This one has a reasonably settled answer, so here it is. Dead space is the answer to the missing dose. A fixed-needle insulin syringe holds a few microlitres in the hub and needle after the plunger bottoms out, and on small draws that is a measurable percentage of every dose. Across ten draws it adds up to most of an eleventh, which is exactly the "nine draws from a ten-dose vial" complaint. Luer-lock syringes are worse; low-dead-space fixed-needle designs are better.

Dr.NateNeph said:

Dead space is the answer to the missing dose.

Agreed, and it is worth saying that rounding to the nearest whole unit is usually acceptable at maintenance doses and is not acceptable at the bottom of a ladder, where one unit can be a fifth of the intended dose.

Dr.GastroMayo said:

Two of us with identical vials and identical diluent arrived at different unit counts, which means one of us has a conceptual error rather than a…

Can confirm the pattern Dr.GastroMayo describes. Do it in two steps and it stops being confusing. First concentration: 10mg into 2ml is 5mg/ml. Then volume: a 0.5mg dose is 0.5 ÷ 5 = 0.1ml. Then units, and this is where people go wrong — a U-100 syringe is graduated in hundredths of a millilitre, so 0.1ml is 10 units. The word "units" has nothing to do with milligrams; it is a volume marking that exists because insulin happens to come at 100 units per ml.

Correct me if the detail matters more than I have assumed.

Clinical perspective, offered as context rather than as advice.

Comparison of compounded dosing arithmetic delivery formats I've tried:

I use lyophilized vials for cost and stability. The reconstitution takes 30 seconds once you've done it a few times.