Conway's Base 13 Function: Explaining its Unique Mathematical Definition

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

Clip title: Conway’s Base 13 Function - Numberphile Author / channel: Numberphile URL: https://www.youtube.com/watch?v=zsBaApyf_Ns

Summary

This Numberphile video introduces Conway’s Base 13 Function, a mathematical concept designed by John Horton Conway, which challenges our intuitive understanding of what a function can represent. The presenter, Asaf Karagila, begins by explaining the common perception of a function: a rule that assigns a unique output (y) to every input (x), often visualized as a continuous or discontinuous graph. He illustrates this with simple examples like f(x) = x^2 + 1 and also highlights step functions, which, despite jumps, still adhere to the unique output rule. The core idea is that for any given input, the function’s output is predetermined and constant.

To understand Conway’s function, the video first delves into number bases, specifically base 13. Unlike the familiar base 10 system with digits 0-9, base 13 requires thirteen unique symbols. Conventionally, this is achieved by using 0-9 followed by A, B, and C to represent values 10, 11, and 12, respectively. Thus, in base 13, the number 13 is written as “10”. Any real number can be expressed in base 13, similar to how irrational numbers like Pi have infinite decimal expansions in base 10. This foundational understanding of alternate number systems is crucial for comprehending the intricate definition of Conway’s function.

Conway’s function operates by taking a real number expressed in base 13 and converting it into a real number in base 10 through a specific, albeit complex, set of rules. The function scans the base 13 representation for specific patterns of digits, particularly looking for the non-standard digits A, B, or C. For example, if the last ‘A’ in the base 13 sequence is followed by a run of only 0-9 digits and then a ‘B’, those 0-9 digits are interpreted as a decimal number, and ‘A’ dictates the sign (positive), while ‘B’ marks the end of this interpretable sequence. If a number’s base 13 expansion doesn’t conform to these specific patterns (e.g., an infinite sequence of A’s, B’s, or C’s after a certain point), the function is defined to output 0. The key takeaway here is that while the algorithm guarantees a unique base 10 output for any given base 13 input, many vastly different base 13 inputs can result in the same base 10 output.

The truly mind-blowing aspect of Conway’s Base 13 Function emerges when considering its graph. Due to its construction, any real number (e.g., Pi or the square root of 2) can be found as the output of the function for an infinite number of different inputs. Moreover, these inputs can be arbitrarily close to each other on the x-axis. If one were to graph this function, zooming out would reveal that the graph appears to fill the entire two-dimensional plane. This defies the intuitive notion of a function having a single, discernible line or curve. Despite this chaotic visual and many-to-one mapping, it remains a mathematically valid function because its rule is consistently applied, producing a unique result for each specific input. The number 13 is not inherently special beyond providing convenient symbols for the algorithm; the same principle could apply to other bases.

Video Description & Links

Description

John Conway’s amazing Base 13 Function, demonstrated by Asaf Karagila. More links & stuff in full description below ↓↓↓

Asaf is a UKRI Future Leaders Fellow. Asaf’s blog - https://karagila.org

More videos and Numberphile podcast featuring Asaf - https://www.youtube.com/playlist?list=PLt5AfwLFPxWJyt0zdvzvDoeL_8pqO0S7p

The late John Conway on Numberphile: https://www.youtube.com/playlist?list=PLt5AfwLFPxWIL8XA1npoNAHseS-j1y-7V

We are also grateful for support from Ben Delo.

NUMBERPHILE

Video by Brady Haran and Pete McPartlan

Brady’s videos subreddit: http://www.reddit.com/r/BradyHaran/

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Conway's Base 13 Function: Explaining its Unique Mathematical Definition