural numbers.
• Rationals are essentially the result of all possible divisions of two integers.

I think most of us would agree (and psychological studies bear this out, if it wasn't totally obvious) that subtraction (inverting addition) is simpler than division (inverting multiplication). So it seems highly coherent to develop the expanding sets of numbers by going up the ladder of operations from simple to more complex, establishing the closure at each step.

In short, the conceptual and psychological benefits seem to weigh in favor of bringing in negatives before quotients, despite the fact of the historical evolution. If anything, it highlights the possibly surprising historically oddity that integers weren't developed before rationals.

Answer 2

At school, quotients come before negative numbers

At least in some school systems. In Finland both decimal numbers and fractions, as well as the single straggler pi, were introduced somewhere in the upper half of ala-aste, so grades four to six, I guess. Memories are vague on details. Negative numbers were introduced during yläaste, so grades seven to nine, but probably fairly early there.

In the current Norwegian curriculum, after fifth grade the pupil is supposed to be able to:

utforske og forklare samanhengar mellom brøkar, desimaltal og prosent og bruke det i hovudrekning

So there come the quotients. (Also other goals related to quotients are there.)

At seventh grade the pupil is supposed to be able to:

utforske negative tal i praktiske situasjonar.

So there come the negative numbers.

This might be different in other countries, but at least my experience suggests the negative numbers indeed do come some years after many quotients.

Presumably this is because of the abstract and finicky nature of negative numbers: is -8 larger or smaller than -2? Natural language makes this ambiguous. The same with the rules of what happens when we add or take away a negative amount of something; this is quite challenging to think about. Not to speak of division with negatives, where intuition and comprehension are mostly exchanged with some variant of negating the minus sign in the divisor or moving it away.

At university level

In most courses all the number systems are presumed known.

• An analysis course might pay attention to completeness of the real numbers and maybe even somewhat carefully go from the rationals to them.
• An algebra course is likely to spend a lot of time on groups such as the integers, and later on fields/rings such as the rationals.
• A course interested in the construction of the number systems is likely to do them in the usual order for reasons of conventions, but certainly quotients could be taken before the negatives. But then one would meet different algebraic structures on the way; whether those are as useful as the standards one I have not thought about. But might also be that the integers make for a simpler equivalence class than positive rationals do, given that most students are more at home with subtraction than fractions.

A general principle

The establishment of mathematical convention should have much less to do with the student's competence and much more to do with providing a logical sequence of concept introductions to promote a sound understanding of mathematics.

When teaching, pupil and student competence is more crucial than the logical structure. Children going to school do not start with the foundations of mathematics, and for a good reason. (The attempts have not been successful.) One needs to grow into abstraction, not be thrown into the lake to either sink or, in a rare few precious cases, swim.

I would also consider the connection to group theory as an important one at university level, and one gets there faster with integers than the alternate way.

Answer 3

There are three possibilities for the introduction of rationals and negatives in school curricula:

• To introduce rationals after negatives.
• To introduce rationals before negatives.
• To introduce rationals alongside negatives.

The first one has been nicely covered in @Daniel R. Collins's answer while the latter seems too much in terms of cognitive load for students. Thus, I should also present here some merits of having things as done so far, i.e., introducing rationals before negatives.

The spirit of this answer is to complete the views that have been expressed by the time this is being written. Personally, I do not have a strong opinion on the matter, as both have significant merits as well as give rise to various problems, from my viewpoint.

Rationals prior to Negatives

What is currently done in most curricula I have in mind is that rationals are introduced some years earlier than negatives. While this appears to be historically accurate, as the OP also observes at some points, it might also appear to have several benefits for the structure and flow of the curriculum as a whole.

To begin with, after having talked about naturals and the four operations between them (addition, subtraction, multiplication and division), introducing fractions facilitates the transition to decimal numbers (hinting, in some sense, to the reals). Moreover, using fractions further facilitates the introduction of proportions and $\%$ percentages, which, among others, are things that are quite relevant to students' everyday lives.

Furthermore, given that an ever-growing number of curricula around the world tend to introduce key concepts of statistics and probabilities quite early (even grades 3 or 4 in some cases), fractions become of crucial importance. While negatives can be "hidden" behind subtraction and they seldomly appear in relevant applications and problems, the concepts of "observation frequency", "percentage" etc cannot be easily conceptualized without the construct of fractions. Given that one of the most common ways to introduce probabilities is through frequency-based definitions (or, better, intuitions), fractions are again a useful construct in that setting.

Other than the above, fractions are also seemingly a prerequisite in introducing functions. Indeed, the latter is usually done through linear functions and proportionally changing variables. In turn, proportions are easily conceived on the basis of fractions and, more precisely, equivalent fractions. Thus, any delay in the introduction of rationals, also affects the way one can talk about functions, at least in the way it is currently done.

There might be more things to say in favor of the status quo, but the above are some indicative cases in which having discussed and learned about fractions early on comes in handy when designing a curriculum.

Answer 4

AFAIK in France there are different levels of introduction of numbers.

1. In primary schools (6-11 years)

   Decimal and other non integer positive numbers (1/3, pi) are presented as simple calculus tools with very little (if any...) theorical explainations. For example, pi is said to be 3.14 or 22/7 without any more detail. And any of those definitions are used as-is to get numeric values. But negative numbers are just left out at this point.

2. In secondary schools (> 11 years)

   It now matters to explain how mathematically things work. And on that point, introducing negative numbers only needs addition/substraction which are the simplest operations. On the other hand, correctly handling fractions (not identifying them to their numeric approximation) is much more complex. So IMHO it seems logical to introduce negative numbers before fractions in this cursus.

That means (IMHO) that the argument that fractions were used centuries before negative numbers has no impact on the way those notions are to be introduced to students. If we just want pupils to be able to do simple calculations fractions should be presented before negative numbers. But if we want to teach some mathematic concepts, then negative numbers should come first.