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Commutation functions pre-combine interest and mortality so that annuity and assurance values become simple ratios. Dₓ = v^x·lₓ is the discounted number of survivors and Nₓ is the running total of Dₓ from age x upwards, giving äₓ = Nₓ/Dₓ.
Commutation columns
Dₓ = v^x·lₓ; Nₓ = ΣDₓ₊ₖ; äₓ = Nₓ/Dₓ
Before computers they turned every annuity valuation into one division, using columns tabulated once per mortality basis.
It models survivors falling by a constant proportion each year, which makes Nₓ a geometric sum and keeps the illustration self-contained.