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Base-13 Representation

Base-13 representation (also known as tridecimal or trinary-ternary hybrid in specific contexts) is a positional numeral system with radix 13. It utilizes thirteen distinct symbols (typically 0–9 and A–C) to represent values.

While rarely used in practical computing, base-13 is mathematically significant due to its role in specific pathological function constructions, most notably conways-base-13-function.

Key Properties

• Radix: 13
• Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C
• Digit Values: A=10, B=11, C=12
• Positional Value: Each position represents a power of 13 ($13^n$).

Mathematical Significance

Conway’s Base 13 Function

The most prominent application of base-13 representation is in the definition of conways-base-13-function, a function $f:\mathbb{R}\to \mathbb{R}$ with extreme discontinuity properties.

• Definition Mechanism: The function interprets the base-13 expansion of a real number $x$.
• Pattern Matching: It scans the expansion for specific patterns (e.g., digits representing signs and decimal points) to determine the output.
• Intermediate Value Property: Despite being discontinuous everywhere, the function satisfies the intermediate value property on every interval.
• Surjectivity: The function maps every open interval onto the entire real line $\mathbb{R}$.

Related Concepts

• real-analysis: Study of functions defined on real numbers.
• Pathological Function: Functions that exhibit counter-intuitive behavior.
• Numeral Systems: General theory of positional notation.

References

• Conway’s Base 13 Function: Explaining its Unique Mathematical Definition: Conway’s Base 13 Function: Explaining its Unique Mathematical Definition
• Conway’s Base 13 Function: Explaining its Unique Mathematical Definition