When is it better to call a thing "modulus" and when "determinant"?

[Anixx]

In what cases it is better to call a thing "modulus" and in what cases "determinant"? In my algebra "determinant" is not a norm, discontinuous, positive for non-zero elements, not abiding triangle inequality. Should I better call it "modulus"?

 

[PeroK]

If a determinant is nothing like a modulus, why call it one?

 

[fresh_42]

What do you mean? A determinant is continuous, not automatically positive for non-zero elements, and who claims it is a norm? Modulus is the term which is not uniquely defined.

 

[mathman]

What are the things that you are using these terms for? When I see "modulus" it usually refers to absolute value (complex number). "Determinant" is a property of a matrix. Obviously your context is different.

 

[Anixx]

What are the things that you are using these terms for? When I see "modulus" it usually refers to absolute value (complex number). "Determinant" is a property of a matrix. Obviously your context is different.

The context: https://math.stackexchange.com/questions/4022965/what-intuitive-meaning-determinant-of-a-divergency-divergent-integral-series

 

[nuuskur]

Determinant is effectively a polynomial. Polynomials are continuous. There exist non-zero matrices with negative or 0 determinant, too.

 

[Anixx]

Thanks, this looks like maybe "modulus" could be more appropriate.

 

[nuuskur]

Are you getting confused with notation such as $|A|$ or $|x|$, where $A$ is a square matrix and $x\in\mathbb C$, say?

 

[fresh_42]

Thanks, this looks like maybe "modulus" could be more appropriate.

You missed the message: It is not!

1. You compare apples and oranges. The terms stem from different areas and must not be compared.
2. Your statements about the properties of the determinant were mostly wrong.
3. The answers you received tried to list evidence why the answer is "No", they do not support your suggestion.
4. Modulus is the more general term, i.e. is context sensitive whereas determinant is not.
5. The terms have been used for centuries. Are you sure the mathematical world waited for someone to change this?
6. There is simply no need, except you want to confuse basically everybody.

 

[Anixx]

> Your statements about the properties of the determinant were mostly wrong.

Which ones?

> The answers you received tried to list evidence why the answer is "No"

"No" to what, modulus or determinant?

> The terms have been used for centuries.

Which one?

> There is simply no need, except you want to confuse basically everybody.

I did not understand what's your advice is. Is it in favor of modulus or determinant?

 

[nuuskur]

You should call determinants determinants and moduli to be moduli. These are not interchangeable. While you may believe mathematics is about confusing people with fancy trickery and play with symbols, it's quite the opposite. We like simplicity in mathematics, as simple as we can make it. That's the dream, at least.

 

[fresh_42]

> Your statements about the properties of the determinant were mostly wrong.

Which ones?

Continuity, sign.

> The answers you received tried to list evidence why the answer is "No"

"No" to what, modulus or determinant?

To substitute one with the other.

> The terms have been used for centuries.

Which one?

Both.

> There is simply no need, except you want to confuse basically everybody.

I did not understand what's your advice is. Is it in favor of modulus or determinant?

Modulus has at least one, probably more defined meanings, depending on the context.
Determinant has one precise meaning, and has nothing to do with modulus.

Both make sense by themselves. Using one for the other creates confusion. Only confusion.

 

[Anixx]

So, you are in favor of calling the thing "modulus", yes?

 

[fresh_42]

No. Non. Nein. Nem. Aʻole. нет. לא. não. 不. لا. Οχι. नहीं।. 番号。

 

[Vanadium 50]

Don't hold back. Tell us what you really think.

 

[nuuskur]

@Anixx why do you keep insisting on this false dilemma?

 

[Anixx]

@Anixx why do you keep insisting on this false dilemma?

Are there other better solutions?

 

[Vanadium 50]

Yes. We continue to call "modulus" modulus and "determinant" determinant.

So far as I can tell, you have not addressed a single point anyone has raised. You are just repeating your assertion over and over and over and over...

 

[Mark44]

Question has been asked and answered, so it's time to close this thread.

 

[fresh_42]

Are there other better solutions?

Let me explain it differently.

There is only one way to understand the word determinant over (for simplicity) a field $\mathbb{F}$. Namely the group homomorphism
$$
\operatorname{det}\, : \,\operatorname{GL}(n,\mathbb{F}) \longrightarrow \mathbb{F}-\{0\}
$$
from the group of regular matrices over $\mathbb{F}$ to the group of units of $\mathbb{F}$. It extents to all square matrices over $\mathbb{F}$ by adding $\operatorname{det}(M)=0$ for singular matrices.

There are at least four ways to understand modulus.

1. The ideal $n\mathbb{Z}$ in $\mathbb{Z}$ is called modulus in modular arithmetic, i.e. in all calculations in $\mathbb{Z}_n.$
2. A modulus is a formal product $\prod_p p^{\nu(p)}\, , \,\nu(p)\geq 0$ in algebraic number theory.
3. A modulus is the absolute value of a real number.
4. A modulus is the absolute value of a complex number.

The only possibility to compare these two terms is the possibility $1$, since it is a group homomorphism, too, in that case. It is even a ring homomorphism. However, whereas we have all kind of fields in the case of a determinant, we have only discrete rings in case of a modulus. This is an important difference. Finite ideals cannot be summarized under infinite groups and vice versa. They are simply two completely different things.

Now you come along and suggest the following: Forget about the uniquely defined term determinant and call it by the term modulus which already has four meanings. Let's give it a fifth.

Why don't we call all and everything a modulus then? Would make life much easier. Or maybe not.