Quantum fractionalism: The Born rule as a consequence of the complex Pythagorean theorem

•Projection factors describe measure contraction in orthogonal projections.•Quantum probabilities equal projection factors of ray on eigenspaces of observable.•Complex Pythagorean theorem corresponds to unit total probability.•In Everettian quantum mechanics projection factors give relative amounts of worlds.•Relative amounts of worlds play all roles associated to quantum probabilities. Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces. Postulating the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds, we show that projection factors give relative amounts of worlds. These appear as relative frequencies of results in quantum experiments, and play the role of probabilities in decisions and inference. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.