Answering the narrow version, because the broad one does not have a single answer. 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.

TrialNerd_Beth said:

Do it in two steps and it stops being confusing.

That is correct as far as it goes, and here is where it stops going. The arithmetic is right and one caveat matters: a U-40 syringe changes the markings, not the volumes. If someone hands you U-40 and you read it as U-100 you will be 2.5 times out, and that error has a direction — it is always an overdose.

B12Beth said:

I reconstituted a 10mg vial with 2ml of bacteriostatic water and I want someone to check my unit arithmetic before I draw anything, because I keep…

Same position here, arrived at the long way round. Concentration choice is a precision decision, not a preference. Reconstitute high and every dose is a tiny volume where one unit of syringe error is a large fraction of the dose. Reconstitute low and you get more graduations per dose, so the same hand tremor costs proportionally less. Against that, more diluent means more benzyl alcohol and a shorter comfortable in-use window.

Clinical perspective, offered as context rather than as advice.

Compounding pharmacy dose accuracy check for dosing arithmetic: I always verify my first dose from a new vial by weighing it on a milligram scale. Draw up your dose, weigh the syringe, subtract empty syringe weight, compare to expected water weight at that volume.

For a 10-unit dose (0.1mL): expected weight ~100mg of water. If it's significantly off, your syringe may be inaccurate. Cheap insulin syringes can have ±5% volume accuracy. Spring for quality brands like BD or Terumo.