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Question 1:

How is the term Number System defined?

Notes:

  • A) I do not mean Numerical System (like Decimal System, Binary System). The two terms are frequently confused. A Numerical System is the representation of numbers through a sequences of digits and other symbols (minus sign, point)

  • B) Number System is sometimes regarded as synonym for Number Set. For me a Number Set is a set with the Natural Numbers as subset. I think it is more than just a set and I would expect additional required properties.

  • C) Number System could be a ring with injection from a subset of Natural Numbers and being compatible with common addition and multiplication. (In this sense Hypercomplex Number Systems are no Number Systems as their multiplication is not commutative)

  • D) The condition can not be to have an injection from all Natural Numbers, as some Numbers Systems are finite (like integers mod n).

  • E) Number System might be a ring as it contains addition and multiplication. I think a Number System is not just a set.

Question 2:

What is the difference between a "number system" and a ring in algebra? Both have a set of elements, addition and multiplication.

Question 3:

Has a number system to be constructed from natural numbers (like complex and hypercomplex number systems do)?

  • F) Having an order can not be a condition as complex numbers are not ordered.

I look for a formal definition and could not find any. References to publicly available sources (preferable pdf textbooks) are most welcome.

I appreciate your help and comments.

Bill Dubuque
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wdo
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  • https://en.wikipedia.org/wiki/Number – Zubin Mukerjee Nov 16 '24 at 10:17
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    Is any ring a number system? No, the ring of smooth functions is not a number ring. But the ring of integers $\mathcal{O}_K$ of a number field $K$ might qualify. You should consider another word than "number system". In algebra and number theory, a ring of integers might be the better option. Note that this is no answer to the question. You have to specify before an answer is possible. – Dietrich Burde Nov 16 '24 at 11:48
  • @DietrichBurde
    Why is the ring of smooth functions not number ring?
    As ring you can add and multiply.

    What is the condition that a ring is a number ring?
    What is the condition that a field is a number field?

     – wdo Nov 16 '24 at 12:13
  • @wdo Perfectly reasonable question. I had the impression that the OP did not want to consider such things as a "number system". Smooth functions are not "generalised numbers". But we need some clarification. – Dietrich Burde Nov 16 '24 at 12:17
  • @DietrichBurde: Thanks for your responses to my question. The main point is that I can hardly clarify the question as the main point is just to get a general common definition of a Number System. I would not like to define it myself, but rather look for a definition. (That could be definition suggested by your) (side topic: what is an OP in your comment) – wdo Nov 16 '24 at 12:24
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    @Dietrich But then you need to define "number". There is no widely accepted definition - it can denote elements of various types of algebraic structures. – Bill Dubuque Nov 16 '24 at 18:47
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    I do not think there is a commonly accepted notion of a "number system". – Moishe Kohan Nov 16 '24 at 19:06
  • Duplicate of What exactly is a number? See also the Linked posts there. – Bill Dubuque Nov 16 '24 at 20:02
  • I will offer an answer I posted on philosophy.se to a similar question https://philosophy.stackexchange.com/questions/118009/what-separates-numbers-from-other-mathematical-objects-and-what-justifies-e-g-t/118019#118019 – JonathanZ Nov 16 '24 at 20:04
  • Here's a good reddit thread on the topic: https://redd.it/1emp79i – Cameron L. Williams Nov 16 '24 at 20:19
  • I personally would consider any commutative ring that includes reals as a subset a number system. As such, quaternions and various cardinals and ordinals are not a number system. – Anixx Mar 09 '25 at 07:41

1 Answers1

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I do not think "number system" has any formal modern definition. I am not a historian, but based on the literature I've seen over the years, "number system" is an antiquated term that predates ring theory, and probably also predates the revolution in foundations of mathematics in the 20th century.

For example, Hamilton referred to quaternions as a "system" with certain axioms in 1843. In Hilbert's Foundations of Geometry, from 1903, he uses komplexes Zahlensystem for what we would call today an ordered division ring. The translation gives it both as "complex number system or just number system."

From references like these I gathered that "system" was the initial term used to describe a set (you will recall that the theory of sets was in its infancy at the time) with axioms governing its elements, and number systems were early definitions of sorts of rings.

If I were you I'd cease wondering any more about what the particular term could mean. You will find plenty of material on rings and algebras that encompass the same ideas, and which are usually set on pretty firm naming conventions.

rschwieb
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  • Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. The question is already almost closed, and already has a duplicate link. – Bill Dubuque Nov 16 '24 at 21:02