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I'm interested in generalizing the underlying mechanism/theory/concept behind subtraction, from natural to negative numbers.

Not tacking on a new concept just for handling the new cases, but applying the same concept to both old and new cases (that in some sense already existed for the old cases). Perhaps identifying some property present for the old cases, and preserving it for the new cases.

This is the farthest I've got:

  1. Natural numbers can be thought of as a physical objects: you can count how many there are, physically add more objects, or remove them (or take them away). But "take away" is not closed over the naturals; you can't physically take away more than you have.

I say that "take away" cannot be generalized to negative numbers, because you cannot have a negative number of physical objects. However, we can look at it in a different way (a different theory, or concept or model), that gives the same answers for the same cases, but that can be generalized to negative numbers. [This is a kind of a cheat, but if the concept is present and active in the original case, I'll allow it.]

  1. We have a set of distinct elements, that are ordered, starting from one, and going on forever. We can go to the "next" one in this order (+1), and go to the previous one (-1). By repeating these, we can add a positive number, and subtract a positive number, and get the same results as above.

But from the first element, we can't go to a previous one. It seems a very natural and logical extension to allow this - it seems more like removing an artificial barrier than introducing a new concept.

If we allow this, we have negative numbers, and, given a positive or negative number, we can subtract a positive number $n$, by going to the previous element, $n$ times. There's also nice a nice symmetry, with both ends being infinite, instead of only one.

Although an improvement, this still isn't closed on this expanded set of numbers (positive and negative), because we can't subtract a negative.

I say that we cannot generalize or extend this mechanism/theory to subtraction of negative numbers, because doing something a negative number of times doesn't make any sense. However, we can again change the figure, and use a a different theory or model which can be generalized in this way. We do this by observing properties in the previous case, and choosing a new model that selectively preserves those properties when it is generalized.

  1. A different way of thinking about "going back" $n$ times to get to the result, or of "taking away" is the difference between two numbers, how many next or previous steps it takes to move from second to the first. We might get to this idea in two steps:

    1. the "going back" $n$ steps is the difference between the starting number and result
    2. it turns out that if we start from the same number, but instead go back by the result number of steps, we get $n$. i.e. in algebraic notation, with $S,R$ for start and result: $S-n=R \iff S-R=n$

This is quite a different concept, but it does generalize smoothly to any pair of numbers, because they all are separated by some number of steps on the ordered elements. It builds on the generalization of positive to negative, and the property that they can both be moved along in steps.

It also changes the role of the operands: instead of one being an instruction (to move left or right), both operands have equal status, and the result is the relationship between them. This, again, has symmetry.

By this model, from any number (positive or negative) can be taken away or subtracted any other number (positive or negative).

To recap, the steps were:

  • extend numbers to negatives (see as "next" and "previous", and "previous" continues past $0$)

  • extend subtraction to negatives (see "take away" as "difference", and "difference" applies between any two numbers).

I think this completes the concept of negative numbers.

But there's another step, to algebraic notation, which has the same properties, but which is different altogether, in that there is no theory or model. It is a purely syntactic system - there is no coherent concept. Just a system of rules. What is most striking to me is that the rules for addition seem overly complex, compared to the above simple and logical model. [Though to anyone proficient in arithmetic, who has internalized the rules, this claim of complexity might be unbelievable].

By Occam's Razor, it is troubling that an unnecessarily complex model is true, that a series of specific cases could he correct. But, unlike the famously complex approximation of "epicycles within epicycles", it is precisely correct.

hyperpallium
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    You want "simple" explanations, but you want rigor and no hand waving? ummm Please try to be more clear on what you are asking. Simple aint necessarily get you rigor; and rigor will likely incorporate foundational ideas. So you need to make up your mind. You want rigor, then be prepared what might not feel "simple" at the start. – amWhy Feb 02 '18 at 01:40
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    Edmund Landau, Foundations of Analysis. Negative numbers are introduced in Chapter IV. – bof Feb 02 '18 at 01:48
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    Check out abstract algebra, at least. The very basics explains your question in a rigorous way. – Kaynex Feb 02 '18 at 02:02
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    The thing is, abstract algebra does make things simpler -- it strips away unnecessary baggage like "positive" and "negative", and everything follows from simple algebraic identities. For instance, if you're willing believe that for every integer $a$ there's an integer called $-a$ so that $a + (-a) = 0$, then it's just a bit of symbol shuffling and looking at definitions to see that $-(-a) = a$. For this reason, you'll get that $b - (-a) = b + a$. Throw in a few beliefs about multiplication, and you can show that $-a$ has to mean the same thing as $-1 \cdot a$. – pjs36 Feb 02 '18 at 02:02
  • @amWhy I hope it gets simpler as we go deeper, so it's eventually understandable. If it keeps getting more complex, it never will. (Or, is school-taught maths an over-simplification, so going deeper gets more complicated before it gets simpler?) Also, if Euclid has simple axoims, shouldn't there be simple axoims for arithmetic? – hyperpallium Feb 02 '18 at 02:03
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    Not sure whether this will be enough, but have you looked at the Formal construction section of the Wikipedia article on negative numbers? The basic idea is to make sense of differences $a-b$ when $b>a$ by defining and investigating the properties of certain equivalence classes of ordered pairs $(a,b)$. – Will Orrick Feb 02 '18 at 02:04
  • @WillOrrick Thanks. I've seen this idea. I can see that it's true; and that it defines $\mathbb Z$ properties by a construction that only use natural numbers and operations... in a sense, that's "extending" one from the other, but I don't see that it's generalizing one from the other. I probably need a detailed, step-by-step account of how each common part corresponds to the analogous part in the other, justifying why they are analogous, and then showing/justifying how the extra features of $\mathbb Z$ are a generalization. – hyperpallium Feb 02 '18 at 02:21
  • @bof The book you recommend is interesting in its own way for its very fundamental analysis, but I wouldn't say it treats what the OP wanted. In any case +1. – Allawonder Feb 02 '18 at 04:52
  • @hyperpallium Perhaps you should consider ElfHog's suggestion below. There are many ways to view the same thing. You may forget about subtraction as an operation, instead define the negative numbers and analyse the addition of additive inverses. See if it helps. – Allawonder Feb 02 '18 at 04:56
  • @Allawonder It starts from basic axioms, and proceeds step by step without skipping any details. And the development includes subtraction of positive and negative real numbers. And if he wants, he can start at the beginning of Chapter IV, with positive reals already established, and see how negative numbers and operations on them are introduced. What else did the OP want??? – bof Feb 02 '18 at 04:57
  • I glanced through it, and you're right. I had meant that it did not treat subtraction as an operation as the OP wanted (it did this, so far as I am aware, for the complex numbers). – Allawonder Feb 02 '18 at 05:00
  • @bof Landau is incredibly tedious - perfect for me! And google shpws it's an authoritative text. Reassuringly, my approach had similarities, using abs value $||$, and going case-by-case. But, both are complicated! I guess that's just how it is. Finally, he rigourously defines subtraction, but doesn't argue for how subtraction of negatives is analogous to subtraction of positives - that's what I meant by "generalize". (he doesn't define subtraction for postives, so can't make the analogy). – hyperpallium Feb 02 '18 at 06:45
  • @bof BTW there is a little dependency on previous chapters: he uses lowercase greek letters for "cuts", which are positive, but it's not mentioned in IV itself. I don't think there are other dependencies (but there could be). – hyperpallium Feb 02 '18 at 06:52
  • Thanks @yh016, I'll have a look. It's been a while since I had a teacher of this, I just used the algebra-precalculus tag because the description seemed close. By "good explanation" I assume you mean the KA one I linked. – hyperpallium Feb 04 '18 at 02:14
  • @yh016 I guess you refer to 2.1.1 Algebraic Properties of R pp.24-25 (40,41 in the pdf) in 4th ed? I searched for "negatives", and that seemed closest. These define, but not show a generalization or extension from natural subtraction over natural numbers. – hyperpallium Feb 04 '18 at 03:04
  • [Also, an aside: since several things(?) can satisfy these properties, isn't it true that they aren't enough to pin down the specific thing (e.g. reals), but a more abstract concept, that enables algebraic manipulation? Sure, anything else that satisfies these is isomorphic to reals... but only in terms of these properties. For example, $\mathbb{Z}$ and $\mathbb{Q}$ also satisfy them. Anyway, I'm interested in negatives... but it's not clear to me that they are enough to pin down $\mathbb{Z}$ as numbers either, since they aren't enough to pin down reals...] – hyperpallium Feb 04 '18 at 03:06
  • @hyper Algebraic properties of numbers are very well understood. In particular, we define different number systems abstractly rather than try to find them in reality. It is then easy to derive, rigorously, in extremely large generality (as in, not just for numbers, but for any associative, commutative operation with inverses) that $-(-x)=x$. – Matt Samuel Feb 04 '18 at 06:58
  • Strictly speaking, real numbers don't exist. There are real numbers that cannot be computed to arbitrary precision by any computer program, no matter how long it runs. Those numbers may as well not exist. So mathematics and reality diverged centuries ago and the two now have only coincidental relation. – Matt Samuel Feb 04 '18 at 07:02
  • @MattSamuel That makes sense (that the advantage of algenraic properties is that they make it easier to prove things rigourously, and that they generalize to other systems that have the same properties). But I take exception to saying it "defines" a number system. If Z, Q and R are different, but have identical algebraic properties, how can those properties be said to define them? (or are there extra properties that distingiish them?) – hyperpallium Feb 04 '18 at 10:41
  • I agree they don't exist. Actually,mI think negatives, rationals, reals and complex numbers don't exist - standard notation suggets this, but not stating the,, but describing how to compute them. e.g. $-1, 1/3, \sqrt{2}, \sqrt{-1}$. "-1" is an expression, a unary operator and a positive. The suprising thing is how useful they are. – hyperpallium Feb 04 '18 at 10:45
  • There are extra properties. You start with natural numbers. Integers come by allowing additive inverses. Rationals cone from allowing division. Then passing to the real numbers is not at all algebraically motivated. It is the topological completion of the rational numbers, what you get when you assert that all Cauchy sequences converge. But then passing to the complex numbers is so that we get an algebraically closed field. – Matt Samuel Feb 04 '18 at 10:47
  • @MattSamuel I don't understand those last two, but if they aren't algebraic properties, then R and C can't be said to be defined by algebraic properties - I think? – hyperpallium Feb 04 '18 at 11:03
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    It depends what you mean by that. In a sense, very few structures are determined solely by their algebraic properties. Usually there are many fundamentally different structures with the same basic properties. So the properties become a tool, but they are not all you have to work with. – Matt Samuel Feb 04 '18 at 11:09
  • You could say the whole basis of algebraic geometry is studying algebraic structures with addition that is commutative and associative with inverses, multiplication that is commutative and associative with identity, such that the multiplication distributes over addition. But that's like saying number theory is the study of strings of digits that terminate. – Matt Samuel Feb 04 '18 at 11:12
  • @MattSamuel Thanks. I though if algebraic properties alone are enough to define (e.g.) numeric addition? i.e. if a system has the same algebraic property as addition over naturals, then there's always a mapping from naturals to that system's elements via which you can do addition? – hyperpallium Feb 05 '18 at 03:14
  • The system has to have at least as many elements (so, infinite for naturals). I saw a mapping to multiplication of naturals, $2^x$, because $2^{i}2^{j}=2^{i+j}$, you can do addition with multiplication. (even the identities match up: $2^0=1$). But it wouldn't work for a sort of "degenerate" system, where every operation gave the same result. Also, I think there's no mapping to do (eg) modulo 10 arithmetic in modulo 12 arithmetic. – hyperpallium Feb 05 '18 at 03:21
  • Anyway, if something like that was possible, (a) it would be amazing (b) I would accept that algebraic properties, in some strange way, "define" arithmetic. [It might be better if I ask this as question in itself] – hyperpallium Feb 05 '18 at 03:21
  • Who is pressuring you that you have to post an incomplete question? (Yes, having a "to be continued" means it is incomplete.) – Asaf Karagila Feb 13 '18 at 08:15
  • @AsafKaragila Thanks. It was my attempt to answer it, which I've now moved to an answer. – hyperpallium Feb 13 '18 at 08:28

2 Answers2

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I will say there is only addition in real numbers, with a notation of "additive inverse".

For example, $3$'s additive inverse is "$-3$", since $3 + (-3) = (-3) + 3 = 0$.

Then subtraction is addition of an additive inverse. E.g. $8-5 = 8 + (-5) = 3$.

[I seem to just restating materials online but I hope it helps]

Tito Piezas III
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ElfHog
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  • If for reals, there is no subtraction, only addition of an inverse, then for naturals, subtraction (or "take away") is completely different (because there is no inverse that could be added)? Although natural subtraction is a subset of real subtraction ($\mathbb N \subset \mathbb R$, and real subtraction of naturals gives the same result as their natural subtraction), that's just a coincidence, and doesn't affect subtraction of non-natural reals? Like a model of naturals as counting physical objects is completely different from the model of reals as geometric directed lengths. – hyperpallium Feb 03 '18 at 01:17
  • I will give my opinion (but in terms of math so I hope it won't get deleted since it is also about construction). We know that $\mathbb{N}$ is constructed by set theory with axioms (5 axioms in my memory). Then we extend it to $\mathbb{Z}$ by defining additive inverse. Here we can already prove that $(\mathbb{Z},+)$ is an abelian group. We can take the field of fractions on $(\mathbb{Z},+,\times)$ (with multiplication making sense as normal $2+2+2=2\times 3$) and use completion method to construct $\mathbb{R}$. – ElfHog Feb 03 '18 at 01:52
  • @hyperpallium In fact we are mapping counting of physical objects ($\mathbb{N}$) to geometric directed lengths ($\mathbb{R}$) with an injective map (embedding) and with construction above, we know that it is not only a coincidence. Consider $p$-adic numbers $\mathbb{Z}_p$. We know that it adapts a different metric as in normal "numbers" or physical "objects". There are many distinctive properties (e.g. non-Archimedean) showing to us that the embedding takes a construction, and different construction (emphasizing on different properties) may give us different spaces. – ElfHog Feb 03 '18 at 01:56
  • Isn't saying there's a injective mapping from $\mathbb N$ to $\mathbb R$ only stating the fact, but not why (i.e. whether it's a coincidence)? [BTW it's just the negatives I'm worried about; we could simplify to "injection from $\mathbb N$ to $\mathbb Z$".] It's extended to negatives "by defining additive inverse", but they're not in $\mathbb N$. It gives the same answers for natural subtraction of naturals, but isn't causally related. It's not an extrapolation of the same mechanism, but a different mechanism, that coincides for the naturals (i.e. it extends the results, but not the cause). – hyperpallium Feb 03 '18 at 03:25
  • @hyperpallium yeah, sorry that I messed up my logic. I was thinking something like "counting loaning of sheeps", for example. Like if I loaned a sheep, then it exert effect of "$-1$" on my number of sheeps. Then we can do the same construction on "$-\mathbb{N}$" and addition on it. That's why I feel that it is kind of normal but I should admit that it doesn't take causal relationship. – ElfHog Feb 03 '18 at 03:33
  • (I googled completion method, p-adic numbers, non-Archimedean, but they're still over my head - hopefully I got the gist anyway) BTW I don't think I'm making sense to anyone here; I don't want to waste your time on a hopeless case. But... am I missing something in your comment, that is a causal link between natural subtraction and real/integer subtraction; or, that shows they use the same mechanism? (e.g. there's a mapping between the 2 eyes of a human and the 2 wheels of a bicycle, but they're not causally related. It's a coincidence). – hyperpallium Feb 03 '18 at 03:39
  • Let us continue this discussion in chat. – ElfHog Feb 03 '18 at 03:40
  • I think notifications aren't sent for chat, so adding this comment to let you know I replied in chat. – hyperpallium Feb 03 '18 at 04:28
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Concerning your comment "My question is about generalizing or extending the underlying mechanism, or theory, or concept - not just generizing or extending the results": what you seem to be groping toward is a general construction called the Grothendieck group construction.

You may also want to consult questions under the tag .

This generalisation is very useful in applications such as topological K-theory.

More specifically, if you start with a monoid $M$ and construct its Grothendieck group $G$ then the operation defined on $G$ will in particular exist for elements that "were not there before" such as negatives in the case $M=\mathbb N$. Subtraction of negatives is a special case of subtraction of arbitrary elements of $\mathbb Z$ which is the Grothendieck group of $\mathbb N$. Needless to say, in the general case there is more going on than merely adding the negatives; for instance, some elements that were distinct in $M$ may become identified in $G$.

Mikhail Katz
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  • Thanks. "Groping toward" is apt. Understanding this "Grothendieck group construction" seems to require more group theory than I currently have. (I don't understand even its significance from the wiki link ) - could you recommend a less technical introduction? I acknowledge that might not be possible. Also, it doesn't seem to involve a generalization from subtraction of positives to subtraction of negatives, in that it doesn't start from the former. (But it may be just that I don't understand it). – hyperpallium Feb 06 '18 at 00:56
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    I tried to address your concern in my answer that I just edited. – Mikhail Katz Feb 06 '18 at 09:47
  • Thanks! I should clarify that I mean a mechanism or theory that was already present in the natural numbers and their subtraction. I think Group theory has its own mechanisms and "theory", so I'm not sure it can answer such a question... However, I still don't understand the Grothendieck construction to see for myself whether it does do what I mean, and how. Maybe it does! BTW you said my "comment", but quoted from my question - since I also discussed it in the comments, I wonder if that's the context you were answering from? I ask because it does seem more related to those comments, to me. – hyperpallium Feb 07 '18 at 03:44
  • The natural numbers $\mathbb N$ are the simplest example of a semigroup, and the Grothendieck construction applied to them produces the group $\mathbb Z$ which is the infinite cyclic group. – Mikhail Katz Feb 07 '18 at 09:37
  • Thanks, that gives me some idea. BTW I meant a mechanism like counting physical objects, physically adding and literally taking away. Viewing $\mathbb{N}$ as a semigroup already views them in terms of group theory, so perhaps not surprising it generalizes to other concepts in group theory. However, it's an interesting generalization - and maybe it does do what I want. I will keep it in mind for when I have the requisite background. Thanks again. – hyperpallium Feb 07 '18 at 23:24