## A little mathematical game

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.

### One Response to “A little mathematical game”

1. [...] commenter on Tuesday’s post wondered what the density function of numbers with low complexity looks like. This seemed like an [...]