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Mathedu Reengineering Mathematics

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The Roman Numerals, an alternative numeration system.

The Roman Numerals System is an alternative numeric system, that was invented by the Romans and is still used in some cases, such as… the Star Wars episodes numbering!
The Roman Numerals System is an alternative numeric system, that was invented by the Romans and is still used in some cases, such as… the Star Wars episodes numbering!

Namely, they are based on a limited series of symbols, that are added or subtracted to each other in order to count from I, that is 1, to MMMCMXCIX, that is 3,999.

So that, for the Romans, the zero did not exist as such, and the biggest possible number was 3,999…

But what are the symbols, and what are their signification?

The first symbol is I, that is one. It is repeated once, to get II, that is two, or three times, to get III, that is three.

Some use of the roman numerals, such as old fashion clocks, write 4 as four times I, that is IIII.

But the most usual way to write 4 is IV, as the subtraction of the 1 of I to the 5 of V, because V, for 5, is the second symbol.

Then you may continue after V, for 5, in an additive logic:
  • VI is 5+1, 6
  • VII is 5+2, 7
  • VIII is 5+3, 8

To get the next number, 9, we need another symbol, because the only allowed repetition of I's is three times, III.

The next symbol is X, for 10, so that 9 is written IX, with the I in front of the X to subtract 1 to 10, and get 9.

With these three symbols, I, V, and X, we may count up to 39.

Indeed, you may add the 9 numbers from I (1) to IX (9), to X (10), by simply putting their representation to the right of X:
  • XI is 10+1, 11
  • XIV is 10+4, 14
  • XVIII is 10+8, 18
  • XIX is 10+9, 19
And you may repeat the X once, to get XX, that is 20=10+10, or three times, to get XXX, that is 30=10+10+10.

Thenyou may add the 9 numbers from I (1) to IX (9), to XX (20), or to XXX (30), by simply putting their representation to the right of XX (respect. XXX):

  • XXV is 20+5, 25
  • XXVII is 20+7, 27
  • XXXiX is 30+9, 39

As the only allowed repetitions of X is three times, and the only allowed additions after a X is IX, for 9, we need another symbol to continue.

That symbol is L, for 50, and we may subtract X, for 10, to it, by placing an X in front of L, to obtain XL, that is 40.

Then we may add the 9 numbers from I (1) to IX (9), to XL (40), or to L (50), by simply putting their representation to the right of XL (respect. L): we count that way up to LiX, for 59.

But in fact, we may also add the numbers up to XXXIX, for 39, to L, to count up to LXXXIX, for 50+30+9=89.

Now that you have understood the principle, we may list the rest of the symbols used for the roman numerals:

  • C is for 100: XC is 90=100-10, XCIX is 99=90+9 (and not IC, because only X can be put to the right of C.
  • D is for 500: CD is 400=500-100, DCCCXCIX is 899=500(D)+300(CCC)+90(XC)+9(IX), the larger possible number with the symbols up to D
  • and M, the last symbol, is for 1,000: CM is for 900=1,000-100, MMXX is for 2020=2,000(MM)+20(XX), and MMMCMXCIX is for 3,999=3,000(MMM)+900(CM)+90(XC)+9(IX).

Because of the limitation to the rules for the construction of roman numerals, including the limited number of symbols, the highest possible number to be written as a roman numeral, is MMMCMXCIX, for 3,999.

So that Romans counted only from I, one, (no zero as a number, only a word 'null') up to3,999.

This was the sense of infinity in those ancient times…
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