6.1 Numerical Systems
Two familiar numerical systems are used in pharmaceutical calculations: Arabic and Roman. Each serves as a kind of numerical abbreviation system for amounts, just as abbreviations are used in texting for words like lol, idk, cya, and others. The Arabic numeral system includes numbers, fractions, and decimals that we use every day (1, 2, 3 , , ; 10.5, 2.75, 3.25; $10.50, $2.75, $3.25). In the Roman numeral system, numerical values are expressed in capital letters or lowercase letters (I or i equals 1; V or v equals 5; and X or x equals 10). See Table 6.1 for common Roman numerals and Arabic values.
Practice Tip
Rule of thumb for Roman numerals: if a small value comes before a larger value, subtract the smaller value. If the larger value comes before the smaller value, add. And there cannot be more than three of the same numerals in sequence.
You may see clock faces and Star Wars movie episode titles using Roman numerals. Similarly, the official number of the Super Bowl has always been marketed in Roman numerals.
Roman numerals are read from left to right like our standard Arabic numbers, starting from the largest number place to the smaller one: I, II, III or i, ii, iii. However, to avoid repeating the same Roman numeral more than three times, there is a different way of noting numbers related to four. You put the numeral for one before the five, subtracting the “i” from “v” to get the value of 4 (or 1 subtracted from 5). So you end up counting, i, ii, iii, iv, v, vi, vii….The same thing happens with the value of nine. You put the numeral for one before the ten, subtracting the “i” from “x” to get the value of 9 (or 1 subtracted from 10). So you end up counting vi, vii, viii, ix, x, xi, xii, xiii, xiv, xv, xvi, xvii, xviii, and ixx.
IN THE REAL WORLD
Many students feel stress and anxiety at the thought of math. Remember that attitude counts! Studies show that students of equal ability who look at learning something different as an important challenge they want to take on as opposed to an intimidating obstacle they have to suffer through do far, far better than those who do not.
Ashanti La Roche was afraid of math and had always failed at it. Then she decided to become a pharmacy technician. She had a very good teacher, asked many questions, and completed every problem in her textbook to build her math skills. She came to learn math and now even teaches math in a pharmacy technician certification program! “Learning math is like building a house; you identify the pieces and their uses,” she says, “but you have to actually do the work and practice to master the skills!”
A similar process occurs for all values related to four and nine, such as forty (xl or XL, because L = 50), four hundred (CD or cd, because c = 100 and d = 500), ninety (XC or xc), or nine hundred (CM or cm, because M = 1,000). Refer again to Table 6.1 and note the derivatives of one, five, and ten as compared to the derivatives of four and nine.
Some clock faces use Roman numerals, giving them an old-fashioned feel.
The general principle is if there is a smaller value before a larger one, as in iv, you subtract the one from the other. If a smaller value comes after, you add it. If you are translating a Roman numeral and are confused about whether to add or subtract a numeral, as in MCM, move from the right and subtract first. MCM equals 1,900 because CM equals 900 (1,000–100), and when you add 900 to the front M, it equals 1,900.
Each level of number (tens, hundreds, thousands) is added or subtracted separately so you get long strings of letters. For instance, 99 ends up as XCIX (90 [XC] + 9 [IX]), never just IC; 999 is CMXCIX (900 [CM] + 90 [XC] + 9 [IX]), not IM. Remember, there are no shortcuts in Roman numerals—you have to use all those letters—but thankfully you will not be using them much in pharmacy practice.
The most frequently used Roman numerals in pharmacy practice are the uppercase letters I, V, and X, which represent the Arabic numbers 1, 5, and 10, respectively. For example, tablet quantities of a narcotic prescription are often written in uppercase letters, such as XXX, indicating 30 tablets. Because they can be confusing, however, pharmacy practice is moving away from them to the standard Arabic numerals.
Table 6.1 Roman Numeral Symbols and Arabic Values
Roman Numeral |
Arabic Value |
Roman Numeral |
Arabic Value |
|
|---|---|---|---|---|
ss |
½ |
XL (xl) |
40 |
|
I (i) |
1 |
XLI (xli) |
41 |
|
II (ii) |
2 |
XLIX (xlix) |
49 |
|
III (iii) |
3 |
L (l) |
50 |
|
IV (iv) |
4 |
LI (li) |
51 |
|
V (v) |
5 |
LIV (liv) |
54 |
|
VI (vi) |
6 |
LXXXIX (lxxxix) |
89 |
|
VII (vii) |
7 |
XC (xc) |
90 |
|
VIII (viii) |
8 |
XCI (xci) |
91 |
|
IX (ix) |
9 |
XCIX (xcix) |
99 |
|
X (x) |
10 |
C (c) |
100 |
|
XI (xi) |
11 |
CD (cd) |
400 |
|
XII (xii) |
12 |
CDXCIX (cdxcix) |
499 |
|
IXX (ixx) |
19 |
D (d) |
500 |
|
XX (xx) |
20 |
DI (di) |
501 |
|
XXIX (xxix) |
29 |
CMXCIX (cmxcix) |
999 |
|
XXXIV (xxxiv) |
34 |
M (m) |
1,000 |
|
XXXIX (xxxix) |
39 |
MI (mi) |
1,001 |
Safety Alert
New safety guidelines discourage the use of Roman numerals in pharmacy practice. Prescribers of controlled sub-stances may use them as a safety precaution to avoid having someone change the 30 tablets to 80 by completing the curves of the 3s, so the prescriber may write XXX tablets or 30 (XXX).
The lowercase Roman numerals i, ii, and iii are also occasionally used in pharmacy practice. These numerals are commonly seen on prescriptions using apothecary measures, and they follow—rather than precede—the unit of measurement. For example, “aspirin gr vi” means “six grains of aspirin.” Lowercase Roman numerals are also sometimes used to express other quantities, such as volume (“tbsp iii”, meaning three tablespoons). However, errors have occurred so often that Roman numerals are discouraged. If they are used, i, ii, and iii are often written with a line above the letters (i, ii, iii). The fractions in Roman numerals follow a different notation system (ss means ), which is part of the reason that Roman numerals are being phased out of pharmacy practice.