  The symbolic representation of a number should be considered as a data
structure, and the choice of data structure depends on the arithmetic
operations that are to be performed. Numbers are almost universally represented
using position based notations based on exponential powers of a base number -
usually 10. This representations is computationally efficient for the standard
arithmetic operations, but it is not efficient for factorisation. This has led
to a common confusion that factorisation is inherently computationally hard. We
propose a new representation of the natural numbers based on bags and using the
prime successor function as a primitive - prime bags (PBs). This data structure
is more efficient for most arithmetic operations, and enables numbers can be
efficiently factored. However, it also has the interesting feature that
addition appears to be computationally hard. PBs have an interesting
alternative interpretation as partitions of numbers represented in the standard
way, and this reveals a novel relationship between prime numbers and the
partition function. The PB representation can be extended to rational and
irrational numbers, and this provides the most direct proof of the
irrationality of the square root of 2. I argue that what needs to be ultimately
understood is not the peculiar computation complexity properties of the decimal
system (e.g. factorisation), but rather what arithmetical operator trade-offs
are generally possible.
