Math Olympiad 3

Week	Topic and action
Week 35	Roman Numerals Practice Evaluation

Roman Numerals

Roman numerals are a numeral system of ancient Rome based on letters of the alphabet, which are combined to signify the sum (or in some cases, the difference) of their values.

Symbol	I	V	X	L	C	D	M
Value	1	5	10	50	100	500	1000

The first ten Roman numerals are: I, II, III, IV, V, VI, VII, VIII, IX, X. The table below also list Roman numerals of tens, hundreds and thousands (up to 3000).
I(1) II(2) III(3) IV(4) V(5) VI(6) VII(7) VIII(8) IX(9)
X(10) XX(20) XXX(30) XL(40) L(50) LX(60) LXX(70) LXXX(80) XC(90)
C(100) CC(200) CCC(300) CD(400) D(500) DC(600) DCC(700) DCCC(800) CM(900)
M(1000) MM(2000) MMM(3000)
- Additive principle: If the symbols are arranged from left to right in descending based on their values, the value of the Roman number is the sum of all of the numerals:For example,
  II = 1 + 1 = 2, VII = 5 + 1 + 1 = 7, XVIII = 10 + 5 + 1 + 1 + 1 = 18.
- Subtractive principle: if a symbol of smaller value precedes a symbol of larger value, the smaller value is subtracted from the larger value, and the result is added to the total.
  For example, IV = 5 − 1 = 4, IX = 10 − 1 = 9, XIX = 10 + (10 − 1) = 19, CMXLIV = (1000 − 100) + (50 − 10) + (5 − 1) = 944.
- The letters should be arranged from the one with the largest value to the one with the smallest. Each letter's value is added to the previous ones.
- Only powers of ten (I, X, C, M) can be repeated. Do not repeat any letter more than three times in a row. For example, 10 is X not VV, and 4 is IV not IIII.
- Because of the preceding rule, certain numbers must be written using subtraction. In this case, a letter with a smaller value precedes one with a larger value and the value of the smaller is subtracted from the larger. The result is then added to the rest of the letters. The following rules apply to subtraction:
  - Only powers of ten (I, X, C, M) can be subtracted. For example, 45 is XLV not VL, 15 is XV not XVX.
  - The smaller letter must be
    1
    5
    or
    1
    10
    of the larger one. For example, 99 is XCIX not IC (since I is not
    1
    5
    or
    1
    10
    of C).
  - The smaller letter must be either the first letter or preceded by a letter at least ten times greater than it.
  - If another letter follows the larger one, it must be smaller than the number preceding the larger one. For example, 140 is CXL not XCL because L (50) isn't smaller than X (10)
To put it most concisely: When converting to Roman numerals, convert each digit separately. For example,
- for 863 convert 800, 60, and 3 to give DCCC, LX, and III to get DCCCLXIII.
- for 386 convert 300, 80, and 6 to give CCC, LXXX, and VI to get CCCLXXXVI.

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I(1)	II(2)	III(3)	IV(4)	V(5)	VI(6)	VII(7)	VIII(8)	IX(9)
X(10)	XX(20)	XXX(30)	XL(40)	L(50)	LX(60)	LXX(70)	LXXX(80)	XC(90)
C(100)	CC(200)	CCC(300)	CD(400)	D(500)	DC(600)	DCC(700)	DCCC(800)	CM(900)
M(1000)	MM(2000)	MMM(3000)