# Tally Counting with Roman Numerals

I found this on my Google Drive. It's a document that I had started years ago but never did anything with. Since it was pretty much done (just had to do a bit of Silvrback markdown), I decided to post it here . Who knows...it might help somebody to remember how to count in Roman numerals!

# Roman Numerals

Roman numeral can seem confusing at first. But, with a little practice, anybody can read and write Roman numerals. This page presents a simplified mental-model that should make it easy for most people to remember how to read and write Roman numerals. It should be noted that the thought exercise presented on this page might or might not have some similarities to the real history or evolution of Roman numerals. Either way, this page is *not* intended to be construed as history page!

## Tally Counting System

Despite their great engineering achievements such as water ducts and the Great Collosseum, the ancient Romans had little need to do complex math. But, they did have a need to count. They took censuses, counted soldiers, kept track of taxes that were paid and owed, kept other records about the current year, etc. And, the whole concept of Roman numerals becomes significantly easier to grasp and remember if one thinks of Roman numerals as nothing more than a simple tally counting system.

You are probably already familiar with one or more tally counting systems. Does “tic, tic, tic, tic, slash” bring anything to mind? In most tally counting systems, some type of tic mark is used to count each item, while some other kind of tic mark is used to identify regular intervals to make it easier to count the total tally when counting is over.

### One, I

In most tally counting systems, some kind of tic mark that can be made with a single pen stroke is added for each item that is tallied. And, Roman numerals are no different. The character `I`

is used to represent one, and `I`

is really nothing more than a stylized single stroke of a pen.

The table below shows tally counting from 1 to 11 using nothing by the “one mark”.

```
1 I
2 II
3 III
4 IIII
5 IIIII
6 IIIIII
7 IIIIIII
8 IIIIIIII
9 IIIIIIIII
10 IIIIIIIIII
11 IIIIIIIIIII
```

### Ten, X

Tally counting using nothing but the `I`

works, but is not very efficient. It does allow one to accurately tally the results, but requires a recount of every `I`

once the tallying has been completed. Introducing a new character every ten tally marks makes recounting significantly easier because one can simply scan for the “ten marks” and count by ten. And, if a one mark is represented by a single stroke, it makes sense that a ten mark could be represented by the next easiest thing, which is a double stroke character. The `X`

character just so happens to be a easier to write, stylized double-stroke.

The table below shows tally counting from 1 to 11 using the one and ten marks.

```
1 I
2 II
3 III
4 IIII
5 IIIII
6 IIIIII
7 IIIIIII
8 IIIIIIII
9 IIIIIIIII
10 IIIIIIIIII
11 IIIIIIIIIXI
```

### Implied Characters

In the table from the tens section above, note how much easier it is to count the 11. One does not even have count the first nine `I`

characters because they are implied by `X`

, which represents ten. And, since they are implied, they can even be omitted when the result is copied from the tally sheet to the record book. Therefore, 11 could be written as `XI`

by dropping the implied characters.

The table below shows a few others, just to drive the point home.

```
10 X ten
21 XXI ten plus ten plus one
32 XXXII ten plus ten plus ten plus one plus one
```

It should be noted that, because of this optimization, Roman numerals will be written such that the characters with the largest numeric value will be on the left and characters with the smallest numeric values will be on the right. This is because any characters with smaller numeric values will always be implied by the characters with the larger numeric values. This allows for the subtraction rule discussed later on this page.

### Five, V

Having a ten mark is nice, but something is sill missing: nine `I`

characters is still a lot to recount, and prone to error since there are so many like characters in a row. It would be more convenient if there was a different mark every five tally marks, not just every ten. And, five is half of ten, so it seems logical to choose `V`

as the five mark because it visually looks like the top half of an `X`

.

Tally counting from 1 to 11 would look like this.

```
11 IIIIVIIIIXI
```

The table below shows some values using the one, five and ten marks.

```
5 V five
6 VI five plus one
17 XVII ten plus five plus one plus one
25 XXV ten plus ten plus five
26 XXVI ten plus ten plus five plus one
```

### Hundred, C

As the tally count gets higher, more characters are needed to make the recount easy. Since the Romans use a base-10 number system and there already one ad ten marks, it make since that there also needs to be 100 mark. And, if the one mark is a single stroke and the ten mark is a double stroke, the logical choice for a hundred is a triple stroke character. If one thinks about it, the character `C`

is kind of a stylized, triple stroke character (top, left & bottom) that can be written quickly with a single “swoop” of he pen. Besides, `C`

is easy to remember for "one hundred" since most be know “century” means 100 years.

Using the hundred mark every one hundred tally strokes looks something like this when counting from 1 to 101:

```
IIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIICI
```

And, as before, there are implied characters because everything between the hundred marks is implied.

```
100 C hundred
101 CI hundred plus one
231 CCXXXI hundred plus hundred plus ten plus ten plus ten plus one
```

### Fifty, L

Recall how the Roman numeral `V`

(5) visually looks like half of `X`

(10). It sure would be nice if the same pattern can be applied to create a character for 50 from the character for 100. Although `C`

can be written in single “swoop” of the pen, recall that it is just a stylized triple-stroke character with a top, left, and bottom. Therefore, half of that would be a right angle bracket similar to ˪. And, the letter `L`

is a stylized right angle bracket (albeit a bit tall).

Using the `L`

, the tally marks from 1 to 101 look something like this.

```
IIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIILIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIIXIIIIVIIIICI
```

The table below shows a few examples with the implied characters removed.

```
50 L fifty
51 LI fifty plus one
60 LX fifty plus ten
66 LXVI fifty plus ten plus five plus one
88 LXXXVIII fifty plus ten plus ten plus ten plus five plus one
```

### Multiples of Ten, CIƆ

It does not seem very practical to continually assign new characters for every new multiple of ten, and for half of that value. But, at some point, the need arouse to count values greater than 1000. And, imagine one was typesetting the tally count, but only had the few letters available that are normally associated with Roman numbers: `I`

, `V`

, `X`

, `L`

, `C`

. A new pattern was created for this.

Roman numerals can use the letter `C`

kind of like parenthesis, and is a letter available to a typesetter whom only has access to the characters generally used for numbers. If `C`

is used as an open parenthesis, then the letter `C`

could be rotated 180° by a typesetter and used as the closed parenthesis; the rotated `Ɔ`

is known as a "apostrophus".

NOTE: Occasionally, for ease of typing, one will sometimes see an open parenthesis `(`

used for `C`

and a closed parenthesis `)`

used for `Ɔ`

.

The first set of parenthesis is used used as a *thousand multiplier*, and each additional set is used as an additional *ten multiplier*. It is important to use all of the closing `Ɔ`

characters to avoid any confusion, as is illustrated in the difference between 101 and 1,000 below.

```
101 CI
1,000 CIƆ
10,000 CCIƆƆ
100,000 CCCIƆƆƆ
1,000,000 CCCCIƆƆƆƆ
```

And, sticking with the pattern that the half marks (`V`

, `L`

) resemble roughly half of the full marks (`X`

, `C`

), we get the following:

```
500 IƆ
5,000 IƆƆ
50,000 IƆƆƆ
500,000 IƆƆƆƆ
```

They can even be combined together! For every opening `C`

there must be a closing `Ɔ`

. But, there can be additional closing `Ɔ`

characters’s:

```
1,500 CIƆƆ thousand plus five hundred
10,500 CCIƆƆƆ ten thousand plus five hundred
15,000 CCIƆƆƆƆ ten thousand plus five thousand
100,500 CCCIƆƆƆƆ hundred thousand plus five hundred
105,000 CCCIƆƆƆƆƆ hundred thousand plus five thousand
150,000 CCCIƆƆƆƆƆƆ hundred thousand plus fifty thousand
1,000,500 CCCCIƆƆƆƆƆ million plus five hundred
1,005,000 CCCCIƆƆƆƆƆƆ million plus five thousand
1,050,000 CCCCIƆƆƆƆƆƆƆ million plus fifty thousand
1,500,000 CCCCIƆƆƆƆƆƆƆƆ million plus five hundred thousand
```

NOTE: Occasionally, some of the more common numbers may even be written with special short-hand characters of similar appearance.

```
1,000 CIƆ ↀ ∞
5,000 IƆƆ ↁ
10,000 CCIƆƆ ↂ
```

### Thousand, M

Even though there is already a way to represent the number 1,000 with `CIƆ`

, it is a bit cumbersome because it takes a great deal of space and those pesky apostrophus characters are not necessarily that easy to type. Therefore, since Roman numerals are very often used for years, having a dedicated thousand character would be pretty handy.

The letter `M`

seems like a good choice because it satisfies all of the patterns set this far:

- It is a four-stroke character, which is what one would expect after the three-stoke
`C`

for 100, two-stroke`X`

for 10, and one-stroke`I`

for 1. - It was previously mentioned that open and closed parenthesis are often used instead of
`C`

and`Ɔ`

, respectively, and so 1,000 could be represented as`CIƆ`

or`(I)`

, and`(I)`

just so happens to resemble an`M`

. Besides, most people know that a millenium is every thousand years.

```
1,000 M
2,001 MI
2,010 MX
```

### Five Hundred, A, D

The previous section mentioned that a dedicated `M`

character can be used for 1,000. Following the same pattern set forth by `V`

for 5 and `L`

for 50, there should be some character assigned to 500 that visually resembles roughly half of an `M`

which is used for 1,000. The `A`

character fits the bill, and some people think that `A`

may have indeed been used for 500. But, it is more common to see a `D`

used for 500. Although the character `D`

does not look like half of an `M`

, it does resemble `IƆ`

which is what it is really replacing anyway.

```
1776 MDCCLXXVI
```

### Subtraction Rule

Implied characters have already been discussed. And, it has already been mentioned that when the implied tally marks are removed so the final number can be recorded in the record book, the characters will be arranged in order of decreasing numeric value with the largest value on the left and the smallest value on the right. And, when that occurs, the smaller value characters on the right are added to the larger value characters on the left.

This actually opens the door for an additional optimization: what happens if smaller value character is placed to the left of or between characters with larger values. The answer is that the smaller value is subtracted from the larger value on the right.

In the table below, take the row number and multiply it by the column multiplier to get the decimal value.

```
x100 x10 x1
----- ----- -----
1 C X I
2 CC XX II
3 CCC XXX III
4 CD XL IV
5 D L V
6 DC LX VI
7 DCC LXX VII
8 DCCC LXXX VIII
9 CM XC IX
10 M C X
11 MC CX XI
12 MCC CXX XII
13 MCCC CXXX XIII
14 MCD CXL XIV
15 MD CL XV
16 MDC CLX XVI
17 MDCC CLXX XVII
18 MDCCC CLXXX XVIII
19 MCM CXC XIX
20 MM CC XX
```

Programmer, Engineer