Chuck Oates

OCCC APPM 1313

08 February 2007

DRAFT Version 0.02

Roman Numerals

An Intuitive Approach

Worksheets 12 and 13 in the worksheet packet at chuckoates.com , URLs http://chuckoates.com/MathHC/Worksheets/Worksheet-12.jpg and http://chuckoates.com/MathHC/Worksheets/Worksheet-13.jpg, present the Roman numeration system in a fairly formal way. Let’s have a look at that system from a bottom-up, somewhat intuitive approach.

Let’s count up from one, very much as we did when we learned the decimal place value system.

The Roman system represents numbers less than ten with the symbols I = 1 and V = 5. The first three numbers are quite easy.

1: I

2: II

3: III

At four, we’re tempted to write IIII, but since this can’t go on forever and still be legible, the Roman system used the form

4: IV (the symbol for one appears before the symbol for five)

meaning “one subtracted from five.” Things go pretty much as expected from five through eight.

5: V

6: VI (five plus one)

7: VII (five plus two)

8: VIII (five plus three)

At nine, once again we encounter the “IIII” problem and solve it pretty much the same way we did at four. We’ll need an extra pair of symbols,

X = 10 and L = 50 to go on to 100. (Notice that in our customary decimal place value system, we just re-used the numerals “1” and “5” moved to the left one place to represent 10 and 50. The Romans didn’t have this luxury!)

9: IX (one subtracted from ten)

10: X

11: XI (ten plus one; compare this with nine, above)

12: XII

13: XIII

Here we are again with the “IIII” problem. We’ll solve it the same way.

14: XIV

15: XV

16: XVI

17: XVII

18: XVIII

The “IIII” problem strikes again.

19: XIX (ten with nine written after it)

20: XX

21: XXI

22: XXII

Things go on up in a similar pattern between tens (10, 20, 30, …), so now let’s count by tens, and you can fill in the numbers in between.

30: XXX

40: XL (the “XXXX” problem is solved using the same trick we used
before at four, IV)

50: L

60: LX

70: LXX

80: LXXX

Now that we’re up near 100, we’ll need to introduce the next “decade” of symbols, C = 100 and D = 500.

90: XC (the “XXXX” problem is solved here again using the trick we
used before at nine, IX)

100: C

110: CX

120: CXX

130: CXXX

140: CXL (one hundred plus forty)

150: CL

Things go up by tens pretty much as before at 50, 60, 70, 80, and 90 at this point. You can figure out the pattern easily. Let’s count by hundreds now to see how that goes.

200: CC

300: CCC

400: CD (100 subtracted from 500 to keep from writing “CCCC”)

500: D

600: DC

700: DCC

800: DCCC

We’re up near 1000 now, so we’ll need to introduce the special symbol for 1000, M = 1000. That’s about as high as we’ll need to go.

900: CM (100 subtracted from 1000 to keep from writing “DCCCC”)

1000: M

1100: MC

1200: MCC

1300: MCCC

1400: MCD (1000 plus 400)

1500: MD

1600: MDC (1000 plus 500 plus 100)

1700: MDCC

1800: MDCCC

1900: MCM (1000 plus 900)

2000: MM

2007: MMVII

That’s about as high as we’ll need to count, since Roman numerals are seldom used for anything other than numbering preface pages in books and disguising copyright dates on movies these days. You’ll still see them used on prescriptions written using apothecaries’ units by private physicians for non-hospital pharmacies, though.

With some study, the information above should give you a pretty good feeling for how to convert our place value number representations (90, 157, 2500) to their Roman equivalents (XC, CLVII, MMD).

Real Soon Now, I’ll add some examples of conversions in the other direction, Roman to place value, otherwise known as Roman to (westernized) Arabic numeration system conversions.

In the mean time, more than you ever wanted to know about Roman numerals and the Roman numeration system can be found at http://home.att.net/~numericana/answer/roman.htm.

Please e-mail me at Chuck@ChuckOates.com if you find errors in the above. It’s a draft version, written quite late at night, and I’d be surprised if there weren’t a few undetected problems remaining.