BMS 1110 · Common Core (Tronc commun)Section 2.1 · Measurement, ratios and conversions

Section 6 of 30

The metric system and mass

Name the metric units of mass, explain the prefix kilo, milli and micro as powers of ten, and convert between kilogram, gram, milligram and microgram by moving the decimal point three places per step.

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Why this matters

Almost every calculation you will do as a nurse begins with a conversion, and a conversion made the wrong way is not a small error: between a milligram and a microgram the gap is one thousand times. That single misread has reached patients as a thousand-fold overdose, because a nurse or a prescriber wrote “mg” where “mcg” was intended, or read “mcg” as “mg”. Countries that all use the same metric units can still harm patients when the conversion step is skipped or guessed. Learning the metric system properly means you no longer guess whether to multiply or divide: the prefix tells you, and you move the decimal point.

What you need to know first

You need to be comfortable reading decimals such as 0.25, 7.5 and 0.005, and multiplying or dividing a number by 1,000. If moving a decimal point still feels uncertain, write the number with a placeholder zero in front (0.25, not .25) and count the places one at a time on paper. Everything else in this section is built from that one skill.

Words to know (5)

Kilogram (kg): The base metric unit of mass, equal to 1,000 grams. It is the unit used for body weight and for any dose written per kilogram. A patient weighs 70 kg, which is 70,000 g and 70,000,000 mg.
Gram (g): A metric unit of mass equal to 1,000 milligrams — one thousandth of a kilogram. Tablet strengths above a gram are written in grams. A 500 g bag of glucose powder weighs 500,000 mg; the same drug given as 1 g is 1,000 mg.
Milligram (mg): One thousandth of a gram, written 10−3 g. Most tablet and capsule strengths, and most oral and injectable orders, are written in milligrams. A paracetamol tablet of 500 mg contains half a gram of drug.
Microgram (mcg): One thousandth of a milligram, written 10−6 g. Very potent drugs — digoxin, levothyroxine, some vitamins — are ordered in micrograms because a milligram would be far too much. Digoxin 0.25 mg is the same amount of drug as 250 micrograms.
Base-ten system: A measuring system in which every unit is ten, one hundred or one thousand times the unit beside it, so conversions are done by moving the decimal point rather than by a separate formula. 1 kg = 1,000 g, 1 g = 1,000 mg, 1 mg = 1,000 mcg: each step is a factor of one thousand.

One idea at a timeIdea 1 / 4

In one sentence Kilogram, gram, milligram and microgram are the four mass units in clinical use, and each step down is a thousand times smaller.

Mass has four metric units, and each is a thousand times the next

Dosage calculation is the process of applying mathematics to work out how much medication to give a patient, and it always begins from the units the order and the stock are written in. For mass, nursing uses four units and only four: the kilogram (kg), the gram (g), the milligram (mg) and the microgram (mcg). Their relationships are fixed and worth memorising as a chain: 1 kg = 1,000 g, 1 g = 1,000 mg, and 1 mg = 1,000 mcg. Following the chain one step further, 1 kg = 1,000,000 mg. Because every link in the chain is the same size — one thousand — the whole system behaves like a ladder with equal rungs. That is why you are never asked to remember a different factor for each pair of units. Knowing where a quantity sits on the ladder tells you whether the number in front of it must become bigger or smaller when you change units.

For example

A vial label reads 1 g of a powder, but the order is written in milligrams. Since 1 g = 1,000 mg, the vial holds 1,000 mg of drug — the same amount, written in a smaller unit.

Pause and check yourself

How many milligrams are there in 1 kilogram, and how did you get that number without a formula?

Show the answer

1,000,000 mg, because kilograms and milligrams are two rungs apart on the chain: kg → g is 1,000 and g → mg is another 1,000, so 1,000 × 1,000 = 1,000,000.

Read detailed explanations

Why moving the decimal point is exact, not a trick: The rule “move the decimal point three places” is often taught as a shortcut, which is why learners drop it under pressure. It is not a shortcut at all: it is the definition of a decimal place-value system. In a positional system, the value of a digit is set by where it stands relative to the units column, and each column is ten times the column to its right. Multiplying by 1,000 therefore multiplies every digit’s weight by 1,000, which sends every digit three columns to the left and moves the separator three places right. Dividing by 1,000 does the mirror image. That is why no arithmetic is needed: the digits themselves are unchanged, only their positions move. Two mechanical details decide whether the move is safe. When a place is empty, it must be filled with a placeholder zero, because a zero in a positional system is what holds the other digits in their correct columns; 4 g converted by writing 4000 mg keeps the columns aligned, while dropping the zeros would silently change the value. When the digits run out at the left, a leading zero is inserted so the separator is unmistakable: 0.25 mg rather than .25 mg, in which a tired reader can easily lose the point. The same reasoning explains why a conversion of two steps moves six places rather than three — each step is a separate multiplication or division by 1,000, and the two combine to 1,000,000, which is six columns. Anyone who can state that sentence has no need to memorise a per-unit conversion table ever again.
Prefix vocabulary and the thousand-step ladder of clinical doses: The prefixes are not arbitrary medical words; they are a shared international vocabulary, and each one is fixed by international agreement so that a prescription written in one country means the same in another. Kilo comes from the Greek for a thousand, and it appears in kilometre, kilojoule and kilogram alike. Milli comes from the Latin for a thousandth, and micro from the Greek for small — specifically one millionth. Because the prefixes are standard, a nurse who has never seen a particular drug can still read its order correctly. The clinical ladder those prefixes build deserves a second look, because it spans a range of about one billion. At the top, a patient’s body mass is measured in kilograms; a newborn is weighed in grams; a scored tablet or a sachet is ordered in grams or milligrams; an oral solid is usually in milligrams; and a potent drug such as digoxin, levothyroxine, vitamin B12 or a hormone is ordered in micrograms. Anaesthetic and hormonal agents can reach nanograms, one thousandth of a microgram, which is 10−9 g. The practical consequence is that the same digit string means wildly different amounts depending on the unit attached. A hundred of something is a body weight in kilograms, a moderate tablet dose in milligrams, and a catastrophic overdose in micrograms. Nurses therefore treat the unit as part of the number, never as a separate annotation, and state the prefix meaning out loud when checking a dose: “micro means one millionth, so this is a thousand times smaller than the milligram version of the same number.”
How unit misreading becomes a megadose, and the habits that stop it: A megadose is a dose many times larger than intended, and the metric system’s own convenience is what makes one possible. Because adjacent mass units differ by exactly one thousand, a single misread letter produces a thousand-fold error, not a ten percent one. The mechanism has been studied closely enough that patient-safety bodies now list the confused abbreviations among their “do not use” list. “mcg” and “mg” are the classic pair. A handwritten “mcg” whose c and g run together can be taken for “mg”; an order typed without a space can be misread; and “μg”, the correct scientific symbol for the microgram, is frequently mistaken for “mg” by a reader who does not recognise the Greek letter mu. The consequences are not symmetrical. If a microgram order is read as milligrams, the patient receives a thousand times too much — the megadose. If a milligram order is read as micrograms, the patient receives a thousandth of the intended amount, and the drug simply fails to work. Paediatric and neonatal wards carry the highest risk, because their doses are small to begin with and a tenfold or thousandfold error is easier to reach. Four habits reduce the risk to near zero, and every one of them is procedural rather than a matter of trying harder. Write “microgram” in full on anything you write by hand, and spell the unit out when you read an order aloud. Keep a leading zero before every decimal point (0.25 mg, never .25 mg) and never write a trailing zero after a decimal point in a dose order or note (2 mg, never 2.0 mg), because a trailing zero invites the reader to add a digit; a point written while counting places, such as 4.0 g, is a working mark to drop before the value is recorded. Read the stock label as well as the order, and compare the two units before calculating rather than after. Finally, have every dose that crosses a unit boundary — especially any microgram dose — calculated by one nurse and independently verified by a second.

Explore the diagram

Look at the whole drawing first. Then select one numbered part and connect it to the idea you just read.

The metric mass ladder: one step is × 1,000

The four clinical mass units drawn as an equal-rung ladder: kilogram, gram, milligram and microgram, each rung a factor of 1,000. Below it, five teaching panels: the decimal point moves three places right per step down and three places left per step up; 0.25 mg becomes 250 micrograms; reading 250 micrograms as 250 mg is a thousand-fold error; the same digit string 250 means different amounts depending on its unit; and a reverse check that converts back and lands on the number you started with.

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01 / 6The four mass units on equal rungs

If the drawing is wider than this panel, scroll it sideways. Labels stay readable; the numbered explanations below show the same ideas.

The four mass units on equal rungs

Clinical mass uses exactly four units: kilogram, gram, milligram and microgram. Each rung of this ladder is a factor of 1,000, so 1 kg = 1,000 g, 1 g = 1,000 mg and 1 mg = 1,000 mcg. Because every step is the same size, the pair you are converting between never changes the factor — only the number of steps changes. Two steps, as from kilograms to milligrams, combine to 1,000,000.

The ladder is drawn with equal rungs so that each step reads as the same factor of 1,000; it is a scale diagram of the steps, not of the drug amounts. The worked figures are the section’s own examples.

Key points to remember

The four mass units are the kilogram, gram, milligram and microgram, and each step down the chain kg → g → mg → mcg is a factor of 1,000.
The prefixes carry the power of ten: kilo means 1,000, milli means one thousandth (10−3) and micro means one millionth (10−6).
Converting to a smaller unit moves the decimal point three places right per step; converting to a larger unit moves it three places left per step.
0.25 mg = 250 micrograms and 7,500 mg = 7.5 g. The number grows when the unit shrinks, and shrinks when the unit grows.
A microgram read as a milligram is a thousand-fold error. Write microgram doses in full, keep a leading zero before the decimal point, and check the conversion in reverse.

Go deeper

Worked example

Converting 4 g into milligrams, three places at a time. A stock label states that a sachet contains 4 g of a powdered drug, but the prescription is written in milligrams. Before the drug can be measured, the nurse must express 4 g in milligrams and be able to say why the number grows.

Write out the chain and locate both units on it: kg → g → mg → mcg. The starting unit is the gram and the target unit is the milligram, one step down.
State the relationship in words with units: 1 g = 1,000 mg. A milligram is smaller than a gram, so there must be more milligrams than grams, and the number must grow.
Write 4 with a visible decimal point: 4.0 g. Moving one step down means three places to the right. Count them on paper: 4.0 → 40 → 400 → 4,000.
Give the answer with its unit: 4 g = 4,000 mg. Check the direction: the number grew, as predicted, and the unit is the smaller one.
Run the check in reverse to confirm: 4,000 mg moved three places left is 4.0 g. When the return trip lands on the number you started with, the conversion is sound.

4 g = 4,000 mg. The conversion needed no formula: the prefix told you the factor, the chain told you the direction, and the decimal point did the arithmetic. Use the same three moves — locate, state with units, move three places — for every mass conversion you meet.

Tutorial 1 — converting 0.25 mg into micrograms for a syringe

A prescription reads: digoxin 0.25 mg by slow intravenous injection. The ampoule is labelled in micrograms, and the syringe you will use is graduated in micrograms, so the dose must be converted before you can draw it up. Convert 0.25 mg into micrograms, showing every move.

Rule used: To convert a mass from a larger unit to a smaller unit, move the decimal point three places to the right for each step down the chain kg → g → mg → mcg. The number grows because the new unit is smaller.

Locate the two units on the chain kg → g → mg → mcg and count the steps between them.
Reveal this step
The starting unit is the milligram and the target is the microgram. They are next to each other on the chain, so there is exactly one step, and one step means three decimal places.
Will the number 0.25 become larger or smaller when you write it in micrograms? Answer before calculating.
Reveal the answer
It becomes larger. A microgram is a smaller unit than a milligram, so the same amount of drug needs more of them. Any conversion that makes the number smaller while the unit also becomes smaller must be wrong.
Write the number with a clear leading zero and a visible decimal point, then move the point three places right, one place at a time.
Reveal this step
Write 0.25 mg. First move: 2.5. Second move: 25. Third move: 250. The digits have shifted three places to the right and no zero had to be added.
Attach the new unit and write the complete statement, then read it aloud with the unit.
Reveal this step
0.25 mg = 250 micrograms. Read aloud: “zero point two five milligrams equals two hundred and fifty micrograms.” Saying the unit out loud is what stops the microgram being dropped.
The syringe is graduated in micrograms. What volume does 250 micrograms represent here, and what must you check on the ampoule label before drawing it up?
Reveal the answer
The 250 micrograms is a mass of drug, not a volume; to know how many millilitres to draw, read the concentration on the ampoule — for example 250 micrograms in 1 mL. Check the strength, the total volume and the expiry date, and have a second nurse verify the calculation before administering.

Answer: 0.25 mg = 250 micrograms. The decimal point moved three places to the right, exactly as the one-step drop from milligram to microgram requires.

Keep this habit: When the target unit is smaller, the number grows and the point moves right, three places per step. Always write the unit in full in your own notes for microgram doses, and never draw up a microgram order until the ampoule concentration has been read and the conversion checked by a second person.

Tutorial 2 — converting a 7,500 mg order back into grams

A patient is prescribed 7,500 mg of a drug that is supplied as a 2.5 g sachet of powder. The ward has only the sachet, and you must know whether one sachet is enough. Convert 7,500 mg into grams and compare the two quantities.

Rule used: To convert a mass from a smaller unit to a larger unit, move the decimal point three places to the left for each step up the chain kg → g → mg → mcg. The number shrinks because the new unit is larger.

Before calculating, will 7,500 become a larger or a smaller number when it is written in grams? Give your reason.
Reveal the answer
Smaller. The gram is a thousand times larger than the milligram, so the same mass needs a thousand times fewer units to express it. Converting 7,500 mg upwards to grams must shrink the number.
Write the number with its decimal point visible. If no point is shown, it sits after the last digit.
Reveal this step
7,500 is written 7500.0 here, with the decimal point after the final zero. Writing the point explicitly is what makes the three-place move safe.
Move the decimal point three places to the left for the single step from milligrams to grams, counting the places one at a time.
Reveal this step
Start at 7500.0. First move left: 750.00. Second: 75.000. Third: 7.5000. The digits have crossed the line and the value is read as 7.5 g.
Is one 2.5 g sachet enough to supply 7,500 mg, and how many sachets will you actually need?
Reveal the answer
One sachet is not enough: 7,500 mg is 7.5 g, and a 2.5 g sachet holds only a third of that. Three sachets supply 7.5 g, and the third would be opened only to its full 2.5 g, so this must be queried with the prescriber before preparing anything.
Finish by checking the answer in the other direction and writing the complete statement with units.
Reveal this step
7.5 g moved three places right is 7500 mg, which is the number we started from, so the conversion holds. The full statement is 7,500 mg = 7.5 g.

Answer: 7,500 mg = 7.5 g. The order therefore requires three 2.5 g sachets, which is a supply problem to raise rather than a calculation to force.

Keep this habit: Moving the point left means moving to a larger unit, so the number shrinks. Always convert the order into the unit printed on the stock before you decide how much to prepare, and always run the conversion back again: if the return trip does not land on the original number, something is misplaced.

Now you try

Attempt each one on paper first. Write the formula, substitute the numbers, then check the unit of your answer.

A prescription is written as 0.5 g of a drug, and the ward stock is labelled in milligrams. Convert the order into milligrams, state the direction and the number of places you moved, and check your answer in reverse.
Hint
The gram and the milligram are neighbours on the chain kg → g → mg → mcg, so one step is enough: three places, and the target unit is smaller, so the number must grow.
Show the worked answer
0.5 g = 500 mg. Writing 0.5 with a visible decimal point and moving it three places right gives 0.5 → 5 → 50 → 500. The number grew because the milligram is the smaller unit. The reverse check: 500 mg moved three places left returns to 0.5 g, so the conversion is correct.
A child is prescribed 7,500 micrograms of a drug supplied in milligram tablets. Convert the dose into milligrams and say whether the order can be given with a whole number of 1 mg tablets.
Hint
Microgram and milligram are one step apart, but this time you are moving up to the larger unit, so the point goes three places left and the number shrinks.
Show the worked answer
7,500 micrograms = 7.5 mg. Moving the point three places left gives 7500 → 750 → 75 → 7.5. A 7.5 mg dose cannot be given with whole 1 mg tablets, because half a tablet would be needed; the nurse must query whether a scored 0.5 mg presentation or a liquid formulation exists before administering anything.
A colleague handwrites an order as “250 mcg” but reads it aloud to you as “250 mg”. Explain the size of the error and describe the two concrete steps you would take before any drug is drawn up.
Hint
Compare the two amounts by converting the microgram figure into milligrams, then recall that the written unit and the stock label are both checkable facts.
Show the worked answer
The error is a factor of 1,000: 250 micrograms is 0.25 mg, so 250 mg is one thousand times the intended dose. Before drawing anything up, read the original written prescription rather than the spoken version, and read the concentration printed on the ampoule or packaging; then have a second nurse verify the figure and, if the written order remains unclear, telephone the prescriber for clarification.
A patient weighing 70 kg has a fluid chart in millilitres but a drug chart in milligrams. Using the chain kg → g → mg → mcg, write out the relationships you would need, and convert 70 kg into milligrams without using a separate formula.
Hint
Two steps separate the kilogram from the milligram, so the decimal point moves six places in total — three places for each step.
Show the worked answer
The relationships are 1 kg = 1,000 g and 1 g = 1,000 mg, so 1 kg = 1,000,000 mg. Two steps down means six places right: 70.0 → 700 → 7,000 → 70,000 → 700,000 → 7,000,000 → 70,000,000. So 70 kg = 70,000,000 mg. The number grew enormously because the milligram is a very small unit, which is why doses per kilogram are normally written in mg per kg rather than as a total in micrograms.

Common misconception

Beginners often believe that converting between metric units means deciding whether to multiply or divide by 1,000, and that the decision is a memory test. It is not. The direction is fixed by the size of the unit you are moving to: a smaller unit needs a larger number, a larger unit needs a smaller number. The second common error is reading a unit as a decoration rather than as part of the number. 0.25 mg and 0.25 micrograms are not the same quantity with different labels — they differ by a factor of one thousand. The habit that prevents both errors is to say the unit out loud with the number and to write microgram doses in full.

Check your reasoning

Explain in your own words why a base-ten measuring system makes conversion both easy and dangerous at the same time, using the milligram and the microgram as your example.

Reveal an answer

The system is easy because every step is the same size: one thousand, so the decimal point moves three places and no formula has to be remembered. It is dangerous for exactly the same reason. Because the steps are large and the abbreviations differ by a single letter, a unit read as its neighbour produces an answer that is wrong by a factor of one thousand rather than by a small amount, and a thousand-fold overdose can be fatal. The system gives no warning inside the arithmetic: 250 is a perfectly ordinary answer whether it is 250 micrograms or 250 mg. The warning has to come from the nurse — from reading the prescription carefully, writing microgram doses out in full, checking the stock strength and having the conversion verified by a second person.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Unit One “Dosage calculations”, lines 377–385: dosage calculation as the application of mathematical principles; the four mass units (kg, g, mg, mcg); the relationships 1 kg = 1,000 g, 1 g = 1,000 mg, 1 mg = 1,000 mcg and 1 kg = 1,000,000 mg; Example 1, 4 g → 4,000 mg; Example 2, 7,500 mg → 7.5 g; Example 3, 0.25 mg → 250 mcg. Lines 376 and 386 (chapter heading and volume measurements) confirm the range boundaries.
Note on the course code · The supplied cover page reads “Course Number: BMS 110” while the curriculum catalogue in this project records “BMS 1110”. The catalogue is treated as authoritative; the notes’ cover value is recorded here for provenance only.
Beyond the supplied notes: decimal place value and the three-place move · The notes state the three conversion factors but never explain the mechanism. The positional place-value account of why moving the decimal point three places per step is exact, the placeholder and leading-zero conventions, and the six-place result for a two-step conversion are standard decimal arithmetic that the notes assume rather than teach.
Beyond the supplied notes: prefix etymology and the clinical dose ladder · The Greek and Latin origins of kilo, milli and micro, the nanogram extension of the mass ladder, and the observation that the same digit string can denote a body weight, a tablet dose or a megadose depending on its unit are not in the supplied notes.
Beyond the supplied notes: safe abbreviation practice · The “do not use” status of “mcg” and “μg”, the ban on trailing zeros after a decimal point, the asymmetric consequences of reading a microgram order as milligrams versus the reverse, the elevated paediatric and neonatal risk, and the two-nurse independent verification of unit-crossing doses are patient-safety practice drawn from standard medication-safety guidance rather than from the notes.

Lesson progress is saved on this device for the current learner. Practice answers stay separate and are not stored.

About this course and its sources

Bilingual lessons organised from the supplied BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (Mr. Ebangan Prosper Achuanoh). Diagrams are original educational schematics; sections that go beyond the supplied notes say so in their sources. Mathematics, clinical and French-language review are pending.