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The Many-Worlds Interpretation of Quantum Mechanics
In the particular case of momentum equals mass times velocity, it is an implication of Hamiltonian mechanics. But I don't think that's the motivation that led Hamilton to invent Hamiltonian Mechanics. Irrational numbers are very, very important in physics and math. If you have some span of real numbers between 1 and 2, almost all of them are irrational.