Transreal Numbers Construction from Real Number Pairs

A 2012 paper by Tiago Soares dos Reis and Walter Gomide from the Instituto Federal do Rio de Janeiro proposes a formal construction of the transreal numbers. This mathematical extension of the real numbers is defined as a class of subsets of ordered pairs of real numbers, establishing closure under addition, subtraction, multiplication, and division, including division by zero.

Use Cases

• Formalizing the transreal number system using subsets of ordered pairs of real numbers.
• Analyzing the axiomatic properties of arithmetic operations, including division by zero, within the constructed set.
• Studying the embedding of the real numbers as a subset within the broader transreal number structure.
• Investigating the logical consistency and consequences of extending the real numbers via ordered pair constructions.

Strengths

• Provides a formal, axiomatic construction from a foundational mathematical object (real numbers).
• Explicitly defines the set and its arithmetic operations, including division by zero.

Limitations

• The dataset is a theoretical paper; it contains no empirical data, sample size, or numerical records for analysis.
• Practical applications or computational implementations are not demonstrated within the provided description.

Provenance

Source

Paper authored by Tiago Soares dos Reis and Walter Gomide, published on Papers with Code.

Collection Method

Theoretical mathematical construction and proof.

Time Range

2012