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Talk:Roman_numerals/Rules

History

[edit]

Ancient Romans: ambiguity (IIII, IIX, IXC XXXX, IIIIII, XXXXXX etc). various symbols for large numbers, sometimes used bars for x1000, but more often to distinguish numerals from letters?

In classical usage, orthography was highly ambiguous, often allowing multiple ways of writing a given value. For example:

  • 4: IV or IIII
  • 6: VI or IIIIII
  • 8: VIII or IIX
  • 9: IX or VIIII
  • 19: XIX, IXX, or XVIIII
  • 40: XL or XXXX
  • 60: LX or XXXXXX
  • 79 1/2: LXXIXS or SXXC
  • 89 1/2: LXXXIXS, IXCS, or SXC
  • 99: XCIX, LXXXXVIIII, or IC
  • 400: CD or CCCC

The Romans generally observed the rule of subtraction to the left and addition to the right, and the rule for only adding one quinary numeral at a time, but much else was unregulated -- even the numerals themselves appeared in varying forms, especially for larger values. It could be said that the Romans often preferred addition to subtraction, but sometimes they emphasized subtraction instead. The vinculum could be used to multiply x1000, but was more commonly used to distinguish words from numbers. Above all, classical orthography was flexible and open-ended.

Medieval?

Modern

[edit]
  • Shaw, Allen A. “Note on Roman Numerals.” National Mathematics Magazine, vol. 13, no. 3, 1938, pp. 127–128. JSTOR, JSTOR, www.jstor.org/stable/3028752.
  • Morandi, Patrick. “Roman Numerals.” Roman Numerals, New Mexico State University, sierra.nmsu.edu/morandi/coursematerials/RomanNumerals.html.
  • “Math Forum: Ask Dr. Math FAQ: Roman Numerals.” Mathforum.org, The Math Forum at NCTM, mathforum.org/dr.math/faq/faq.roman.html.

The modern era has seen the emergence of a standardized orthography for roman numerals, which permits only one permutation for any given value.[1][2][3][4] While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be described by the following ruleset:

Old Version

[edit]
  • Repeated decimal numerals (I, X, C, M) are additive.
  • Up to three (3) decimal numerals may be repeated sequentially per power (there may be up to four (4) non-sequential repetitions per power).
  • Smaller numerals placed to the right are also additive.
  • Smaller decimal numerals placed to the left are subtractive.
  • Quinary numerals (V, L, D) may not be repeated.
  • Quinary numerals may not be subtracted, only added.
  • Only one (1) decimal numeral may be subtracted per power.
  • Subtraction must be by 1/5 or 1/10 of the adjacent value (i.e. the next lower decimal numeral relative to the adjacent value).
  • Addition may be by any amount that is equal to or less than the adjacent value (not greater).
  • Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right.
  • Addition/subtraction operations are to be applied to the rightmost numeral per power.
  • Fractions may only be added to the right of all numerals (not subtracted).
  • Addition/subtraction should not be redundant (eg X should not be IXI).
  • If there is no adjacent value, a numeral may be entered ex nihilo.
  • Roman numerals are positive integers with pre-defined values.
  • Placing a bar over a numeral multiplies it x1000.
  • To add to or subtract from a barred numeral, either a barred I or a standard M may be used for 1,000. ??? ambiguous.

Revised

[edit]
  • Repeated decimal numerals (I, X, C, M) are additive.
  • Up to three (3) decimal numerals may be repeated sequentially per power (there may be up to four (4) non-sequential repetitions per power).
  • Smaller numerals placed to the right are also additive.
  • Smaller decimal numerals placed to the left are subtractive.
  • Quinary numerals (V, L, D) may not be repeated.
  • Quinary numerals may not be subtracted, only added.
  • Only one (1) decimal numeral may be subtracted per power.
  • Subtraction must be by 1/5 or 1/10 of the adjacent value (i.e. the next lower decimal numeral relative to the adjacent value).
  • The additive value must be less than the subtractive value in a given power.
  • Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right.
  • Fractions may only be added to the right of all numerals (not subtracted).
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999); and may be iterated up to three (3) times.
Current orthography===

The modern era has seen the emergence of a standardized orthography for roman numerals, which permits only one permutation for any given value.[5][6][7][8] While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be described by the following ruleset:

  • Repeated decimal numerals (I, X, C, M) are added together.
  • Up to three (3) decimal numerals may be repeated in sequence per power (there may be up to four (4) non-sequential repetitions per power).
  • Smaller numerals placed to the right are also added.
  • Smaller decimal numerals placed to the left are subtracted.
  • Quinary numerals (V, L, D) may not be repeated.
  • Quinary numerals may not be subtracted, only added.
  • Only one (1) decimal numeral may be subtracted per power.
  • Subtraction must be by 1/5 or 1/10 (i.e. the next lower decimal numeral).
  • Addition must be less than subtraction for any given numeral.
  • Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right.
  • Fractions may only be added to the right of all numerals (not subtracted).
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999); and may be iterated up to three (3) times.
Modern orthography===

Roman Numerals are described above as a decimal pattern, but may also be defined by a logical set of rules.[9][10][11][12][13] While exceptions can be made (notably IIII instead of IV on clockfaces), the modern standard is widely adhered to, and is prescribed as follows:

Final

[edit]
  1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right.
  2. Repeated 'tens' numerals (I, X, C, M) are added together, with up to three (3) permitted in sequence (subtraction allows up to four (4) non-sequential repetitions per power).
  3. 'Fives' numerals (V, L, D) may only be added once per power (not repeated nor subtracted).
  4. Numerals placed to the right of a larger value are added, while those placed to the left are subtracted (and must be 'tens').
  5. Only one (1) 'tens' numeral may be subtracted from a single numeral per power, and this must be by 1/5 or 1/10 (i.e. the next lower 'tens' numeral).
  6. Addition must be less than subtraction for any given numeral (i.e. the value to the right must be less than the subtractor).
  7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.

The above describes the basic pattern of integers from 1 - 3,999. The system can be extended further by employing fractions and vinculums:

  • Fractions may only be added to the right of all numerals (not subtracted), and are duodecimal.
  • For fractions above 6/12, the S (semi) is placed first, then the dots (unciae).
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999), and may be iterated up to three (3) times.
  • Starting at 4,000, and at every thousandth multiple thereafter, a new vinculum is added, with the pattern beginning at IV.

With fractions and vinculums employed, the lowest possible value is 1/12, while the highest is (((MMMCMXCIX))) or 3,999,999,999,999.

Alternate

[edit]
  1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right.
  2. 'Tens' numerals (I, X, C, M) may be repeated up to three (3) times in sequence (subtraction allows up to four (4) non-sequential repetitions per power), while 'fives' numerals (V, L, D) may only be used once per power.
  3. Numerals are added together, except when placed to the left of a larger value, in which case they are subtracted.
  4. Only one (1) 'tens' numeral may be subtracted from a single numeral per power ('fives' numerals may not be subtracted).
  5. Subtraction must be by 1/5 or 1/10 of the adjacent larger value (i.e. the next lower 'tens' numeral).
  6. Addition must be less than subtraction for any given numeral (i.e. the value to the right must be less than the subtractor).
  7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.

Final+

[edit]
Standard orthography===

The modern era has seen the emergence of a standardized orthography for roman numerals, which permits only one permutation for any given value. This system may be described as a decimal pattern, as above, but also as a logical set of rules. While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be prescribed by the following ruleset:[14][15][16][17][18][19][20]

Basic Rules
  1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right, with numerals either added or subtracted.
  2. Repeated 'tens' numerals (I, X, C, M) are added together, with up to three (3) permitted in sequence (subtraction allows up to four (4) non-sequential repetitions per power).
  3. 'Fives' numerals (V, L, D) may only be added once per power (not repeated nor subtracted).
  4. Lesser numerals placed to the right of a greater value are added, while those placed to the left are subtracted (and must be 'tens').
  5. Only one (1) 'tens' numeral may be subtracted from a single numeral per power, and this must be by 1/5 or 1/10 (i.e. the next lower 'tens' numeral).
  6. Addition must be less than subtraction for any given numeral (i.e. the added value to the right must be less than the subtractor).
  7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.
Examples

Following the above rules:
90 must be XC. It cannot be LXXXX because that would break rule 2 (up to three in sequence), and it cannot be LXL because that would break rule 3 ('fives' not repeated).
45 must be XLV. It cannot be VL because that would break rule 3 ('fives' not subtracted).
99 must be XCIX. It cannot be IC because that would break rule 5 (subtract by one fifth or tenth).
18 must be XVIII. It cannot be IIXX or IXIX because these would break rule 5 (subtract once per power).
19 must be XIX. It cannot be IXX because that would break rule 6 (subtract>add).
10 must be X. It cannot be IXI because that would break rule 6 (subtract>add).
14 must be XIV. It cannot be VIX because that would break rule 7 (at least 10x).

1894 is MDCCCXCIV. Consider how this agrees with the rules: powers of ten are arranged properly (rule 1), C is used three times sequentially and a fourth time non-sequentially (rule 2), 'fives' numerals D and V are added once each (rule 3), X and I are subtracted by being placed to the left of larger values while the rest are added (rule 4), they are subtracted once per power by one tenth and one fifth respectively (rule 5), IV is less than X (rule 6), and C is 10x X and 100x I (rule 7).

Fractions & Vinculums

The following rules extend the system further:

  • Fractions may only be added to the right of all numerals (not subtracted), and are duodecimal.
  • For fractions above 6/12, the S (semi) is placed first, then the dots (unciae).
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999), and may be iterated up to three (3) times.
  • Starting at 4,000, and at every thousandth multiple thereafter, a new vinculum is added, with the pattern beginning at IV.

With fractions and vinculums employed, the lowest possible value is • or 1/12, while the highest is (((MMMCMXCIX))) or 3,999,999,999,999.

Final++

[edit]
Standard orthography===

The modern era has seen the emergence of a standardized orthography for roman numerals, which permits only one permutation for any given value. This system may be described as a decimal pattern, as above, but also as a logical set of rules. While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be prescribed by the following ruleset:[21][22][23][24][25][26][27][28]

Basic Rules
  1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right, with numerals either added or subtracted.
  2. Repeated 'tens' numerals (I, X, C, M) are added together, with up to three (3) permitted in sequence (subtraction allows up to four (4) non-sequential repetitions per power).
  3. 'Fives' numerals (V, L, D) may only be added once per power (not repeated nor subtracted).
  4. Lesser numerals placed to the right of a greater value are added, while those placed to the left are subtracted; if placed between two larger numerals, the 'tens' value is subtracted (i.e. subtraction takes precedence).
  5. Only one (1) 'tens' numeral may be subtracted from a single numeral per power, and this must be by 1/5 or 1/10 (i.e. the next lower 'tens' numeral).
  6. Addition must be less than subtraction for any given numeral (i.e. the added value to the right must be less than the subtractor).
  7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.
Examples

Following the above rules:
90 must be XC. It cannot be LXXXX because that would break rule 2 (up to three in sequence), and it cannot be LXL because that would break rule 3 ('fives' not repeated).
45 must be XLV. It cannot be VL because that would break rule 3 ('fives' not subtracted).
99 must be XCIX. It cannot be IC because that would break rule 5 (subtract by one fifth or tenth).
18 must be XVIII. It cannot be IIXX or IXIX because these would break rule 5 (subtract once per power).
19 must be XIX. It cannot be IXX because that would break rule 6 (subtract>add).
10 must be X. It cannot be IXI because that would break rule 6 (subtract>add).
14 must be XIV. It cannot be VIX because that would break rule 7 (at least 10x).

1894 is MDCCCXCIV. Consider how this agrees with the rules: powers of ten are arranged properly (rule 1), C is used three times sequentially and a fourth time non-sequentially (rule 2), 'fives' numerals D and V are added once each (rule 3), X and I are subtracted by being placed to the left of larger values while the rest are added (rule 4), they are subtracted once per power by one tenth and one fifth respectively (rule 5), IV is less than X (rule 6), and C is 10x X and 100x I (rule 7).

Fractions & Vinculums

The following rules extend the system further:

  • Fractions may only be added to the right of all numerals (not subtracted), and are duodecimal.
  • For fractions above 6/12, the S (semi) is placed first, then the dots (unciae).
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999), and may be iterated up to three (3) times.
  • Starting at 4,000, and at every thousandth multiple thereafter, a new vinculum is added, with the pattern beginning at IV.

With fractions and vinculums employed, the lowest possible value is • or 1/12, while the highest is (((MMMCMXCIX))) or 3,999,999,999,999.

Final+++

[edit]
Standard orthography===

The modern era has seen the emergence of a standardized orthography for roman numerals, which permits only one permutation for any given value. This system may be described as a decimal pattern, as above, but also as a logical set of rules. While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be prescribed by the following ruleset:[29][30][31][32][33][34][35][36]

Basic Rules
  1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right, with numerals either added or subtracted.
  2. Repeated 'tens' numerals (I, X, C, M) are added together, with up to three (3) permitted in sequence (subtraction allows up to four (4) non-sequential repetitions per power).
  3. 'Fives' numerals (V, L, D) may only be added once per power (not repeated nor subtracted).
  4. Lesser numerals placed to the right of a greater value are added, while those placed to the left are subtracted; if placed between two larger numerals, the 'tens' value is subtracted (i.e. subtraction takes precedence).
  5. Only one (1) 'tens' numeral may be subtracted from a single numeral per power, and this must be by 1/5 or 1/10 (i.e. the next lower 'tens' numeral).
  6. Addition must be less than subtraction for any given numeral (i.e. the added value to the right must be less than the subtractor).
  7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.

The above describes the basic pattern of integers from 1 - 3,999.

Examples

Following the above rules:
90 must be XC. It cannot be LXXXX because that would break rule 2 (up to three in sequence), and it cannot be LXL because that would break rule 3 ('fives' not repeated).
45 must be XLV. It cannot be VL because that would break rule 3 ('fives' not subtracted).
99 must be XCIX. It cannot be IC because that would break rule 5 (subtract by one fifth or tenth).
18 must be XVIII. It cannot be IIXX or IXIX because these would break rule 5 (subtract once per power).
19 must be XIX. It cannot be IXX because that would break rule 6 (subtract>add).
10 must be X. It cannot be IXI because that would break rule 6 (subtract>add).
14 must be XIV. It cannot be VIX because that would break rule 7 (at least 10x).

1894 is MDCCCXCIV. Consider how this agrees with the rules: powers of ten are arranged properly (rule 1), C is used three times sequentially and a fourth time non-sequentially (rule 2), 'fives' numerals D and V are added once each (rule 3), X and I are subtracted by being placed to the left of larger values while the rest are added (rule 4), they are subtracted once per power by one tenth and one fifth respectively (rule 5), IV is less than X (rule 6), and C is 10x X and 100x I (rule 7).

Fractions & Vinculums

The following rules extend the system further:

  • Fractions may only be added once to the right of all numerals (not subtracted), without redundancy, and are duodecimal.
  • Beginning at 6/12, the S (semi) is used, followed by marks (unciae) for additional quantities up to 11/12.
  • Placing a bar (vinculum) over a numeral multiplies it x1000.
  • The vinculum is used for values of 4,000 and greater (it is not used up to 3,999), and may be iterated up to three (3) times.
  • Starting at 4,000, and at the next two powers of a thousand thereafter, a new vinculum is added, with the pattern beginning at IV.
  • Once vinculums are employed, the higher power is modified in preference to the lower (eg IV, not MV).
  • Vinculums are always contiguous and are placed leftmost (ie no broken overlines or lower powers to the left per basic rule 1).

Thus, 2 2/3 would be rendered as IIS••, while 6,986 would be (VI)CMLXXXVI. Note that the symbol V is used more than once in that example, this is only permissible with the vinculum setting them apart. Also note that while M is used, 7,000 would be (VII).

(I), (II), and (III) are not used, with M, MM, MMM being preferred under 4,000. SS is never used in place of I (ss is sometimes seen as a variant of semi, but this is strictly pharmaceutical notation).

With fractions and vinculums employed, the lowest possible value is • or 1/12, while the highest is (((MMMCMXCIX)))((CMXCIX))(CMXCIX)CMXCIX or 3,999,999,999,999 (within standard usage). Four trillion, rendered as (((MMMM))), would be non-standard since it breaks basic rule 2, although the four M's can be used as a variant form, and it is more aesthetically pleasing as a maximum figure. Another alternative is ((((IV)))), but since the general principle is three sequential iterations, this should also apply to vinculums. The reason for the three-rule is that the eye can readily distinguish between one, two, or three marks, while four or greater become illegible. Thus, 4 M's are preferable as a variant form.


new sources, useful:

  1. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. JSTOR 3028752.
  2. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  3. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  4. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  5. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. doi:10.2307/3028752. Retrieved November 27, 2018.
  6. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  7. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  8. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  9. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. doi:10.2307/3028752. Retrieved November 27, 2018.
  10. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  11. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  12. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  13. ^ Seitz, Lee K. (December 8, 1999). "LURNC: How Roman Numerals Work". home.hiwaay.net. Retrieved May 24, 2020.
  14. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. doi:10.2307/3028752. Retrieved November 27, 2018.
  15. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  16. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  17. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  18. ^ Seitz, Lee K. (December 8, 1999). "LURNC: How Roman Numerals Work". home.hiwaay.net. Retrieved May 24, 2020.
  19. ^ "Roman Numerals". factmonster.com. FactMonster Staff. February 21, 2017. Retrieved July 29, 2020.
  20. ^ "Roman Numerals: Educational Articles". us.edugain.com. Edugain USA. July 2, 2016. Retrieved August 3, 2020.
  21. ^ Seitz, Lee K. (December 8, 1999). "LURNC: How Roman Numerals Work". home.hiwaay.net. Retrieved May 24, 2020.
  22. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. doi:10.2307/3028752. Retrieved November 27, 2018.
  23. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  24. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  25. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  26. ^ "Roman Numerals". factmonster.com. FactMonster Staff. February 21, 2017. Retrieved July 29, 2020.
  27. ^ "Roman Numerals: Educational Articles". us.edugain.com. Edugain USA. July 2, 2016. Retrieved August 3, 2020.
  28. ^ https://www.mytecbits.com/tools/mathematics/roman-numerals-converter
  29. ^ Seitz, Lee K. (December 8, 1999). "LURNC: How Roman Numerals Work". home.hiwaay.net. Retrieved May 24, 2020.
  30. ^ Shaw, Allen A. (December 1938). "Note on Roman Numerals". National Mathematics Magazine. 13 (3). Taylor & Francis, Ltd. on behalf of the Mathematical Association of America: 127–128. doi:10.2307/3028752. Retrieved November 27, 2018.
  31. ^ Morandi, Patrick. "Roman Numerals". nmsu.edu. New Mexico State University. Retrieved November 20, 2018.
  32. ^ "Math Forum: Ask Dr. Math FAQ: Roman Numerals". Mathforum.org. The Math Forum at NCTM. 1994–2018. Retrieved November 21, 2018.{{cite web}}: CS1 maint: date format (link)
  33. ^ Lewis, Paul (October 4, 2005). "ROMAN NUMERALS: How They Work". clarahost.co.uk. Retrieved December 8, 2018. Version 3.11
  34. ^ "Roman Numerals". factmonster.com. FactMonster Staff. February 21, 2017. Retrieved July 29, 2020.
  35. ^ "Roman Numerals: Educational Articles". us.edugain.com. Edugain USA. July 2, 2016. Retrieved August 3, 2020.
  36. ^ https://www.mytecbits.com/tools/mathematics/roman-numerals-converter
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