Half-Life Calculator

Remaining quantity after radioactive decay.

Percentage change is the operation most often misapplied in daily life, because a rise and a fall of equal size do not cancel: down 20% and back up 20% leaves the original figure 4% lower. A percentage point and a percent are different quantities.

These follow standard order of operations — exponents first, then multiplication and division, then addition and subtraction, with brackets overriding everything. Logarithms invert exponents, roots invert powers, and a quadratic has two solutions unless the discriminant collapses them into one.

A half-life is the time for half a quantity to decay, and the process is exponential rather than linear: after two half-lives a quarter remains, not none. The same arithmetic describes radioactive decay, drug clearance and some kinds of depreciation.

Rounding happens only at the end, so a long chain of steps can drift by a unit in the final digit.

Order of operations is a convention rather than a mathematical truth, and calculators, spreadsheets and programming languages all follow the same one. Percentage points, percentages and ratios each answer a different question even when they use identical numbers.

Half-Life Calculator sits with the rest of the library's maths work, and it rests on the same algebraic arithmetic the way a textbook or an exam paper does. Its answer is only as good as the assumptions standing behind it.

The quantities it works from — initial quantity, half-life and elapsed time — are the ones a real algebraic question actually turns on, and each carries a convention of its own, set out below.

Mathematics is the one subject here that depends on nothing local: the same theorem holds in Ohio and Osaka, and the same fraction carries the same value everywhere. What changes is how results are written — American schooling teaches decimal notation and long division, and business arithmetic runs on fractions of a dollar.

Precision and rounding are where most practical errors creep in.