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This new Wikipedia article may look different by the time the reader of this question sees it. For now, it says $6\div 2$ can be construed in either of two ways:

  • "How many parts of size $2$ must be added to get $6$?" (Quotition division)
  • "What is the size of each of $2$ equal parts whose sum is $6$?" (Partition division)

The numerical answer is of course the same either way, and that is in a trivial sense equivalent to the commutativity of multiplication.

I never heard of any particular names for this distinction until I saw this article about an hour ago. It is alleged that the terminology has some currency in the field of education.

In the Oxford English Dictionary I find the related terms ''quotient'', ''quotity'', and ''quotum'', but not ''quotition''.

So:

  • Is this a familiar concept to everyone except me (for some reasonable values of "everyone")?
  • Does it occur in other contexts? E.g. might set theorists doing ordinal arithmetic think about it?
  • Are there things of interest to mathematicians to say about this distinction?

Later edit: Someone has since edited the article further to cite a book published almost 100 years ago.

Michael Hardy
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  • For what it's worth: I read $a\cdot b$ as '$a$, $b$ times'. In ordinal arithmetic this useful: $2\cdot \omega$ is $2$, $\omega$ times, that is, $\omega$. On the other hand $\omega \cdot 2$ is $\omega$ two times,that is $\omega+\omega$. – Git Gud Jan 14 '14 at 23:42
  • "quotition" is new to me. – Gerry Myerson Jan 15 '14 at 03:08
  • I don't know if I am a reasonable value of "everyone", but I never heard of this. – Betty Mock Jan 15 '14 at 04:21
  • The two ways of thinking about division are familiar, but the word is new to me. – Sammy Black Jan 15 '14 at 06:04
  • There's a nice analogy between the partition vs quotition models for solving $a = b \cdot c$ for $b$ or $c$ and the logarithm vs root tools for solving $a = b^c$ for $b$ or $c$. – Sammy Black Jan 15 '14 at 06:14
  • I am aware that it seems to differ from the young pupil's perspective (e.g. some kid might quickly "see" that dividing 12 into 2 equal parts makes the parts size 6, but to check ho many 2s fit into 12 he/she has to do some counting); thus in math education a distinction between the two types of text problem may be important. But I still haven't heard of the expression "quotition" (nor any equivalent in my native langugage - I learned English a few years later than division). – Hagen von Eitzen Jan 17 '14 at 15:56

1 Answers1

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In ZF set theory without the axiom of choice, these two notions can differ for infinite sets. Partition division and quotition division are not necessarily the same in ZF.

See for example the answers to my question, On the difference between two concepts of even cardinalities: Is there a model of ZF set theory in which every infinite set can be split into pairs, but not every infinite set can be cut in half?

The main question there is still open, but Asaf Karagila gives a central example: under ZF there can be an infinite set that can be partitioned into pairs, but admits no partition into two sets of equal cardinality. This shows that quotition division and partition division are not necessarily the same thing for infinite sets in ZF.

Meanwhile, it is easy to see that they are the same for infinite sets in ZFC.

JDH
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  • Can anything be said about ordinals? – Michael Hardy May 14 '17 at 23:32
  • BTW I have this fantasy that Neal Stephenson writes a sequel or prequel to his novel Anathem, and in that later novel we will read that the worldwide leader of the Bazian Orthodox Ark has died and the princes of the Ark have assembled in the city of Baz to elect his successor, and -- lo and behold -- we learn that those august personages are called Ordinals.

    (Those who haven't read Anathem are instructed not to read this comment unless they want to.)

     – Michael Hardy May 14 '17 at 23:34
  • @MichaelHardy For ordinals, the two concepts are same. Indeed, for every linearly ordered set, it can be partitioned into pairs if and only if at can be partitioned into two sets of the same cardinality. Given pairs, take the lower set and the upper set. Given two sets and a correspondence, turn it into pairs. – JDH May 15 '17 at 00:02
  • oh . . . . . I hadn't actually thought about this, but the non-commutativity of multiplication of ordinals prompted the question. – Michael Hardy May 15 '17 at 00:04
  • Ah, wanting to do with ordinal arithmetic is a different story. Every other ordinal is $2\cdot\alpha$ for some $\alpha$, and all limit ordinals are even in this sense. But many fewer ordinals have the form $\alpha+\alpha$. For example, $\omega=2\cdot\omega$, but it cannot be written as $\alpha+\alpha$ for any $\alpha$. See for example the discussion at https://math.stackexchange.com/a/49046/413. – JDH May 15 '17 at 00:11
  • After posting above, I actually thought about this for 30 seconds and realized you may not have known this was that particular "different story". – Michael Hardy May 15 '17 at 01:45