Conway's Base 13 Function - Numberphile

Key Concepts:

• Function: A mapping that assigns a unique output to each input.
• Base 10: The standard decimal number system.
• Base 13: A number system using 13 digits (0-9, A, B, C).
• Real Numbers: All rational and irrational numbers.
• Conway's Base 13 Function: A function that takes a real number expressed in base 13 and maps it to a real number in base 10 using a specific algorithm.
• Fractal: A self-similar pattern that repeats at different scales.
• Measure: A way of assigning a "size" or "volume" to sets.

1. Understanding Functions

• A function, denoted as f(x), maps an input (x) to a unique output. For example, f(x) = x² + 1.
• The graph of a function visually represents this mapping, with the x-axis representing the input and the y-axis representing the output (y = f(x)).
• Key property: A function must provide the same output for the same input consistently. f(1) will always be 2 in the example f(x) = x² + 1.
• Functions can be discontinuous, like step functions, which have jumps in their graph.
• Example of a discontinuous function: f(1/n) = 1 for positive integers n, and f(x) = 0 otherwise. This creates a graph with points at 1, 1/2, 1/3, 1/4, etc., approaching zero.

2. Base 13 Representation

• Base 10 uses digits 0-9. Base 13 extends this with symbols for 10, 11, and 12, typically A, B, and C.
• In base 13, the number 13 is represented as 10.
• Conversion: To convert 99 (base 10) to base 13: 99 / 13 = 7 with a remainder of 8, so 99 (base 10) = 78 (base 13).
• Real numbers can be represented in base 13, similar to decimal representation (e.g., 3.14 in base 10 is 3 * 10⁰ + 1 * 10⁻¹ + 4 * 10⁻²).

3. Conway's Base 13 Function Algorithm

• The function takes a real number expressed in base 13 as input.
• The algorithm looks for the last occurrence of the digit 'A' or 'C' in the base 13 expansion.
• If the last 'A' is found, everything before it is discarded. The 'A' is replaced with a positive sign, the subsequent digit is treated as the integer part, 'B' is treated as the decimal point, and the remaining digits are interpreted as a base 10 decimal.
• If the last 'C' is found, everything before it is discarded. The 'C' is replaced with a negative sign, the subsequent digit is treated as the integer part, 'B' is treated as the decimal point, and the remaining digits are interpreted as a base 10 decimal.
• Example 1: AC13.07BC7AB010101000... becomes +0.010101000...
• Example 2: 77349A.7CBAC49B141414... becomes -49.141414...
• If there is no 'A' or 'C' in the base 13 expansion, or if the expansion consists only of infinite sequences of 'A' or 'C', the function's behavior is different (see section 4).

4. Properties and Implications of Conway's Function

• The function maps base 13 numbers to base 10 numbers using a specific algorithm.
• The mapping is not unique; many different base 13 numbers can map to the same base 10 number.
• For any real number, one can find numbers arbitrarily close to it that, when fed into Conway's function, will output any chosen real number. This means that if you pick any real number like pi, you can find a base 13 number extremely close to any other base 13 number that will output pi when run through Conway's function.
• Graphing the function: If you were to graph this function, and zoom out far enough, it would appear to color the entire plane because of the density of different values.
• Measure Theory: Almost every number is mapped to zero. The set of numbers that map to non-zero values has a negligible measure. This means if you randomly pick a number, it will almost definitely be mapped to zero.

5. Base 13 is Not Special

• The choice of base 13 is not crucial. The same principle can be applied with other bases, as long as there's a sufficient difference (at least 3) to represent the positive/negative sign and the decimal point.
• You can start in base 10 and convert to base 7, or start in base 5 and convert to binary.
• The algorithm can be modified to incorporate the discarded values in the calculation.

6. Notable Quotes

• "The map is not the territory. The menu is not the food. This video is not me. It's a faximile of me." - Emphasizing the distinction between representation and reality.

7. Conclusion

Conway's base 13 function is a mind-blowing example of a function with highly unusual properties. It demonstrates how a seemingly simple algorithm can create a mapping that is non-unique, dense, and fractal-like. The choice of base 13 is merely a convenient implementation detail, and the underlying principle can be generalized to other bases. The function highlights the counterintuitive nature of real numbers and the complexities that can arise in mathematical mappings.