This is the version of the explanation I wish somebody had given me, written down before I forget what confused me. It is about the dosing arithmetic, and it is deliberately narrow — everything I am not confident about is marked as such.
What is actually established
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.
The condition it depends on
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.
The practical version
The three numbers to write on the vial: total mg, total ml, and mg per ml. Everything else is division. And the sanity check is that dose volume times number of doses should be less than the volume you put in, because dead space takes the difference.
What I am not sure about
The narrow version of the question is how to check the arithmetic without trusting a website calculator, since three calculators gave me three answers. Not looking for reassurance. Looking for the part I have got wrong.
— SallyK_inj · corrections welcome and will be edited into this post with credit