What a label does not say

The arithmetic here is exact. That is a property of the calculation and says nothing whatever about what is in the vial it was performed on.

Educational arithmetic only. Not medical advice and not a dosing protocol.

Where every calculation on this bench begins

With a number printed on a container. The mass of solid, or the strength per millilitre, or the potency in international units: whichever tool is in use, the first field is a figure somebody else wrote on a label, and nothing downstream can check it.

That is not a criticism of labels. It is a statement about where the uncertainty in an answer actually lives. The arithmetic contributes almost none of it. The label, the preparation, the dissolution and the reading contribute all of it.

Four things a correct answer still does not know

Whether the vial holds what the label claims

A vial marked 5 mg contains what its manufacturer says it contains. Assay figures, where they exist at all, belong to a batch rather than to a vial, and no arithmetic recovers them.

Whether the solid dissolved

Strength is mass over volume on the assumption that the mass is in solution. Undissolved material on the shoulder of the vial is mass that is not in the liquid being drawn, and it makes the real strength lower than the computed one by an amount nobody has measured.

Whether the barrel is accurate

A graduation is a printed mark with a manufacturing tolerance behind it. The barrel pages compute what the printed resolution costs; they cannot say anything about how closely a given barrel matches its own printing.

Whether the reading was read correctly

The meniscus, the angle of the eye, the light in the room. All real, none visible to software, and all larger in effect than the difference between an exact calculation and a rounded one.

What the bench will not tell you

This is educational arithmetic on figures entered by the reader. It is not medical advice, not a dosing protocol, and not a recommendation that anything be prepared, obtained or administered.

Nothing here states what quantity is appropriate for anyone, and there is no function in it that produces an amount to administer. A question about anyone’s health belongs with a licensed clinician who can examine the person asking.

Compounds of the kind this arithmetic is used on are largely research chemicals that are not approved by the FDA for human use. An exact calculation is not an accurate measurement: what is actually in a syringe depends on the preparation and the device, neither of which this page can see.

No qualified clinical reviewer is recorded against this site, and no credential is claimed anywhere on it. the limits of this bench.

Why the bench says this out loud

Because precision is persuasive. A figure printed to three decimal places reads as authoritative whether or not the quantity behind it deserves three decimal places, and a tool that prints one without comment is borrowing credibility it has not earned.

So the answers here carry their basis with them. A strength derived without a measured volume says "at most". A volume derived from it says "at least". A reading that does not land on a mark names the mark it does not land on. A curve computed in floating point says it is approximate. None of that is hedging; it is the difference between what was calculated and what is known.

What is left after all that

Quite a lot, actually. The arithmetic being reliably right removes one whole category of error from a situation that has several, and it makes the remaining uncertainty honestly about the preparation rather than about the maths. That is the job this bench is for, and it is the whole of it.

Everything beyond it - whether a figure is appropriate, whether a preparation is what it says, whether any of this should be happening at all - sits outside arithmetic entirely. The limits of this bench page says so at length, and means it.