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Wikipedia constructs an integer as the difference of natural numbers denoted by an

oriented pair. Negation is a flip of the orientation. A positive integer is a

difference, but the literature generally says it can be taken as a natural number.

A positive integer has an additive inverse, a natural number does not. How am I to

to understand a positive integer as a natural number?

Richard Fuller
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  • Natural numbers can be "embedded" into integers mapping $n$ into $(n,0)$. – Mauro ALLEGRANZA Mar 17 '18 at 14:41
  • A positive integer $(n,0)$ has an additive inverse because it is an iteger. If we consider the corresponding natural $n$, it has no additive inverse. – Mauro ALLEGRANZA Mar 17 '18 at 14:44
  • According to your comments on the answers, it seems the question you asked is not the one you want answered. An answer explaining integers as equivalence classes of differences of natural numbers would not be at all a suitable way to explain negative numbers to schoolchildren. Instead, you would get (and have gotten) answers pitched to a much more advanced understanding. I don't know if you can fix this at this point; perhaps you should ask a new question, but this time ask the question you actually care about. – David K Mar 18 '18 at 03:06

2 Answers2

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The set of natural numbers is not a group because additive opposites of natural numbers are not natural numbers.

When we extend the natural numbers to the set of integers, then we allow negative integers and the set of integers will be a group.

Then we can consider the set of natural numbers as a subset of the group of integers, thus a positive integer has an additive opposite which is an integer but is not a natural number.

Thus when we say natural numbers do not have additive opposites, we mean the additive opposite is not a natural number.

Mohammad Riazi-Kermani
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  • Sorry, I still don't get it? What is the additive inverse of a negative integer defined as the difference of two natural numbers? It is not a difference? What is the equivalence collection? Thanks – Richard Fuller Mar 17 '18 at 16:39
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The important thing to understand is that there exists not the natural numbers, in the sense that each natural number is a specific object. The natural numbers can just as well be given as digit strings, as strings repeating one character (|, ||, |||, …), or as a collection of apples. What is a natural number is determined not by the objects, but by the structure put on the objects.

In particular, with natural numbers, you must have a certain minimal element ($0$ or $1$, depending on your definition), for each element an unique next element, addition and multiplication which fulfills certain rules, and you get all of them by just starting at the first and repeatedly going to the next (this is a quite informal description; the more formal description is given by the Peano axioms).

It can be verified that all those conditions are fulfilled for the positive (or non-negative) integers, therefore those integers describe the natural numbers. Or again, more formally: The positive (or non-negative) integers fulfill the Peano axioms if you identify the natural number $n$ with the integer $+n \equiv (n,0)$, the successor with $n+1$, and the addition and multiplication with the corresponding operations on the integers.

About the additive inverse: A positive integer has an additive inverse in the integers, but it does not have an additive inverse in the positive integers. So when identifying the natural numbers with the positive integers, there does not exist a natural number that is the additive inverse of another natural number.

celtschk
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  • Isn't this more an equivalence, (n,0) is the canonical name of an equivalence class of integers. You then identify n with that name. I came – Richard Fuller Mar 17 '18 at 17:17
  • I came to this question when looking for a way for grade schoolers to understand negative numbers. Integers as differences suggest integers are different from natural numbers. As you point out the natural numbers are "relationship" numbers, they exist by virture of their relations. On the other hand integers are "interior" numbers, an oriented difference of natural numbers, each can exist independently of another. You can see my problem. Is there anyway out of it that can be explained to school children? Thanks – Richard Fuller Mar 17 '18 at 17:44
  • @RichardFuller: About the equivalence: Yes, in the standard construction the integer $+n$ actually is not the pair $(n,0)$ but the equivalence class of pairs. One could, of course, construct them differently so that indeed the pair itself represents the number. About your second comment: No, there's no conceptual difference; the integers are not the differences of natural numbers, that is just one way to introduce them (just as “|,||,|||,…” is a way to introduce the natural numbers). You can also define them as the unique ordered ring that is generated by its multiplicative identity. – celtschk Mar 17 '18 at 18:49
  • Actually when first introducing the negative numbers to school children, I would use neither construction. Instead I'd simply take a scale (as the one of a thermometer) with marks for the natural numbers, and then ask, is there a reason not to continue it to the other side? That's the level at which you need to understand the negative numbers; the mathematical construction then is only a way to convince oneself that doing so does not lead to unforeseen problems. – celtschk Mar 17 '18 at 18:55
  • I agree "continuation to the other side" can be suitable for a first contact. My concern with it, and with other definitions by association, is they are taken as the definitions, and remain as the intuition. Students can be left with the infamous mystery "subtraction of a minus". More generally they see problem solution as retrieval of associations of numbers with objects. It appears they have little contact with abstraction and deduction before seeing them as math undergrads. Is there a middle ground? I thought the oriented pair might work with effort to render it appropriate for (?) graders. – Richard Fuller Mar 19 '18 at 17:08
  • @RichardFuller: What is the problem with subtraction of a minus? (Note that I'm not a teacher, so I naturally have limited knowledge about what children have trouble with.) As soon as you've understood that subtraction is the same as addition of the negative, and that the negative of the negative is the original positive (both of which can be nicely seen on the scale), IMHO the subtraction of a negative should be obvious: Subtraction of the negative is addition of the negative of the negative, that is, addition of the positive. – celtschk Mar 20 '18 at 08:09
  • As far as I can tell subtraction is generally taught before integers, often as "removal". In any case the solution is generally offered as remembering "minus of a minus is a plus". Coming to appreciate the point you make led me to looking for a simple, and principled approach. The riff on a "positive integer is a natural number" goes back to an attempt to clearly avoid privileging positive as somehow normal/natural and negative as different/special. With your help I think I have a presentation that does that. – Richard Fuller Mar 20 '18 at 15:11
  • @RichardFuller: “As far as I can tell subtraction is generally taught before integers, often as "removal".” — Sure, but as soon as you teach integers, it's time to teach that subtraction is actually the same as the addition of negatives. Remember, the more models you know, the better! — “With your help I think I have a presentation that does that” — Glad I could help. – celtschk Mar 20 '18 at 15:43