Split rational numbers

Patrick Reany

20 May 2024

Introduction:
One of the favorite topics of beginning ring theory is to extend the rational numbers by some irrational number, like the square root of 2. But what does that look like? Let $a$ and $b$ be any two rational numbers, then the extension of the rationals by the $\sqrt{2}$ looks like

$a + b\sqrt{2}$
I don't know if this extension has a specific name, but I thought I could invent one for it. I modeled this `new name' after the so-called `split complex numbers'. Let $A$ and $B$ be any real numbers and let $j$ be a noncomplex number that squares to unity, or $j^2 = 1$ (this $j$ is sometimes referred to as a `hyperbolic unit'). Then the split complex numbers are of the form
$A + Bj$

Patrick:
I defined the `split-rational numbers' as follows: let $a,b$ be any rational numbers and let $\xi$ be any irrational number. Then a + b\xi is a split-rational number. For example, consider the ring of all split-rational numbers when $\xi = \sqrt{2}$. Comment, please?

ChatGPT-4-turbo:
The concept of split-rational numbers that you defined, \(a + b\xi\) where \(a\) and \(b\) are rational numbers and \(\xi\) is an irrational number, is quite interesting. This definition essentially forms a ring structure that extends the field of rational numbers \(\mathbb{Q}\) by adjoining an irrational element \(\xi\).

To analyze this structure more formally, let's consider the case where \(\xi = \sqrt{2}\):

Ring Structure

The set of all split-rational numbers in this context can be denoted as \(\mathbb{Q}[\sqrt{2}] = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}\).

Addition

Addition in this ring is straightforward:
\[ (a + b\sqrt{2}) + (c + d\sqrt{2}) = (a + c) + (b + d)\sqrt{2} \] Since \(a, b, c,\) and \(d\) are all rational numbers and the sum of rational numbers is rational, the result is still of the form \(a' + b'\sqrt{2}\) where \(a' = a + c\) and \(b' = b + d\) are rational.

#### Multiplication
Multiplication also follows standard rules:
\[ (a + b\sqrt{2})(c + d\sqrt{2}) = ac + ad\sqrt{2} + bc\sqrt{2} + bd(\sqrt{2})^2 = (ac + 2bd) + (ad + bc)\sqrt{2} \] Here, both \(ac + 2bd\) and \(ad + bc\) are rational, so the product remains in the form \(a' + b'\sqrt{2}\) where \(a' = ac + 2bd\) and \(b' = ad + bc\).

Properties

- Commutativity: Both addition and multiplication are commutative because they adhere to the commutative properties of rational numbers.
- Associativity: Both operations are associative for the same reason.
- Distributivity: Multiplication is distributive over addition.
- Identity Elements: The additive identity is \(0 + 0\sqrt{2} = 0\), and the multiplicative identity is \(1 + 0\sqrt{2} = 1\).

Ring Extension

\(\mathbb{Q}[\sqrt{2}]\) can be seen as a quadratic extension of \(\mathbb{Q}\), denoted by \(\mathbb{Q}(\sqrt{2})\). It is an example of an algebraic number field, specifically a quadratic field. This extension is generated by adding \(\sqrt{2}\) to \(\mathbb{Q}\), and any element of \(\mathbb{Q}(\sqrt{2})\) can be uniquely written as \(a + b\sqrt{2}\).

Integral Domain

\(\mathbb{Q}[\sqrt{2}]\) is an integral domain because it is a subset of the field of real numbers \(\mathbb{R}\), and there are no zero divisors.

Noetherian Ring

\(\mathbb{Q}[\sqrt{2}]\) is also Noetherian, as it is a finitely generated \(\mathbb{Q}\)-module.

Applications

These types of rings are useful in number theory and algebraic geometry. For instance, studying the solutions of Diophantine equations often involves such rings. They are also essential in the study of algebraic integers and units in quadratic fields.

Conclusion

The ring of split-rational numbers \(\mathbb{Q}[\sqrt{2}]\) is a well-structured algebraic entity. It maintains many of the desirable properties of \(\mathbb{Q}\) while extending its capabilities by incorporating irrational elements. This extension leads to rich mathematical structures and provides a fertile ground for further exploration in algebra and number theory.

Patrick:
I got the idea for the name 'split-rational numbers' from the name 'split complex numbers'.

ChatGPT-4-turbo:
The term 'split-rational numbers' derived from 'split complex numbers' is an intriguing choice. It suggests a similar structural extension from the familiar set of rational numbers to a new set incorporating an irrational component, much like how split complex numbers extend real numbers.

Split Complex Numbers

To draw a parallel, let's first briefly summarize split complex numbers. The split complex numbers are of the form \( z = x + yj \), where \( x \) and \( y \) are real numbers, and \( j \) is a unit with the property \( j^2 = 1 \) (unlike the imaginary unit \( i \) in complex numbers where \( i^2 = -1 \)).

Split-Rational Numbers

In the case of split-rational numbers, defined as \( a + b\xi \) where \( a, b \in \mathbb{Q} \) and \( \xi \) is an irrational number (e.g., \(\sqrt{2}\)), we have a different structure, yet the naming convention suggests a similar extension approach.

Key Differences and Similarities:

1. Number Systems:
- Split Complex Numbers: Extend the real numbers \(\mathbb{R}\) by adjoining \( j \) with \( j^2 = 1 \).
- Split-Rational Numbers: Extend the rational numbers \(\mathbb{Q}\) by adjoining \(\xi\) where \(\xi\) is an irrational number.

2. Algebraic Structure:
- Split Complex Numbers: Form a 2-dimensional algebra over \(\mathbb{R}\) with specific rules for multiplication based on \( j^2 = 1 \).
- Split-Rational Numbers: Form a 2-dimensional vector space over \(\mathbb{Q}\) with multiplication influenced by \(\xi\).

Example with \(\xi = \sqrt{2}\)

Let's revisit the ring of split-rational numbers when \(\xi = \sqrt{2}\):
\[ \mathbb{Q}[\sqrt{2}] = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\} \]

Algebraic Operations

- Addition:
\[ (a + b\sqrt{2}) + (c + d\sqrt{2}) = (a + c) + (b + d)\sqrt{2} \]
This operation is straightforward, preserving the form \(a' + b'\sqrt{2}\) with rational \(a'\) and \(b'\).

- Multiplication:
\[ (a + b\sqrt{2})(c + d\sqrt{2}) = (ac + 2bd) + (ad + bc)\sqrt{2} \]
This operation also preserves the form \(a' + b'\sqrt{2}\) with rational coefficients.

Ring Properties

- Commutativity and Associativity: Both operations inherit these properties from the rational numbers.
- Distributivity: Multiplication distributes over addition.
- Identity Elements: The additive identity is \(0\), and the multiplicative identity is \(1\).

Algebraic Significance

The split-rational numbers \(\mathbb{Q}[\sqrt{2}]\) are significant in algebra as they form an example of a quadratic field. Such fields are vital in the study of algebraic numbers and have applications in solving polynomial equations, number theory, and algebraic geometry.

Conclusion

The analogy between split-rational numbers and split complex numbers is appropriate in the sense that both extend a familiar number system by incorporating a new element that introduces additional structure and properties. The term 'split-rational numbers' effectively captures this idea of extension and the new algebraic framework it creates. The exploration of these numbers can lead to deeper insights into algebraic extensions and their applications.

Afterwards:

The name ``split complex numbers" (also referred to as perplex numbers, hyperbolic numbers, or double numbers) lives inside that realm of mathematics laden with some confusing names. Let's start with the real numbers ℝ, which are not confusing. We can define the complex numbers (also not confusing) as the extension of the ℝ by the non-real number $i$, where $i^2 = -1$. So, the complex numbers look like
A + Bi,
where $A$ and $B$ are real numbers. If we extend the complex numbers by a second imaginary number $i'$ (i.e., $i'^2 = -1$), we get what's called the `bicomplex numbers', which have the form (we assume that $ii' = i'i$)
A + Bi + Ci' + Dii',
where $ii'$ squares to +1 and thus it acts as a hyperbolic unit.

By the way, the quaternions represent a second way to extend the real numbers to get a four-component extension, but this ring is noncommutative, whereas the ring of bicomplex numbers is commutative. The reason that the quaternion ring is noncommutative is because the extension elements by which they are based do not commute. We can represent the quaternions as the double extension of the reals, first by the unit imaginary $i$ and then by a different unit imaginary $j$, where $ij = -ji$. So, a quaternion looks like

A + Bi + Cj + Dij,
which can be written as
A + Bi + Cj + Dk,
where $k = ij$, and where ijk = -1$.

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 $u$. By the way, in the unipodal number system, the hyperbolic unit element $u$ is referred to as being `unipotent' (since $u^2 = +1$).

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 $i^2 = -1$.