Roman Numerals

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Basics
How do I read and write Roman numerals?
What are the rules for subtracting letters? Can I write MIM? What about IIII?
How do I write large numbers?
Web resources: introductions, charts, calculators, and converters
Calculations
Did the Romans use fractions?
How can I use Roman numerals to do arithmetic problems?
Uses Past and Present
How did the Romans use math?
How are Roman numerals used today?
Resources for other numeral systems

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### Basics

How do I read and write Roman numerals?

A numeral is a symbol used to represent a number. (Our digits 0-9 are often called Arabic numerals.) Each letter used in Roman numerals stands for a different number:

 Roman Numeral Number I 1 V 5 X 10 L 50 C 100 D 500 M 1000

A string of letters means that their values should be added together. For example, XXX = 10 + 10 + 10 = 30, and LXI = 50 + 10 + 1 = 61. If a smaller value is placed before a larger one, we subtract instead of adding. For instance, IV = 5 - 1 = 4.

You can use these rules to write a number in Roman numerals. Convert one digit at a time. Let's try 982:

982
= 900 + 80 + 2
= CM + LXXX + II
= CMLXXXII.

#### From the Dr. Math archives:

What are the rules for subtracting letters? Can I write MIM for 1999? What about IIII for 4?

Here are the official rules for subtracting letters:

• Subtract only powers of ten, such as I, X, or C. Writing VL for 45 is not allowed: write XLV instead.
• Subtract only a single letter from a single numeral. Write VIII for 8, not IIX; 19 is XIX, not IXX.
• Don't subtract a letter from another letter more than ten times greater. This means that you can only subtract I from V or X, and X from L or C, so MIM is illegal.

These rules only became official in the Middle Ages. Even today, not everybody follows them: you might notice that some clocks say IIII instead of IV.

#### From the Dr. Math archives:

How do I write large numbers?

The biggest Roman numeral is M, for 1000, so one easy way to write large numbers is to line up the Ms: MMMMMMM would be 7000, for instance. This system gets cumbersome quickly. When they needed to work with many large numbers, the Romans often wrote a bar above a numeral. The bar meant to multiply by 1000. Using this method, 7000 would be .

#### Web Resources:

Calculators and Converters

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### Calculations

Did the Romans use fractions?

The Romans didn't have a standard way to write fractions using their numerals. Instead, they just wrote out the word for the fraction: for example, two-sevenths was "duae septimae" and three-eighths was "tres octavae." The Romans did not have a word for every imaginable fraction: how often do you need to say thirty-three seventieths? If necessary, they would probably have said something like, "thirty-three seventieth parts," or "triginta tres septuagensimae partes."

The Romans did most of their practical calculations with fractions by using the uncia. The uncia started out as 1/12 of the as, a unit of weight (the word uncia is related to our word "ounce"), but it soon came to mean 1/12 of anything. You can add up twelfths to make halves, thirds, or quarters, so the uncia was fairly versatile. When they wanted smaller fractions, the Romans usually cut the uncia into smaller parts. The system is very similar to measuring length in inches and fractions of the inch: you might not measure an object's length exactly, but you can still come very close.

There were Roman and medieval symbols for multiples of the uncia. The semis, which was six unciae, or one-half, was often represented by this symbol: . However, uncia symbols were never standardized, and not everybody used them. Some late medieval writers even substituted the modern fraction bar.

#### From the Web:

How can I use Roman numerals to do arithmetic problems?

Let's start with an addition problem: 23 + 58. In Roman numerals, that's XXIII + LVIII. We'll begin by writing the two numbers next to each other: XXIII LVIII. Next, we rearrange the letters so that the numerals are in descending order: LXXVIIIIII. Now we have six Is, so we'll rewrite them as VI: LXXVVI. The two Vs are the same as an X, so we simplify again and get LXXXI, or 81, as our final answer. (We can check this answer using Arabic numerals.)

Now let's try another addition problem: 14 + 17, or XIV + XVII. Notice that the I in XIV is being subtracted, so this problem is going to be a little more complicated. We begin the way we did before, by writing the numbers side by side: XIV XVII. The subtracted I in XIV cancels out another I, so we cross them both out: X I V XVI I . Next we put the remaining letters into the right order: XXVVI. Simplifying gives us XXXI, or 31.

There are similar methods for subtracting, multiplying, and even dividing Roman numerals. However, they can be frustrating, and for a good reason. Even the Romans didn't use them. When they wanted to do complicated arithmetic problems, the Romans used a special counting board or an abacus. A Roman counting board looked something like this:

Counters, such as pebbles, were placed in each column. The columns were often grooved, so that the pebbles wouldn't roll away. A pebble in the bottom half of the board meant one, ten, one hundred, or one thousand, depending on its placement. A pebble in the top half had its value multiplied by five.

 For example, here's 2564 (MMDLXIV) on the counting board. Note that the Romans didn't worry about the subtraction principle unless they were actually writing their numbers down (and not always then).

There were many variations on the Roman counting board, such as extra columns for larger numbers. People in the Middle Ages turned the columns the other way and drew lines down the middle, so that the board could hold several numbers at once. This board was designed for flat counters. Counters were placed on the line for I, X, C, and M, and between lines for V, L, and D. The little x on the M line is like the modern comma in 1,000: it helps you remember the meaning of each numeral's position.

 Let's use this board to add 23 and 58. The first step is to place the two numbers (XXIII and LVIII) on the board: Next, we slide the counters together: We can replace five of the counters on the I line by a counter in the V space, and the two counters in the V space by another counter on the X line, for a final answer of LXXXI.

Here's an example of multiplication.

 We'll multiply 116 by 32 (CXVI times XXXII). This board has space for three numbers, so we can keep track of partial results. We have to break the multiplication into steps. 32 = 30 + 2, so we're going to start by multiplying by 30. The first step is to make a copy of the bigger number, 116, on the other side of the board. Next we multiply our copy by 10, because 30 = 10*3. We can do this by sliding the counters up one full line. Now we have to multiply by 3: just triple the number of counters in the copy. We simplify our result, and remove the three X counters from the 32 section, to show that we have multiplied by 30. The next step is to multiply 116 by 2. We can do this by doubling the counters for 116 on the left-hand side of the board. We simplify again, and remove the 2 I counters, because we have finished multiplying by 2. As soon as we push the counters together, and simplify one last time, we are done. Our answer is MMMDCCXII, or 3712. This can be checked using a calculator, or by hand.

Try making your own board and doing calculations with stones or pennies. By practicing on a counting-board or abacus, you can become quite fast. However, some multiplication and most division problems are still very complicated. People who needed to do many multiplications or divisions probably looked the answers up in a table, or hired someone who could.

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### Uses Past and Present

How did the Romans use math?

Unlike the ancient Greeks, the Romans weren't particularly interested in "pure" math, such as theorems about abstract geometry. They used their mathematics for more practical purposes, such as building roads, bridges, and temples out of stone, keeping accounts, and supplying their armies. People went on using Roman numerals for hundreds of years after the Roman empire fell. You might have needed Roman numerals in the Middle Ages to build a fortress, calculate the date of Easter, or sell a shipment of wool.

#### From the Web:

How are Roman numerals used today?

People use Roman numerals

• to make writing look fancy (on clocks and official documents),
• to make writing look old, and
• to avoid confusion with ordinary numbers (in outlines and the introductions of books.)

#### From the Dr. Math archives:

Roman Numerals
What are three ways Roman numerals are used today (in modern days)?

### Resources for other numeral systems

- Ursula Whitcher, for the Math Forum