The thousand-fold error

Not an arithmetic mistake. A unit mistake, which produces an answer that is out by a factor of 1,000 and looks exactly as confident as a correct one.

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

Why this particular error and not another

Two units are in constant use in this arithmetic and they differ by a thousand. Vials are labelled in milligrams: 5 mg, 10 mg. Amounts are usually talked about in micrograms: 250 mcg, 500 mcg. Every calculation crosses between them, and both numbers are small, plausible and typeable.

What makes it dangerous is that nothing about the output changes. A volume of 0.2 mL and a volume of 200 mL are both numbers; one of them is a draw and one of them is a jug. But 0.02 mL and 0.2 mL are both draws, both fit a barrel, and only one of them is right.

The three things that stop it here

The unit is part of the field

Every quantity on this bench is entered as a number and a unit together, as one control. There is no field that accepts a bare figure and assumes which unit it is in, because that assumption is the error. The selector sits against the input rather than above the form, so the unit is read at the same time as the number.

Conversions are exact integer ratios

A milligram is a thousand micrograms by definition, so the conversion is a multiplication by exactly 1,000 and nothing is approximated on the way through. Quantities are held as fractions of whole numbers and rounded once, at the point of display, so a chain of four conversions cannot accumulate drift in the digit a reader acts on.

Mismatched dimensions are refused

Micrograms into millilitres is not a conversion: it needs a strength, and inventing one would be worse than refusing. So the tools return a refusal with the reason rather than a number, and the two tools that do have a strength to work through are named in it.

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.

The same amount expressed four ways, and the volume it occupies at one strength. Reading across a row is safe; reading down a column and assuming a unit is where the factor of a thousand comes in.
In mcg	In mg	At 2.5 mg/mL	On a U-100
100 mcg	0.1 mg	0.040 mL	4 units
250 mcg	0.25 mg	0.100 mL	10 units
500 mcg	0.5 mg	0.200 mL	20 units
1,000 mcg	1 mg	0.400 mL	40 units
2,500 mcg	2.5 mg	1.000 mL	100 units

The one combination that is safe to do in your head

A solution at exactly 1 mg/mL puts one microgram in every microlitre, so an amount in micrograms is the same number of microlitres. 250 mcg is 250 microlitres, which is 0.25 mL, which is 25 units on a U-100. No factor to apply and nothing to get the wrong way round.

That is why 1 mg/mL turns up in so many worked examples, and why a 2 mg vial in 2 mL is the combination most of them use. It is the one arrangement where the units do the work for you.

Related notes

The mass and volume tool
Every factor, exactly, with no intermediate rounding.
What a label does not say
The uncertainty that survives correct arithmetic.