## Introduction

Although within the Roman Empire itself it was enforced a set of stricter rules that would have lead to the standardization of Roman numerals writing, in the last hundred years some rules were applied to eliminate further confusion.

We concluded that there is **a set of six rules** to be remembered about when writing with Roman numerals.

Read them below, in order. Don't worry if you don't understand something right away. When you finish reading all the six rules, things will become clearer.

### Table of content / Rules summary:

#### 1) The first rule - Roman numerals set of basic symbols

#### 2) The second rule - numerals repetition

#### 3) The third rule - numerals subtraction

#### 4) * The fourth rule - what numerals are allowed to be subtracted from larger value numerals

#### 5) The fifth rule - numerals addition

#### 6) The sixth rule - the decomposition of Arabic numbers (decomposing, breaking Arabic numbers) into place value subgroups, in expanded notation, before converting to Roman numerals

## 1) The first rule - Roman numerals set of basic symbols

#### The major numerals, the symbols that were used to build the rest of the numbers in Roman system:

- 1 = I (one)
- 5 = V (five)
- 10 = X (ten)
- 50 = L (fifty)
- 100 = C (one hundred)
- 500 = D (five hundred)
- 1,000 = M (one thousand)
- * The numerals below were a later addition to the Roman numeral system; these numerals were also written with a bar above them or between two vertical bars, to indicate multiplication by 1,000. We prefer writing in brackets because: 1) it is more accessible to computer users and 2) for avoiding any possible confusion with the symbol for one - I.
- 5,000 = (V) (five thousand) *
- 10,000 = (X) (ten thousand) *
- 50,000 = (L) (fifty thousand) *
- 100,000 = (C) (one hundred thousand ) *
- 500,000 = (D) (five hundred thousand) *
- 1,000,000 = (M) (one million) *

At the beginning, the Romans did not use numbers larger than 3999, having no representation for numbers such as: 5,000, 10,000, 50,000, 100,000, 500,000, 1,000,000. These were added later on and for them various different notations were used, not necessarily the ones above. Thus, initially, the maximum number that could be written with Roman numerals was: MMMCMXCIX (3999).

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## 2) The second rule - numerals repetition

#### Repetition of certain symbols in a number:

- these symbols:
**I**(1),**X**(10),**C**(100),**M**(1,000),**(X)**(10,000),**(C)**(100,000) and**(M)**(1,000,000) may occur no more than three times__in a row__in any number. - these symbols:
**V**(5),**L**(50),**D**(500),**(V)**(5,000),**(L)**(50,000),**(D)**(500,000) may appear only once in any number.

- Number 4 is written as a numeral: IV and not IIII (though the IIII form circulated as an additive form)
- Number 5 is written as a numeral: V and not IIIII
- Number 10 will be written as: X and not VV
- Number 100 will be written as: C and not XXXXXXXXXX, nor LXXXXX, nor LL etc.

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## 3) The third rule - numerals subtraction

#### Numerals subtraction rule:

A numeral of lesser value placed in front of a larger value numeral (immediately to the left of it) is subtracted from the latter:- only these symbols are allowed to subtract from larger value numerals:
**I**(1),**X**(10),**C**(100),**M**(1,000),**(X)**(10,000),**(C)**(100,000),**(M)**(1,000,000). - numerals V (5), L (50), D (500), (V) (5,000), (L) (50,000), (D) (500,000) are not allowed to be used to subtract from larger value numerals.

- Number four (4) is written by using two important symbols listed under the first rule above: I (1) and V (5), subtracting I from V, by placing I ahead of the V symbol. We thus obtain: IV (4).
- Number nine (9) is written by using two important symbols listed under the first rule above: I (1) and X (10), subtracting I from X by placing I ahead of X symbol. We thus obtain: IX (9).

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## 4) * The fourth rule - what numerals are allowed to be subtracted from larger value numerals

* This rule is an addition to the third rule of subtraction; we preferred to treat it separately since it's important to understand it correctly.

#### What numerals are allowed to be subtracted from larger value numerals: numerals magnitude when subtracting from larger value numerals:

Romans, as a rule, placed a smaller numeral in front of another larger numerals (immediately to the left of it) only if the latter were whithin the set of the next two larger numerals (in another words, they placed in front of a larger numeral (immediately to the left of it) only another smaller one that was one step or maximum two steps below in the Roman numerals basic set of symbols, as described in the Rule no. 1):- I (1) is placed only in front of a units or tens value numeral, more specifically: I (1) can be placed only in front of V (5) and X (10), the only subtractions in this category can be: IV = 5 - 1 = 4 and IX = 10 - 1 = 9;
- X (10) can be placed only in front of a tens or a hundreds value numeral, more specifically: X (10) only in front of L (50) and C (100), the only subtractions in this category can be: XL = 50 - 10 = 40 and XC = 100 - 10 = 90;
- C (100) can be placed only in front of a hundreds or thousands value numeral, more specifically: C (100) only in front of D (500) and M (1,000), the only subtractions in this category can be: CD = 500 - 100 = 400 and CM = 1,000 - 100 = 900;
- M (1,000) only in front of one of the thousands or tens of thousands, which is exactly: M (1,000) only in front of (V) (5,000) and (X) (10,000), the only subtractions in this category: M(V) = 5,000 - 1,000 = 4,000 and M(X) = 10,000 - 1,000 = 9,000;
- (X) (10,000) only in front of a tens of thousands or hundreds of thousands: (X) (10,000) only in front of (L) (50,000) or (C) (100,000), the only subtractions in this category can be: (X)(L) = 50,000 - 10,000 = 40,000 and (X)(C) = 100,000 - 10,000 = 90,000;
- (C) (100,000) only in front of a hundred of thousands or a million: (C) (100,000) only in front of (D) (five hundred thousand) or (M) (1 million), the only subtractions in this category can be: (C)(D) = 500,000 - 100,000 = 400,000 and (C)(M) = 1,000,000 - 100,000 = 900,000
- and so on ... they placed in front of a larger numeral only another smaller one that was one step or maximum two steps below in the basic set of symbols, as described above
- numerals V (5), L (50), D (500), (V) (5,000), (L) (50,000), (D) (500,000) were not used to decrease numbers' value by placing them in front of other larger numerals.

- 99 is written as XCIX and not IC (thus subtracting I from X to get 9, X from C to get 90, then add IX to XC by placing it after XC and getting XCIX - and not subtracting I directly from C (since between I and C there is V, X and L, only V and X could be used for subtraction by placing them after I). Correct form: 99 = XCIX / Wrong forbidden form: 99 = IC
- 95 is written as XCV, and not VC.

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## 5) The fifth rule - numerals addition

#### Addition rule

A numeral placed immediately after another one of larger or equal value (immediately to the right of it), is added to the latter. The fifth rule outweigh the third rule, that of subtraction, when writing Roman numerals. Only if a number can not be built by using the addition rule, subtraction rule applies. Also please review Rule no. 2 (numerals may not occur more than three times in a row in any numeral.- Number two (2) is written by using a single symbol listed under the first rule: I (1). Placing the I symbol after another I symbol leads to: II (2)
- Number three (3) is also written by placing I symbol after I (1) twice, so we get: III (3)
- Number six (6) is written by placing the I symbol (1) after the V symbol (5), so we get: VI (6)
- Number seven (7) is written by placing the symbol I (1) after the symbol V (5) twice, so we get: VII (7)
- Number eight (8) is written by placing the symbol I (1), after the symbol V (5), three times, to get: VIII (8) - and NOT IIX
- Number eleven (11) is written by adding I (1) to X (10), by placing the symbol I (1) after the symbol X (10), to obtain: XI (11)
- Number Twenty (20) is written by adding X (10) after another X (10), to obtain: XX (20)
- Number two hunderd and one (201) is written by adding I (1) after CC (200), to obtain: CCI (201)

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## 6) The sixth rule - the decomposition of Arabic numbers (decomposing, breaking Arabic numbers) into place value subgroups, in expanded notation, before converting to Roman numerals

#### Rule of decomposition (decomposing, breaking numbers) into place value subgroups, in expanded notation, before converting Arabic numbers into Roman numerals

To turn any Arabic number into a Roman numeral, the rule is to break that Arabic number (decompose it) into place value subgroups, in expanded notation, of units, tens, hundreds, thousands, ten of thousands, hundred of thousands, millions, etc.. and to transform each of these subgroups separately into Roman numerals, then to reassemble them into the end Roman numeral according to the rule of addition.- 19 = 10 + 9 = 10 + (10 - 1) = X + IX = XIX (not IXX)
- 39 = 30 + 9 = (10 + 10 + 10 ) + (10-1) = XXX + IX = XXXIX (not IXL)
- 42 = 40 + 2 = (50 - 10) + 2 = XL + II = XLII
- 79 = 70 + 9 = (50 + 20) + (10 - 1) = LXX + IX = LXXIX (not ILXXX)
- 99 = 90 + 9 = (100 - 10) + (10 - 1 ) = XC + IX = XCIX (not IC)
- 104 = 100 + 0 + 4 = C + IV = CIV
- 120 = 100 + 20 + 0 = C + XX = CXX
- 150 = 100 + 50 = CL
- 173 = 100 + 70 + 3 = 100 + (50 + 20) + 3 = C + LXX + III = CLXXIII
- 199 = 100 + 90 + 9 = 100 + (100 - 10) + (10 - 1) = C + XC + IX = CXCIX (not ICC, not CIC)
- 200 = CC
- 207 = 200 + 0 + 7 = 200 + (5 + 2) = CC + VII = CCVII
- 267 = 200 + 60 + 7 = 200 + (50 + 10) + (5 + 2) = CCLXVII
- 448 = 400 + 40 + 8 = (500 - 100) + (50 - 10) + 8 = CD + XL + VIII = CDXLVIII
- 503 = 500 + 3 = D + III = DIII
- 944 = 900 + 40 + 4 = (1,000 - 100) + (50 - 10) + (5 - 1) = CM + XL + IV = CMXLIV
- 1,973 = 1,000 + 900 + 70 + 3 = 1,000 + (1,000 - 100) + (50 + 20) + 3 = M + CM + LXX + III = MCMLXXIII
- 2,012 = 2,000 + 10 + 2 = MM + X + II = MMXII
- 3,999 = 3,000 + 900 + 90 + 9 = 3,000 + (1,000 - 100) + (100 - 10) + (10 - 1) = MMM + CM + XC + IX = MMMCMXCIX
- 4,000 = 5,000 - 1,000 = (V) - M = M(V)

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## More examples of converting Arabic numbers into Roman numerals:

- 2 = II, 3 = III, 4 = IV, 6 = VI, 7 = VII, 8 = VIII, IX = 9, 11 = 10 + 1 = XI, 12 = 10 + 2 = XII, 13 = 10 + 3 = XIII, 14 = 10 + 4 = XIV
- 47 = 40 + 7 = XL + VII = XLVII
- 79 = 70 + 9 = LXX + IX = LXXIX
- 469 = 400 + 60 + 9 = CD + LX + IX = CDLXIX
- 2,000 = MM
- 2,010 = 2,000 + 10 = MMX
- 2,234 = 2,000 + 200 + 30 + 4 = MM + CC + XXX + IV = MMCCXXXIV
- 4,787 = 4,000 + 700 + 80 + 7 = M (V) + DCC + LXXX + VII = M(V)DCCLXXXVII
- 6,787 = 6,000 + 700 + 80 + 7 = (V)M + DCC + LXXX + VII = (V)MDCCLXXXVII
- 30,924 = 30,000 + 900 + 20 + 4 = (X)(X)(X) + CM + XX + IV = (X)(X)(X)CMXXIV
- 3,999,893 = 3,000,000 + 900,000 + 90,000 + 9,000 + 800 + 90 + 3 = (M)(M)(M) + (C)(M) + (X)(C) + M(X) + DCCC + XC + III = (M)(M)(M)(C)(M)(X)(C)M(X)DCCCXCIII
- Date 1: 2012-Oct-17 = 2012-10-17 = XXII-X-XVII
- Date 2: 1863-Aug-29 = 1863-8-29 = (1,000 + 800 + 60 + 3)-8-29=(M + DCCC + LX + III)-VIII-XXIX = MDCCCLXIII-VIII-XXIX