Historically, the application of phase-change materials and devices has been limited to the provision of non-volatile memories. Recently, however, the potential has been demonstrated for using phase-change devices as the basis for new forms of brain-like computing, by exploiting their multilevel resistance capability to provide electronic mimics of biological synapses. Here, a different and previously under-explored property that is also intrinsic to phase-change materials and devices, namely accumulation, is exploited to demonstrate that nanometer-scale electronic phase-change devices can also provide a powerful form of arithmetic computing. Complicated arithmetic operations are carried out, including parallel factorization and fractional division, using simple nanoscale phase-change cells that process and store data simultaneously and at the same physical location, promising a most efficient and effective means for implementing beyond von-Neumann computing. This same accumulation property can be used to provide a particularly simple form phase-change integrate-and-fire neuron, which, by combining both phase-change synapse and neuron electronic mimics, potentially opens up a route to the realization of all-phase-change neuromorphic processing.