Given five things:
- The number 0, denoted 0
- The ability to find ex for any number x, denoted {x}
- The ability to find the principle natural logarithm log(x) for any number x, denoted [x]
- The ability to find the additive inverse -x for any number x, denoted <x>
- The ability to find the sum x+y for any pair of numbers x and y, denoted (x+y)
What numbers can you make and how long are their denotations? This gives some sort of metric to how "complicated" a number is. Write the length of the smallest possible denotation for number x as L(x). Then:
- Subtraction: a-b is denoted as (a+<b>) and L(a-b) <= 5+L(a)+L(b)
- Multiplication: ab is denoted as {([a]+[b])} and L(ab) <= 9+L(a)+L(b)
- Division: a/b is denoted as {([a]+<[b]>)} and L(a/b) <= 11+L(a)+L(b)
- Exponentiation: ba is denoted as {{([a]+[[b]])}} and L(ba) <= 13+L(a)+L(b)
Some interesting numbers, with their complexities:
| 1 | {0} | 3 |
| e | {{0}} | 5 |
| -1 | <{0}> | 5 |
| 2 | ({0}+{0}) | 9 |
| i | {{([[<{0}>]]+<[({0}+{0})]>)}} | 29 |
| π | <{([[<{0}>]]+[{{([[<{0}>]]+<[({0}+{0})]>)}}])}> | 47 |
Some interesting questions:
- How does the complexity function L grow with its argument?
- What interesting numbers do not have finite complexity?
- How could the game be changed to include them?
Related: Fine structure constant update.
[...] commenter on Tuesday’s post wondered what the density function of numbers with low complexity looks like. This seemed like an [...]