Roman Numbers - Conversion to decimal number

Question

I have read that if a smaller number is to the left of a larger number means that the smaller number has to be subtracted from the larger number.

Ok I can understand quickly for below Roman Numbers :

IX = X - I = 10 - 1 = 9

But I have difficulty in understanding Roman Numbers that have odd Roman Numerals - say :

XIX 

So if we go from Left to right we get

XIX = 10 + 1 + 10 = 21

But if we go from right to left we get

XIX = 10 + 10 - 1 = 19

So which direction we should consider before applying the rule of smaller followed by bigger Roman Numerals - left to right or right to left?

Answer 1

When reading roman numerals, I prefer to think in the following way:

Read from left to right, and if at any point the value of a character decreases, put a comma between the decrease. Then, add each block together.

MCMXCVI $\mapsto$ M,CM,XC,V,I $\mapsto$ $1000+900+90+5+1=1996$

MDCCCLXXIV $\mapsto$ M,D,CCC,L,XX,IV $\mapsto$ $1000+500+300+50+20+4=1874$

MCMXCIX $\mapsto$ M,CM,XC,IX $\mapsto$ $1000+900+90+9 = 1999$

Note that 1999 was not written as IMM or MIM. Converting from arabic numbers to roman numerals, one only uses powers of ten one apart to denote subtraction.

Answer 2

XIX is read left to right, the "I" is always applied to the final X.

XIX = X + IX = 10 + 9

XXI = X + XI = 10 + 11

Answer 3

When you are reading Roman numerals, start from the left-most character. Read rightward until the value of the character increases. Then, section those two characters off, and repeat.

That sounds really complicated, and I wrote it somewhat poorly, so here are some examples.

In $XIX$, we start with the left $X$ which is $10$. Then we move to the $I$ which is $1$. We decreased from $10$ to $1$, so we'll move on. Then we get to the right $X$ which is $10$. We increased from $1$ to $10$, so we need to section off the $IX$. It would look like this:

$$XIX = X + IX = 10 + 9 = 19$$

Here's a more complicated example:

$$MCMXXIX = M + CM + XX + IX = 1000 + 900 + 20 + 9 = 1929$$

Here we had to section off the $CM$ since we increased in value from $C$ to the second $M$. We had to section off the $IX$ since we increased in value from the $I$ to the last $X$.

One last really complicated example:

$$MMCDXLIV = MM + CD + XL + IV = 2000 + 400 + 40 + 4 = 2444$$

Answer 4

You can apply the following logics:

• the units are denoted I, II, III, IV, V, VI, VII, VIII$^*$, IX; the tenths are denoted X, XX, XXX, XL, L, LX, LXX, LXXX, XC; the hundredths C, CC, CCC, CD, D, DC, DCC, DCCC, CM; the thousands, M, MM, MMM.

• numbers are written in thousands, hundredths, tenths and units from left to right.

• no other pattern is allowed.

The Roman Empire wasn't designed to last longer than until 3999 AD :)

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$^*$Sometimes IIX is used for VIII.