Why bottle mortality in a formula?
By now you have two ways to describe how a life fades. The [[life-table|life table]] is the empirical one — a long column of numbers, one survivorship figure per integer age, read straight off the data. The [[force-of-mortality|force of mortality]] is the smooth one — the instantaneous rate of dying at each exact moment, the quantity that, woven through the survival–density–hazard relationship, generates the whole [[survival-function|survival function]]. A *mortality law* is the marriage of the two: a single compact formula for the force of mortality that, once chosen, reproduces an entire survival curve.
Why bother, when a table already holds the truth? Three reasons. A formula is *portable* — a couple of constants travel in a textbook problem where a thousand-row table cannot. It is *smooth*, so it hands you an answer at any age, including the fractional ages between a table's rungs, with no extra assumption. And it is *transparent*: when you can see mortality climbing as a clean exponential, you understand the shape of human death in a way a wall of digits never reveals. The price is honesty — no tidy curve fits real data perfectly, so a law is a deliberate simplification, never a fact.
Four classic laws, from toy to plausible
The family of [[analytic-mortality-laws|analytic mortality laws]] runs from charmingly crude to genuinely usable. The simplest is De Moivre (1729): assume deaths are spread *perfectly evenly* across ages up to some ultimate age ω, say 100. Then survival falls in a straight line and the force of mortality is 1 ÷ (ω − x) — rising gently, blowing up only at the very end. It is a toy: real deaths are not uniform. But it is wonderfully easy to integrate, so it earns its keep in first exercises and quick sanity checks.
Next is constant force: assume the force of mortality is the same number μ at every age. This makes the future-lifetime an exponential variable — the famously *memoryless* one, where a 20-year-old and an 80-year-old face the identical chance of dying next year. Plainly false for humans, whose risk climbs with age, yet it is the natural baseline for a short window where mortality barely moves, and, as we will see, it doubles as one of the fractional-age assumptions. Its survival curve decays as e to the power of −μ times t.
The leap to realism comes from Gompertz (1825), who noticed something profound: across much of adult life, the force of mortality grows *geometrically* — it multiplies by a fixed factor each year. Write it as μ(x) = B·c^x, with c a touch above 1. That single exponential captures the central truth of human ageing: your risk of dying roughly doubles every eight years or so from your thirties onward. Makeham (1860) then added one constant: μ(x) = A + B·c^x. The extra A is an age-independent floor — accidents, infections, sheer bad luck that strike young and old alike — and it makes the fit to real data markedly better, especially at younger ages where the pure exponential dips too low.
Lined up, the four are one rising sequence of force-of-mortality formulas, each buying a little more truth with a little more algebra. De Moivre uses 1 ÷ (ω − x), uniform deaths up to age ω. Constant force uses a flat μ, memoryless across age. Gompertz uses B·c^x, ageing as a clean exponential. Makeham uses A + B·c^x — Gompertz plus that accident floor A. Notice that constant force is just Gompertz with c set to exactly 1 (no ageing), and De Moivre is the odd one out, built on uniform deaths rather than a multiplying force. For real adult human mortality, the Gompertz–Makeham pair is the one to reach for.
What a law buys you — and what it cannot
The payoff is leverage. Because a law gives the force of mortality everywhere, you can integrate it once to get the survival function, and from there every probability you need — the chance of surviving any span, the distribution of the [[future-lifetime-random-variable|future-lifetime random variable]], the [[life-expectancy|life expectancy]] as a clean integral rather than a long summation. Under constant force, for instance, the complete expectation of future lifetime is simply 1 ÷ μ: a force of 0.02 implies a mean remaining life of 50 years, no table required. That kind of closed-form answer is gold in an exam hall and a sharp intuition-builder anywhere.
Between the rungs: fractional-age assumptions
Now the everyday problem. A life table tells you the chance a 60-year-old reaches 61, but a policy might pay on death within the *next six months*, or a premium might fall due at age 60-and-a-third. The table is mute between integer ages. [[fractional-age-assumptions|Fractional-age assumptions]] are the rules we adopt to interpolate — to invent a defensible shape for mortality across each single year of age, using only the two table values that bracket it. There are two workhorses, and the difference between them is exactly the difference between two of the laws above, applied one year at a time.
The first is the uniform distribution of deaths (UDD): within each year of age, assume deaths are spread evenly across the twelve months — De Moivre, scoped down to a single year. It makes the probability of dying within a fraction t of the year simply t times the whole-year death probability: the chance a 60-year-old dies in the first half-year is just half the chance of dying in the full year. UDD is the most popular assumption precisely because it is so easy to apply, and it interpolates the survivorship column linearly between integer ages.
The second is the constant force of mortality within each year: hold μ flat between integer ages — the constant-force law, scoped down to a single year. Here survivorship decays exponentially, not linearly, across the year, so the half-year survival probability is the square root of the whole-year one. The two assumptions usually give answers within a whisker of each other for the small probabilities of ordinary ages; they part company only when death rates are high or you slice the year very finely. A subtle but real contrast: under UDD the force of mortality actually *creeps upward* through each year, while constant force, by definition, holds it dead level.
A half-year, worked two ways
Make it concrete. Suppose the table says a 60-year-old has a 0.01 chance of dying before 61 — so the one-year survival probability is 0.99. What is the chance of dying within the first *half* year? Under UDD it is half the annual rate: 0.5 × 0.01 = 0.005. Under constant force the half-year survival probability is √0.99 ≈ 0.99499, so the death probability is about 0.00501. The two answers differ in the fifth decimal place. That tiny gap is the whole story of fractional-age assumptions: a real but usually negligible choice, made visible only when you zoom in close.
Table fact: q_60 = 0.01 (so p_60 = 0.99)
Half-year death probability, 0.5 * q_60 :
UDD : 0.5 * 0.01 = 0.00500
Constant force : 1 - sqrt(0.99) = 0.00501
^ differ in the 5th decimal