problem solving with roman numerals

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Arithmetic with Roman Numerals

Have you ever wondered how to add, subtract, multiply, and divide using Roman numerals? On one hand, we’ll give the simple answer that the Romans didn’t actually do what you think; on the other hand, we’ll consider what they actually did.

Adding and subtracting

We’ll start with a question from 1999:

Doctor Rick answered:

We discussed this relationship of Roman numerals to the abacus last time. The FAQ (before showing how to use the abacus!) gives two examples of how one might do addition, which I’ll reformat here for readability:

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 : LXXV IIIIII . Now we have six Is, so we’ll rewrite them as VI: LXXV VI . The two Vs are the same as an X, so we simplify again and get LXX X I, or 81, as our final answer.

Note that this had no subtractive parts, so the sum is just the sum of all the symbols, which we could rearrange at will. All the work was mere simplification.

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: XX VV I. Simplifying gives us XX X I, or 31.

Would you enjoy doing this? Doctor Rick showed a subtraction, offering some ways to make both addition and subtraction easier:

The Romans probably never wrote work like this, but did equivalent things on the abacus:

We’ll be taking that advice soon! But first we’ll try other ways.

Adding on the Roman (and later) counting board

Here is what the FAQ shows for addition:

It’s not hard to see how to follow the same procedure on the Roman version of the board, or on the abacus (which has attached beads or buttons rather than pebbles or counters). Here is a picture, from last time, of a replica abacus:

Multiplying?

Here’s a question from a teacher in 2001, which is not about Roman numerals at all, but will lead to them:

I answered:

So the Romans, if they tried to multiply as we do, would have needed to memorize huge multiplication tables. That’s why they considered the very idea ridiculous.

Multiplying on the abacus

The FAQ. after showing how to add on the Roman counting board, shows how to use it to multiply.

I imagine you still prefer the modern way, whether on paper or with a calculator!

Multiplying the Egyptian way

As mentioned before, the Romans likely also used the Egyptian method, which is the subject of this 1996 question:

Doctor Jodi answered:

All of this is true of the Romans, too!

All the arithmetic you need is doubling, in each column, and then subtraction to find which doubles add up to our multiplier!

We found that \(41=32+8+1\), so $$59\cdot41=59(32+8+1)\\=59\cdot32+59\cdot8+59\cdot\\=1888+472+59=2419$$

How is this binary? In splitting 41 into 32 + 8 + 1, we are effectively writing it as the binary number 101001. We’ll look at this more next week, in examining a related method in more depth.

It doesn’t matter which number you put on each column, though one way may involve more additions:

Here, \(58=32+16+8+2+1\), so $$41\cdot59=41(32+16+8+2+1)\\=41\cdot32+41\cdot16+41\cdot8+41\cdot2+41\cdot1\\=1312+656+328+82+41=2419$$

Dividing the Egyptian way

A 1998 question is about Egyptian division, which the Romans would also have used:

Doctor Mateo answered:

Here, rather than find which rows in the first column add up to the multiplier , we are finding which rows in the second column add up to the dividend .

Here, we found that the dividend is $$153=144+9=9\cdot16+9\cdot1=9(16+1)=9\cdot17,$$ so \(153\div9=17\).

What if we didn’t get an exact sum to 153?

I’m going to skip the rest, because the Egyptians had their own way to handle fractions, which is irrelevant to Roman numerals. I imagine they would find a way to use uncia (twelfths).

Egyptian division in Roman numerals

We’ll close with a 2004 question on division:

(The reference to halving is to a variant of the method, called Russian Peasant Multiplication ; we’ll look into that next time.)

But what would it look like to use the Egyptian method for division, using Roman numerals ? Let’s have fun!

I have doubled repeatedly, starting with 1 and 9, then subtracted repeatedly from 153 to find that \(153=144+9\), just as in the example above. The doubling, and the subtractions, would undoubtedly be done on an abacus. I also suspect that they would skip the subtractive notation, and just write IIII and CXXXXIIII rather than IV and CXLIV. That takes less thought.

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Roman Numerals

Explore a variety of different solutions to this tricky exercise, ranging from recursion to table-based lookups, one solution that makes use of mixed radix numbers, and some simplistic cheating by Common Lisp.

Today, most people in the world use Arabic numerals (0–9). But if you travelled back two thousand years, you'd find that most Europeans were using Roman numerals instead.

To write a Roman numeral we use the following Latin letters, each of which has a value:

A Roman numeral is a sequence of these letters, and its value is the sum of the letters' values. For example, XVIII has the value 18 ( 10 + 5 + 1 + 1 + 1 = 18 ).

There's one rule that makes things trickier though, and that's that the same letter cannot be used more than three times in succession . That means that we can't express numbers such as 4 with the seemingly natural IIII . Instead, for those numbers, we use a subtraction method between two letters. So we think of 4 not as 1 + 1 + 1 + 1 but instead as 5 - 1 . And slightly confusingly to our modern thinking, we write the smaller number first. This applies only in the following cases: 4 ( IV ), 9 ( IX ), 40 ( XL ), 90 ( XC ), 400 ( CD ) and 900 ( CM ).

Order matters in Roman numerals! Letters (and the special compounds above) must be ordered by decreasing value from left to right.

Here are some examples:

And a final more complex example:

Your task is to convert a number from Arabic numerals to Roman numerals.

For this exercise, we are only concerned about traditional Roman numerals, in which the largest number is MMMCMXCIX (or 3,999).

There are lots of different ways to convert between Arabic and Roman numerals. We recommend taking a naive approach first to familiarise yourself with the concept of Roman numerals and then search for more efficient methods.

Make sure to check out our Deep Dive video at the end to explore the different approaches you can take!

Roman Numerals is featured as part of #48in24 challenge . Solve it in Elixir , Julia , and Pharo to earn the gold award.

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Roman Numerals

Ancient Romans used a special method of showing numbers

Examples: They wrote C instead of 100 And wrote IX instead of 9

Read on to learn about Roman Numerals or go straight to the Roman Numeral Conversion Tool .

The Roman Symbols

Romans Numerals are based on the following symbols:

Basic Combinations

Which can be combined like this:

Forming Numbers - The Rules

When a symbol appears after a larger (or equal) symbol it is added

• Example: VI = V + I = 5 + 1 = 6
• Example: LXX = L + X + X = 50 + 10 + 10 = 70

But if the symbol appears before a larger symbol it is subtracted

• Example: IV = V − I = 5 − 1 = 4
• Example: IX = X − I = 10 − 1 = 9

To Remember: A fter Larger is A dded

Don't use the same symbol more than three times in a row (but IIII is sometimes used for 4, particularly on clocks)

How to Convert to Roman Numerals

Break the number into Thousands, Hundreds, Tens and Ones, and write down each in turn.

Example: Convert 1984 to Roman Numerals.

Break 1984 into 1000,900, 80 and 4, then do each conversion

1000 + 900 + 80 + 4 = 1984, so 1984 = MCMLXXXIV

How To Remember

Think "MeDiCaL XaVIer". It has the roman numerals in descending order from 1000 to 1.

I, for one, like Roman numerals!

Really Big Numbers

Numbers greater than 1,000 are formed by placing a dash over the symbol, meaning "times 1,000", but these are not commonly used:

Conversion Tool

You can convert to/from Roman Numerals here:

Roman Numerals Questions

Roman numerals questions with solutions are given here to practise and learn Roman numerals. Roman numerals are a way to write general decimals or base ten numbers. Romans from ancient Europe devised this numeral system. Roman numerals have specific symbols designated to different numerals of the number system. These symbols are given in the following chart.

For example, if we want to write 156 in Roman numerals , it will be CLVI.

Video Lesson on Roman Numerals

Roman Numerals Questions with Solutions

Let us solve a few questions on Roman numerals.

Question 1:

Write the following in Roman numerals:

Question 2:

Convert the following Arabic numerals into Roman numerals.

(vii) 98100

(viii) 87107

(ix) 109826

Roman Numerals for Larger Numbers

Question 3:

Write the following Roman numerals in Arabic numerals:

(iii) DXLVI

\(\begin{array}{l}(v)\:\overline{XIX}DCCLIV\end{array} \)

• Roman Numerals Conversion
• Roman Numerals Calculator
• Roman Numerals from 1 to 50
• Roman Numerals from 1000 to 2000

Question 4:

Write the following Roman numerals in words:

\(\begin{array}{l}(i).\:\overline{V}MMMDCXLIII\end{array} \)

(ii) MMDCXLVII

(iii) MCCLVIII

Let us first convert the Roman numerals to Arabic numerals.

∴ 8643 = Eight thousand six hundred forty-three.

∴ 2647 = Two thousand six hundred forty-seven.

∴ 1258 = One thousand two hundred fifty-eight.

Question 5:

Arrange the following Roman numerals in ascending order:

CCXCVIII, CCCLXI, CCVII, CCLXXXIX, CCCXVI, CCVI.

CCVI < CCVII < CCLXXXIX < CCXCVIII < CCCXVI < CCCLXI.

Question 6:

Arrange the following Roman numerals in descending order:

DLXXVI, DCCVI, DLXXX, DVIII, DLXVII, DLX.

706 > 580 > 576 > 567 > 560 > 508.

∴ DVIII < DLX < DLXVII < DLXXVI < DLXXX < DCCVI

Question 7:

Match the following:

a. – (iv)

b. – (v)

c. – (ii)

d. – (vi)

e. – (i)

f. – (iii)

Question 8:

Use the following Roman numerals to form 5 different numerals: X, V, I

Five different numerals are: XI, XV, IX, VI, XVI.

Question 9:

(i) XCII + CXII

(ii) CCXXXIV – CCII

(iii) LVI × XVI

(i) XCII = 92 and CXII = 112

Then, 92 + 112 = 204

∴ XCII + CXII = CCIV.

(ii) CCXXXIV = 234 and CCII = 202

Then, 234 – 202 = 32

∴ CCXXXIV – CCII = XXXII.

(iii) LVI = 56 and XVI = 16

Then, 56 × 16 = 896

∴ LVI × XVI = DCCCXCVI.

Question 10:

State ‘True’ or ‘False’ for the following:

(i) V is never written left to X.

(ii) The roman numeral for 500 is CCCLLLL.

(iii) There is no sign for 0 in the Roman numerals.

(iv) D, L, and M stand for 500, 50 and 1000, respectively.

(i) V is never written left to X. (True)

(ii) The Roman numeral for 500 is CCCLLLL. (False)

(iii) There is no sign for 0 in the Roman numerals. (True)

(iv) D, L, and M stand for 500, 50 and 1000, respectively. (True)

Practice Questions on Roman Numerals

1. Write the following numbers in Roman numerals:

2. Write the following Roman numerals in Arabic numerals:

(i) MMMCCLXXXVII

(ii) MMMCDLXXXVI

(iii) MMMXL

\(\begin{array}{l}(iv)\:\overline{V}DCII\end{array} \)

(v) MLXVIII

\(\begin{array}{l}(vi)\:\overline{XXX}MMDCCCLXXVI\end{array} \)

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Roman numerals: fun teaching strategies for Year 4

Although Roman Numerals are not used frequently today, it’s important for children to see numbers represented in different ways.

Children will still see Roman numerals in lots of real-life contexts as they are a rich part of our cultural heritage, they teach basic maths facts, and they can be fun to learn!

Why do children need to learn these cumbersome relics from the past? Well, firstly it’s one of the statutory requirements to ‘read Roman numerals to 100 (I to C)’ in the Year 4 maths programme of study for Number. But, I also find that learning Roman numerals is a great way to help children increase their number sense, better understand how numbers work, and is particularly good for increasing mental maths skills.

Where to begin

It’s worth talking to the children about the history of Roman numerals to help them realise the various counting systems in the ancient world before the creation of Roman numbers and our own numerical system. For example, you could talk to them about the Etruscans, who lived in central Italy before the Romans and had their own numeral system with different symbols.

Give the children some context on how Roman numbers were widely used throughout the Roman Empire in everyday life and, following the fall of the Roman Empire, how they were used throughout Europe up until the 1600’s. Ask them to think about where they may have seen Roman numerals recently and show examples used on clocks, buildings and monuments.

Learning the basics

Initially it’s vitally important for children to learn what Roman Numerals look like, how they combine together to make different numbers and how to read/convert them. Understanding what Roman numerals represent can feel like deciphering a code. It can be challenging but can also present some really fun options for learning.

To get started, show children ‘the basics’. While Year 4 are only required to learn 1-100, there’s no reason why you can’t also introduce 500 = D and 1000 = M and share with them all 7 symbols.

Remembering I, V, X, L, C, D and M can be tricky, so ask the children to make up a sentence to help such as:

“ I V alue X ylophones L ike C ats D evour M ilk”

“ I f V era’s X -ray L ooks C lear D on’t M edicate”

“ I V iewed X avia L eaping C arelessly D own M ountains”

Another great way to help children learn these numerals is to play a simple recognition game such as a classic ‘Four In A Row’ game:

What you will need:

• Sticky labels to cover a 1-6 die in the Roman numerals V, X, L, C, D and M
• Some coloured counters
• A playing board

What to do:

• Find a partner, this is a game for two players.
• Take turns to roll the die. When you see your number, cover it up using one of your counters.
• The first player to get four numbers in a row (horizontally, vertically or diagonally) is the winner.

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How to combine numerals

It is important to help your pupils understand that numerals can be combined to make lots of different numbers. I tend to show them a 1-100 Roman numeral square then I explore various number combinations and explain how the numerals ‘work’.

The key rules are that when a symbol appears after a larger symbol, it is added (eg: VI = V + I = 5 + 1 = 6), and when a symbol appears before a larger symbol it is subtracted (eg: IX = X – I = 10 – 1 = 9). You can make up a rhyme or saying that can help with the learning:

When “left” is small and “right” is bigger, Subtract the “left” from “right-hand” figure.

Arguably, the first 12 Roman numerals are the most important to focus on as these are the numerals need for telling the time. Students should then also become aware that they shouldn’t use the same symbol more than three times in a row (e.g. 4 is written as IV not IIII).

You may decide to use a partially completed 1-100 square for children to complete themselves so they become familiar with the combinations or you play the Roman Numeral Bingo game:

• Read out numbers for children to spot on their Bingo cards. You can give each child a 1-100 number square of Roman numerals to help them.
• When children see the number on their card they cross it off.
• The first child to mark off a winning pattern is the winner.

Games that teach, build or strengthen maths skills and concepts while having fun is essential and a really exciting way for children to recognise and learn Roman numerals.

Taking the learning further

‘Doing’ maths with Roman numerals is a bit of a mixed bag. Adding and subtracting using Roman numerals is actually very easy but multiplication, division and fractions are much more of a challenge. The lack of a zero is a huge disadvantage so it’s little surprise that Indo-Arabic numerals slowly replaced Roman ones over the years.

Getting hands-on and minds-on is fundamental to really ‘getting’ Roman Numerals. I often find using safe matchsticks is a great way for children to not just form the numerals but also practise some addition and subtraction. Why not set a few word problems using them? Or you could play a sequence game where children can time each other to see how fast they can put Roman numerals in the correct sequence starting with the smallest number. I often set up some Roman numerals multiplication or division problems but just be careful, make sure the ‘times’ sign is in lower case to ensure there’s no confusion!

Roman numerals can be practised playing hopscotch, loop games, using input/output machines and number balances, as part of Venn diagrams and Carroll diagrams, to make magic squares, used as coordinates, for area and perimeter calculations, for codebreaking, exchanging money, playing calendar games, dice games, number snap, addition and subtraction grids and much more.

While learning to read and write, Roman numerals might not be the most important maths children will learn. There is still value in it because it is a fun way to integrate maths into history lessons, reinforce other maths concepts and develop children’s maths stamina, resilience and interest.

John Dabell

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Roman Numerals Problem Solving

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

2 December 2014

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