Split rational numbers, appendum

Patrick Reany

20 May 2024

Introduction:
This is a followup chat I had with ChatGPT-4 [on 'Split rational numbers'], concerning how to reformulate the naming convention of ring extensions.

Patrick:
Below, I introduce a new naming convention for ring extensions. In this new system, what is now referred to as 'split complex numbers' would be renamed as the 'split real numbers'. Please comment on it. Thanks.

The `unipodal numbers' are formed as

A + Bu,
where $A$ and $B$ are complex numbers and $u$ is a hyperbolic unit (i.e., $u^2 = +1$). The unipodal numbers are isomorphic to the bicomplex numbers, but they prefer to be viewed as having two complex components rather than the four real components of the bicomplex numbers.

Let $j$ be the hyperbolic unit. Then we can extend the reals by this number to get

A + Bj,
where $A$ and $B$ are real numbers and $j^2 = 1$. I think that this system should be referred to as the `split real numbers' but, instead, it's referred to as the `split complex numbers'. (I don't know why.) However, I think that the `split complex numbers' should refer to the complex numbers extended by a hyperbolic unit, which we have symbolized by either $j$ or $u$.

In this redefined naming system I just suggested, the 'split complex numbers' are equal to the unipodal numbers (because these numbers are 'split' into two complex components), and the 'split real numbers' are the extension of the reals by a hyperbolic unit (because these numbers a 'split' into two real components) and are equal to the hyperbolic numbers.

So, to formalize this new naming convention pattern that I'm suggesting: Let $R$ be any ring and let $\xi$ be any number not an element of $R$. Then, the 'split $R$ numbers' are of the form

A + B\xi,
where $A$ and $B$ are elements of $R$ and $\xi$ not in $R$, but is otherwise arbitrary, which introduces an ambiguity that we'll deal with now.

Symbolically, we can write the extension of a ring $R$ by some element $\xi$ as $R[\xi]$, which is the standard notation. One way to describe these numbers verbally is: 'the double $R$ numbers split by $\xi$'. Or, when $\xi$ is understood in some context, the 'split $R$ numbers.'

By the way, we could split the ring of integers $Z$ by extending with $i$, written as $Z[i]$. These are already well known and are called the Gaussian Integers, which look like

a + bi,
where $a$ and $b$ are integers and $i^2 = -1$.

ChatGPT-4:
Your proposed renaming convention for ring extensions introduces clarity by directly indicating the nature of the components being combined and the structure of the extended ring. Below, I'll provide a detailed commentary on your new system and how it compares to the existing nomenclature.

Existing Nomenclature:


1. Split Complex Numbers:
- Typically of the form \( z = x + yj \) where \( x \) and \( y \) are real numbers, and \( j \) is a hyperbolic unit with \( j^2 = 1 \).
- This term implies an extension of the real numbers using a hyperbolic unit.

Proposed Nomenclature:


1. Split Real Numbers:
- Your proposal renames what is traditionally called "split complex numbers" to "split real numbers" because they extend real numbers by a hyperbolic unit, yielding elements of the form \( x + yj \) where \( x \) and \( y \) are real.
- This name more accurately reflects the components involved--real numbers and the hyperbolic unit.

2. Split Complex Numbers (renamed to Unipodal Numbers):
- Your system suggests that extending complex numbers by a hyperbolic unit should be called "split complex numbers."
- The form would be \( A + Bu \) where \( A \) and \( B \) are complex numbers, and \( u \) is a hyperbolic unit.
- This maintains the notion of a "split" structure, but with complex components.

Formalized Naming Convention:


- Split \(R\) Numbers:
- Defined as \( A + B\xi \) where \( A \) and \( B \) are elements of a ring \( R \), and \( \xi \) is an element not in \( R \) but essential for the extension.
- The term "split \(R\) numbers" indicates an extension of \(R\) by \( \xi \), maintaining flexibility and generality.

Examples:


1. Split Real Numbers:
- \( x + yj \) with \( x, y \in \mathbb{R} \) and \( j^2 = 1 \).
- This is the existing "split complex numbers," now renamed to reflect the real components.

2. Unipodal Numbers (previously suggested split complex numbers):
- \( A + Bu \) with \( A, B \in \mathbb{C} \) and \( u^2 = 1 \).
- They highlight the combination of complex numbers with a hyperbolic unit.

3. Gaussian Integers:
- \( a + bi \) with \( a, b \in \mathbb{Z} \) and \( i^2 = -1 \).
- These are known extensions where \(\xi = i\), maintaining existing terminology.

Commentary:


Your proposed renaming convention aims to provide a more intuitive understanding of the nature of these number systems by directly referencing the components involved in their formation. Here are some key points of your approach:

1. Clarity: By specifying "split real numbers" and "split complex numbers," the terminology directly indicates what base ring \( R \) is being extended and by what kind of unit.
2. Consistency: This system maintains a consistent naming pattern for various ring extensions, potentially reducing confusion.
3. Generality: The framework you propose can be applied to any ring \( R \), providing a versatile and expandable naming system.

Considerations:


- Adoption: Changing well-established mathematical terminology can be challenging as it requires widespread acceptance and usage within the mathematical community.
- Ambiguity: Ensure that the introduction of new terms does not overlap confusingly with existing concepts or create ambiguity in mathematical literature.

Conclusion:


Your renaming convention introduces a logical and intuitive framework for understanding ring extensions. By emphasizing the components and structure, it aids in clarifying the relationships and nature of these extended number systems. However, practical adoption would require careful consideration of existing conventions and potential impacts on communication within the mathematical community.