9.1 Roman Numerals: Additive, Subtractive & Apothecary Rules

Roman numerals represent an ancient numeric system developed during classical antiquity that still appears in modern health care. Roman numerals are a standard HESI A2 math review topic, so aim for complete fluency. In clinical environments, Roman numerals are routinely encountered in controlled substance schedules (Schedule $\text{I}$ through Schedule $\text{V}$), cranial nerve assessments (Cranial Nerves $\text{I}$ through $\text{XII}$), blood clotting factors (such as factor $\text{VIII}$), New York Heart Association functional classes $\text{I}$ through $\text{IV}$, and legacy prescription orders utilizing apothecary units. Misreading or miswriting a Roman numeral in clinical practice can lead to a tenfold dosing discrepancy or severe medication toxicity.

The Seven Fundamental Roman Symbols

Unlike the modern Hindu-Arabic numeral system, the Roman numeral system does not utilize positional place-value zeros. Instead, it builds all numerical quantities from seven foundational uppercase Latin letters, each anchored to a permanent integer value:

Mnemonic Devices for Rapid Recall

To memorize the descending order of Roman numerals from largest ($1000$) to smallest ($1$), memorize the classic sentence:

$\mathbf{M}\text{y } \mathbf{D}\text{ear } \mathbf{C}\text{at } \mathbf{L}\text{oves } \mathbf{X}\text{-tra } \mathbf{V}\text{itamins / } \mathbf{I}\text{njections}$

$\text{M } (1000) \longrightarrow \text{D } (500) \longrightarrow \text{C } (100) \longrightarrow \text{L } (50) \longrightarrow \text{X } (10) \longrightarrow \text{V } (5) \longrightarrow \text{I } (1)$

Notice that the symbols alternate between powers of ten ($\text{M}, \text{C}, \text{X}, \text{I}$) and five-based sub-bases ($\text{D}, \text{L}, \text{V}$). This distinction dictates which symbols can be repeated and subtracted.

The Rule of Repetition

Numbers are generated by writing symbols sequentially. However, strict syntactical rules limit repetition:

1. Powers of Ten May Repeat up to Three Times: The symbols $\text{I}$, $\text{X}$, $\text{C}$, and $\text{M}$ may be written consecutively up to three times to represent additive multiples:
   • $\text{I} = 1$, $\text{II} = 2$, $\text{III} = 3$ (Never $\text{IIII}$)
   • $\text{X} = 10$, $\text{XX} = 20$, $\text{XXX} = 30$ (Never $\text{XXXX}$)
   • $\text{C} = 100$, $\text{CC} = 200$, $\text{CCC} = 300$ (Never $\text{CCCC}$)
   • $\text{M} = 1000$, $\text{MM} = 2000$, $\text{MMM} = 3000$ (Standard Roman numerals do not exceed $3{,}999$ without vinculum horizontal overbars)
2. Five-Based Symbols Never Repeat: The symbols $\text{V}$, $\text{L}$, and $\text{D}$ can never be repeated consecutively:
   • $\text{VV}$ is strictly invalid because $5 + 5 = 10$, which is correctly written as $\text{X}$.
   • $\text{LL}$ is strictly invalid because $50 + 50 = 100$, which is correctly written as $\text{C}$.
   • $\text{DD}$ is strictly invalid because $500 + 500 = 1000$, which is correctly written as $\text{M}$.

The Rule of Addition

When a Roman numeral of equal or smaller value appears immediately after (to the right of) a numeral of greater or equal value, add the values together:

$\text{Combined Value} = \text{Larger Value} + \text{Smaller Value}$

Consider these clinical and mathematical examples:

• $\text{VI} = 5 + 1 = 6$
• $\text{VIII} = 5 + 1 + 1 + 1 = 8$
• $\text{XII} = 10 + 1 + 1 = 12$ (Cranial Nerve $\text{XII}$: Hypoglossal nerve)
• $\text{LV} = 50 + 5 = 55$
• $\text{CLXX} = 100 + 50 + 10 + 10 = 170$
• $\text{MDCC} = 1000 + 500 + 100 + 100 = 1700$

The Rule of Subtraction & Strict Positional Constraints

To prevent strings of four identical characters, subtractive notation is used. When a smaller numeral appears immediately before (to the left of) a larger numeral, subtract the smaller value from the larger value:

$\text{Combined Value} = \text{Larger Value} - \text{Smaller Value}$

However, subtraction is governed by three constraints:

Constraint 1: Only Powers of Ten Can Be Subtracted

Only the symbols $\text{I}$, $\text{X}$, and $\text{C}$ can precede a larger numeral to perform subtraction. The five-based symbols ($\text{V}$, $\text{L}$, and $\text{D}$) are never subtracted. For example:

• $45$ is never written as $\text{VL}$ ($50 - 5$). The correct representation is $\text{XLV}$ ($40 + 5$).
• $495$ is never written as $\text{VD}$ ($500 - 5$). The correct representation is $\text{CDXCV}$ ($400 + 90 + 5$).
• $950$ is never written as $\text{LM}$ ($1000 - 50$). The correct representation is $\text{CML}$ ($900 + 50$).

Constraint 2: The Two-Step Subtraction Hierarchy

A subtractive power-of-ten numeral can only be subtracted from the next two higher symbols in the hierarchy:

• $\text{I}$ ($1$) can only be subtracted from $\text{V}$ ($5$) and $\text{X}$ ($10$): $\text{IV} = 5 - 1 = 4$ $\text{IX} = 10 - 1 = 9$ $\text{I}$ cannot be subtracted from $\text{L}, \text{C}, \text{D},$ or $\text{M}$. For example, $99$ is not $\text{IC}$ ($100 - 1$); it must be written as $\text{XCIX}$ ($90 + 9$). Similarly, $49$ is not $\text{IL}$ ($50 - 1$); it must be written as $\text{XLIX}$ ($40 + 9$).
• $\text{X}$ ($10$) can only be subtracted from $\text{L}$ ($50$) and $\text{C}$ ($100$): $\text{XL} = 50 - 10 = 40$ $\text{XC} = 100 - 10 = 90$ $\text{X}$ cannot be subtracted from $\text{D}$ ($500$) or $\text{M}$ ($1000$). For example, $490$ is not $\text{XD}$ ($500 - 10$); it must be written as $\text{CDXC}$ ($400 + 90$). And $990$ is not $\text{XM}$; it must be written as $\text{CMXC}$ ($900 + 90$).
• $\text{C}$ ($100$) can only be subtracted from $\text{D}$ ($500$) and $\text{M}$ ($1000$): $\text{CD} = 500 - 100 = 400$ $\text{CM} = 1000 - 100 = 900$

Constraint 3: Exactly One Subtractive Numeral

You may place only one smaller numeral before a larger numeral. Multiple subtractive prefixes are strictly prohibited:

• $8$ is written as $\text{VIII}$ ($5 + 3$), never as $\text{IIX}$ ($10 - 2$).
• $18$ is written as $\text{XVIII}$ ($10 + 5 + 3$), never as $\text{IIXX}$ ($20 - 2$).
• $70$ is written as $\text{LXX}$ ($50 + 20$), never as $\text{XXC}$ ($100 - 20$).

Side-by-Side Analysis: Valid versus Invalid Numerals

The table below contrasts commonly mistaken invalid constructions with their correct Roman numeral counterparts:

Place-Value Decomposition (Converting Numbers up to 3,999)

The most dependable strategy for writing any Arabic number up to $3{,}999$ in Roman numerals is place-value decomposition. Break the number into thousands, hundreds, tens, and units, translate each group independently, and concatenate the results.

Standard Place-Value Building Blocks

• Thousands: $1000 = \text{M}, \; 2000 = \text{MM}, \; 3000 = \text{MMM}$
• Hundreds: $100 = \text{C}, \; 200 = \text{CC}, \; 300 = \text{CCC}, \; 400 = \text{CD}, \; 500 = \text{D}, \; 600 = \text{DC}, \; 700 = \text{DCC}, \; 800 = \text{DCCC}, \; 900 = \text{CM}$
• Tens: $10 = \text{X}, \; 20 = \text{XX}, \; 30 = \text{XXX}, \; 40 = \text{XL}, \; 50 = \text{L}, \; 60 = \text{LX}, \; 70 = \text{LXX}, \; 80 = \text{LXXX}, \; 90 = \text{XC}$
• Units: $1 = \text{I}, \; 2 = \text{II}, \; 3 = \text{III}, \; 4 = \text{IV}, \; 5 = \text{V}, \; 6 = \text{VI}, \; 7 = \text{VII}, \; 8 = \text{VIII}, \; 9 = \text{IX}$

Step-by-Step Conversion Walkthroughs

Example 1: Convert $1{,}984$ to Roman Numerals

1. Decompose into expanded place values: $1{,}000 + 900 + 80 + 4$
2. Translate thousands: $1000 = \text{M}$
3. Translate hundreds: $900 = 1000 - 100 = \text{CM}$
4. Translate tens: $80 = 50 + 10 + 10 + 10 = \text{LXXX}$
5. Translate units: $4 = 5 - 1 = \text{IV}$
6. Concatenate: $\text{M} + \text{CM} + \text{LXXX} + \text{IV} = \mathbf{\text{MCMLXXXIV}}$

Example 2: Convert $2{,}449$ to Roman Numerals

1. Decompose: $2{,}000 + 400 + 40 + 9$
2. Translate thousands: $2000 = \text{MM}$
3. Translate hundreds: $400 = 500 - 100 = \text{CD}$
4. Translate tens: $40 = 50 - 10 = \text{XL}$
5. Translate units: $9 = 10 - 1 = \text{IX}$
6. Concatenate: $\text{MM} + \text{CD} + \text{XL} + \text{IX} = \mathbf{\text{MMCDXLIX}}$

Example 3: The Maximum Standard Value ($3{,}999$)

1. Decompose: $3{,}000 + 900 + 90 + 9$
2. Translate: $\text{MMM} + \text{CM} + \text{XC} + \text{IX} = \mathbf{\text{MMMCMXCIX}}$

Decoding Roman Numerals to Arabic Numbers

To translate a Roman numeral back into an Arabic integer, scan the expression from left to right:

1. If a symbol is followed by a symbol of equal or lesser value, add its value.
2. If a symbol is followed by a symbol of greater value, subtract the current symbol from the next symbol, add the difference to the total, and skip past the paired symbol.

For instance, to evaluate $\text{MCDXLVII}$:

• $\text{M} = 1000$
• $\text{CD} = 500 - 100 = 400$
• $\text{XL} = 50 - 10 = 40$
• $\text{V} = 5$
• $\text{II} = 1 + 1 = 2$
• Sum: $1000 + 400 + 40 + 5 + 2 = \mathbf{1{,}447}$

Clinical Apothecary Usage & Prescription Conventions

In pharmacology and nursing practice, Roman numerals appear in handwritten and legacy electronic medical orders, particularly within the apothecary system of measurement.

Lowercase Notation

Prescriptions and medication administration records (MARs) traditionally express Roman numerals in lowercase letters: $\text{i}, \text{ii}, \text{iii}, \text{iv}, \text{v}, \text{vi}, \text{vii}, \text{viii}, \text{ix}, \text{x}$.

The Anti-Tampering Terminal 'j' Rule

In handwritten orders, physicians historically replaced the final letter '$\text{i}$' in a sequence with a letter '$\text{j}$' (often with a horizontal line or dots placed above the numerals):

• $\text{ij} = 2$
• $\text{iij} = 3$
• $\text{vij} = 7$
• $\text{viij} = 8$

The tail on the terminal '$\text{j}$' served as a vital clinical safety feature. It marked the definitive end of the number, preventing unauthorized individuals from adding an extra vertical line to alter the quantity of a controlled narcotic prescription (e.g., transforming $\text{ii}$ into $\text{iii}$).

The Half-Fraction Symbol: 'ss'

In apothecary prescription writing, the fraction one-half ($\frac{1}{2}$ or $0.5$) is represented by $\text{ss}$ (or $\overline{\text{ss}}$), derived from the Latin semis:

• $\text{gr ss} = \frac{1}{2}\text{ grain} = 0.5\text{ grain}$
• $\text{tabs iss} = 1\frac{1}{2}\text{ tablets} = 1.5\text{ tablets}$
• $\text{caps iiss} = 2\frac{1}{2}\text{ capsules} = 2.5\text{ capsules}$
• $\text{tabs iiiss} = 3\frac{1}{2}\text{ tablets} = 3.5\text{ tablets}$

Apothecary Grain Conversions

The foundational unit of mass in the apothecary system is the grain (abbreviated $\text{gr}$): $1\text{ grain } (\text{gr } \text{i}) \approx 60\text{ mg} \quad (\text{or } 65\text{ mg depending on the drug formulation})$

Patient Safety & Contemporary Practice Standards

The Institute for Safe Medication Practices (ISMP) and The Joint Commission (TJC) strongly discourage the ongoing clinical use of apothecary units and handwritten Roman numerals due to frequent transcription errors. For example, the apothecary symbol $\text{gr}$ (grain) is easily misread as $\text{g}$ (gram), representing an error factor of over $15$-fold ($1\text{ g} = 1000\text{ mg}$ versus $1\text{ gr} = 60\text{ mg}$). Similarly, lowercase '$\text{iv}$' written on a chart can be misread as 'IV' (intravenous route).

Modern documentation uses metric units and Arabic numerals, but Roman numerals remain a HESI A2 math review topic and are still needed to read controlled-substance schedules, clotting factors and older records.