Exact and approximate
This site does two kinds of arithmetic with two different guarantees, and mixing them up would be a way of overstating one of them. So every figure is labelled.
Educational arithmetic only. Not medical advice and not a dosing protocol.
The exact kind
Every quantity in the vial arithmetic is held as a fraction of two whole numbers and rounded once, at the point of display. Five milligrams in three millilitres is held as exactly five thirds of a milligram per millilitre, not as 1.6666666666666667, and a chain of four conversions cannot accumulate drift in the digit a reader acts on.
That covers the draw count, the remainder, the exact ratio, every strength, every volume, every unit reading, and the fractions at whole multiples of a half-life - because one half to the power of a whole number is rational. Those figures are not estimates and are not labelled as such.
The approximate kind
The fraction remaining at an arbitrary time is
0.5t ÷ t½, and for most inputs the exponent is fractional and
the value is irrational. There is no exact decimal to print. The same applies to everything
derived from it: the retained fraction across an interval, the accumulation limit, the
number of intervals to approach it, and every point on a plotted curve.
Those are computed in double-precision floating point and every one of them is labelled about in the worksheets and approximate in the tables. The error is far below anything that matters at the precision these figures are quoted to; the label is there because the distinction is real, not because the numbers are shaky.
| Figure | Kind | How it is labelled |
|---|---|---|
| Draws in a vial, and the remainder | Exact | no qualifier |
| Strength from a measured finished volume | Exact | no qualifier |
| Strength from a mass and a diluent volume | Exact, but a bound | at most |
| Volume derived from a bounded strength | Exact, but a bound | at least |
| Unit reading on a barrel | Exact | with the nearest printed mark named |
| Fraction remaining at whole half-lives | Exact | no qualifier |
| Fraction remaining at any other time | Approximate | about |
| Accumulation limit and intervals to settle | Approximate | about |
A bound is a third category and the most useful one
A strength computed from a mass and a diluent volume is exact arithmetic on the wrong denominator. The dissolved solid takes up room, so the finished solution is larger than the diluent and the true strength is lower than the figure. The calculation has no error in it; the premise does.
Which makes the honest label "at most" rather than "approximately". The figure is not uncertain in both directions - it is an upper bound, true for every possible displacement, and the volume derived from it is correspondingly a lower bound. Getting the direction of an uncertainty right is worth more than widening it symmetrically.
Why this distinction is worth a page
Because precision is persuasive. A figure printed to three decimal places reads as authoritative whether or not the quantity behind it deserves three decimals, and a site that prints both kinds in the same typeface with no label is borrowing credibility from the half it earned.
It also matters for checking. An exact figure can be reproduced by hand and must match to the last digit; an approximate one will not, and knowing which you are checking saves concluding there is a bug when there is only a rounding difference in the twelfth decimal place.
Related
- Vial yield is a divisionThe exact half of the site, and the one place a floor matters.
- Where the series comes fromThe approximate half, derived.
- How the figures are madeWhere every number here comes from and how it is checked.