Numeration: Roman Numerals
The number system we primarily use in our daily lives is called the Hindu-Arabic system. It is
a base 10, place-value system. That is, every number we can dream up is composed of some
mixture of ten different digits – 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9 – and the position of the digit
determines its value. A 4 in the tens place stands for 40, but a 4 in the hundreds place stands for
400.
You may, however, be surprised to know that there are many other systems of numeration.
Roman Numerals
We see Roman Numerals on clocks, in dates, and in every Super Bowl. The system uses capital
letters for symbols.
Symbol
M
D
C
L
X
V
I
Value
1000
500
100
50
10
5
1
Next, here are the rules,
•
Never use more than three consecutive identical symbols.
•
If a smaller value precedes a larger value, subtract the smaller from the larger to find the
represented value. This is called the subtraction property for Roman numerals.
•
Each instance of subtraction must be with exactly two digits, as follows.
o I can only be subtracted from V or X (IV = 4 & IX = 9)
o X can only be subtracted from L or C (XL = 40 & XC = 90)
o C can only be subtracted from D or M (CD = 400 & CM = 900)
Living in simpler times, the Romans rarely had a need for large numbers, so there is a fair
amount of disagreement about how large numbers (over 3999) were written. For values over
3999, we will use an overscore to indicate multiplication by 1000. For example, the number 7000
will be written VII .
Once the value for each character (or pair of characters if the subtraction property is used) is
determined, we simply add those values together. Thus, the Roman system is referred to as an
additive system.
Converting from Roman to Hindu-Arabic
To correctly read and write a Roman numeral, we need to remember the system is an additive
system.
Example 1: Write the Roman numeral XI as a Hindu-Arabic number.
Break the number apart into individual symbols, and then add the values together.
XI = X + I = 10 + 1 = 11
Example 2: Write the Roman numeral CCCXII as a Hindu-Arabic number.
CCCXII = C + C + C + X + I + I = 100 + 100 + 100 + 10 + 1 + 1 = 312
Be sure to look for the subtraction property. Remember, when a smaller digit precedes a larger
one, group those two symbols together.
Example 3: Write the Roman numeral XCVII as a Hindu-Arabic number.
XCVII = XC + V + I + I = 90 + 5 + 1 + 1 = 97
Example 4: Write the Roman numeral MCMXCI as a Hindu-Arabic number.
MCMXCI = M + CM + XC + I = 1000 + 900 + 90 + 1 = 1991
Example 5: Write the Roman numeral IVDII as a Hindu-Arabic number.
IVDII =!IV + D+I+I!=!4000+500+1+1!=!4502
Converting from Hindu-Arabic to Roman
The most common mistake made when converting one of our numbers to Roman is considering
more than one digit at a time. When converting 95 to Roman, many people incorrectly assume
it to be VC. After all, C is 100, V is 5, and since the V is placed in front of the C, we are to
subtract: 100 - 5 = 95. Unfortunately, VC is not 95.
In fact, VC is meaningless. It would be like counting by saying "... eighteen, nineteen, tenteen,
eleventeen, ..." which, by the way, is a common mistake for children to make when they are first
learning to count. To convert 95, we need to think of it as 90 + 5, first. Then find the Roman
equivalent for the 90 (XC) and the 5 (V), and, finally, put them together. Thus, 95 would be
XCV.
Next, we need to remember we will never have more than three consecutive identical symbols
in a row. For example, for the value 40, we would use the subtractive rule and write XL, instead
of XXXX.
Essentially, to convert a number to Roman, we must remember the rules and ONLY consider
one digit at a time - in its correct place value. The best way to start is to write the number in
expanded form and then assign the corresponding Roman numerals.
Example 6: Write 13 as a Roman numeral.
13 = 10 + 3 = X + III = XIII
Example 7: Write 76 as a Roman numeral.
76 = 70 + 6 = LXX + VI = LXXVI
Example 8: Write 945 as Roman numeral.
945 = 900 + 40 + 5 = CM + XL + V = CMXLV
Example 9: Write 17,401 as Roman numeral.
17, 401!=!10,000 + 7000 + 400 + 1!=!X!+!VII!+!CD!+!I! =!XVIICDI!
As we become more efficient in converting to/from Roman numerals, shortcuts are easily
recognizable and the conversions can be done more quickly. For example, in Example 9, if we
see 17,401 as 17,000+400+1 and convert it to XVII!+!CD!+!I , we would be OK.
Be sure, however, not to sacrifice accuracy for sake of expedience.