Topogical properties of the set of algebraic numbers

[invite286face9]

Hello,

Question is: Modulus or Module?

Context comes from a French book and concerns the construction of "the number system and (structural) topological properties".

(1). Set of natural numbers -> (2). Set of integers -> (3). Set of rational numbers -> [(4). Set of real numbers , (5). Set of algebraic numbers] -> (6). Set of complex numbers

Consider the "topological properties" of such sets:

(3). Topological property of the "Set of rational numbers" is: "Non complete metric space for the topology of the absolute value".

(4). Topological property of the "Set of real numbers" is: "Complete metric space".

(5). Topological property of the "Set of algebraic numbers" is: "Non complete metric space for the topology of the modulus". (Is it Module or Modulus here?)

(6). Topological property of the "Set of complex numbers" is: "Complete metric space (any Cauchy sequence converges)".

*In french module is used to express both English terms Modulus and Module, so to translate the text from french to english I meet a problem. Furthermore, the term "Module" regarded as a "vector space" in which the scalar is a ring rather a field, is used in particular in Abstract algebra , Homology theory and Extension field.

I share my doubt with you, is it Modulus or Module?

(In spite of what precedes:*) I think it could be rather "Modulus" referring to the metric notion.

Is there someone to help me to resolve this question?
Thank you so much.
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[Médiat]

I do believe it is "Module", the complex equivalent of the the "absolute value".

[invite9cf21bce]

Salut.

The absolute value on the set of real numbers is still called "absolute value" in English, when extended to the set of complex numbers. But the term "modulus" is a synonym :

Wikipedia (En)

In French, the situation is different, as we usually use the word "module" for that notion (and also in the context of "modules over a ring" = "modules sur un anneau", so I agree with you, it's ambiguous), and prefer to say "valeur absolue" only in the case of real or p-adic numbers.

Taar.

[invite286face9]

Thanks. The context and remarks comes from a French book in which unfortunately there is no comment. Such an approach seems very interesting, but a doubt appears to me about this question. I think it is modulus rather than module, but doubt persists.

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[invite286face9]

I do believe it is "Module", the complex equivalent of the the "absolute value".

Math Oxford dictionary says: "Modulus of complex number". It's why I would tend to use rather modulus. What do you think?

[Médiat]

Math Oxford dictionary says: "Modulus of complex number". It's why I would tend to use rather modulus. What do you think?

Undoubtedly, my English is a bit less accurate than the Math Oxford dictionary’s one, so I have to admit it is modulus .
The important part is that it is the complex equivalent of the "absolute value".

[invite286face9]

Undoubtedly, my English is a bit less accurate than the Math Oxford dictionary’s one, so I have to admit it is modulus .
The important part is that it is the complex equivalent of the "absolute value".

Thanks. Th.

[invite286face9]

So, is there someone who can decide between Modulus and module within such a framework?

[invite9cf21bce]

Salut.

So, is there someone who can decide between Modulus and module within such a framework?

Mediat and I actually just did this.

It's "modulus".

Taar.

[invite286face9]

Salut.



Mediat and I actually just did this.

It's "modulus".

Taar.

Thanks Taar and Mediat. Th.