Chalmers talks about quantum mechanics, which work with real numbers. Complex numbers had been used in physics before quantum mechanics but always as a kind of algebraic trick to make the math easier. Two-dimensional complex numbers are exactly what one needs to describe the fuzzy, smeared world. Within quantum theory, things like electrons and photons are represented by wave functions that completely describe all the many possible states of a single particle. These multiple personalities are depicted by a series of complex number within the wave function that describe the probability that a particle has a particular property such as a certain location or momentum.
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