  Complex numbers are an intrinsic part of the mathematical formalism of
quantum theory, and are perhaps its most mysterious feature. In this paper, we
show that the complex nature of the quantum formalism can be derived directly
from the assumption that a pair of real numbers is associated with each
sequence of measurement outcomes, with the probability of this sequence being a
real-valued function of this number pair. By making use of elementary symmetry
conditions, and without assuming that these real number pairs have any other
algebraic structure, we show that these pairs must be manipulated according to
the rules of complex arithmetic. We demonstrate that these complex numbers
combine according to Feynman's sum and product rules, with the modulus-squared
yielding the probability of a sequence of outcomes.
