Roman Numerals

Introduction

The numerals 0, 1, 2, 3 are used in writing numbers. These numbers are Indian in origin and the Arabs picked them up and spread to Europe.  Therefore, these are called Hindu-Arabic numerals systems. This is not the only system available. One of the early systems on numeration still in common use today was developed by Romans and is called Roman numerals systems. Let us learn about this system.

What are Roman Numerals?

The Roman numeral system is one of the early numeral systems that are still in use today in many places. For example, we can see its use in the numbering of different volumes or parts of books, chapter numbers, numbers of issues of magazines, numbers on clock faces, etc. 

They are a special kind of numerical notations that were earlier used by the Romans. The Roman numeral is an additive and subtractive system in which letters are used to denote certain base numbers and arbitrary numbers in the number system. Although the Roman numeral system provided for easy addition and subtraction, other arithmetic operations proved more difficult. 

History of Roman Numerals

The Roman numeral system for representing numbers was developed around 500 B.C. As the Romans conquered much of the world that was known to them, their numeral system spread throughout Europe, where Roman numerals remained the primary manner for representing numbers for centuries. Around 1300 A.D, the Roman numerals were replaced throughout most of Europe with the more effective Hindu-Arabic system that is still in use today.

The Distinct Symbols in Roman Numerals System

 We know that in the Hindu-Arabic numeral system we have 10 distinct digits that form the entire numeral system, namely, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Similarly, there are 7 distinct symbols that form the entire Roman numeral system. These symbols are I, V, X, L, C, D and K. So how do we use these symbols to form values in this system? 

Relation between Roman Numerals and Hindu-Arabic Numerals

The following is the relation between the Roman Numerals and the Hindu-Arabic Numerals

Roman Numerals	Hindu-Arabic Numerals
I	1
V	5
X	10
L	50
C	100
D	500
K	1000

How can these seven numerals be used to write different numbers?  Certain rules define the writing of numerals under the Roman numeral system. Let us learn about these rules.

Rules that define the Roman Numeral System 

Roman numerals are usually written in order, from largest to smallest and from left to right, but more than three identical symbols never appear in a row. The following rules determine the way the numbers are written in the Roman numeral system.

1. If a symbol is repeated, its value is added as many times as it occurs.

For example, 

II = 1 + 1 = 2

III = 1 + 1 + 1 = 3

XX = 10 + 10 = 20

XXX = 10 + 10 + 10 = 30

2. If a symbol of a smaller value is written to the right of a symbol of greater value, we add its value to the value of the greater symbol.

For example,

VI = 5 + 1 = 6

VII = 5 + 2 = 7

VIII = 5 + 3 = 8

XI = 10 + 1 = 11

XII = 10 + 2 = 12

LXV = 50 +10 +5 = 65

3. If a symbol of a smaller value is written to the left of a symbol of greater value, then its value is subtracted from the value of the greater symbol. 

For example,

IV = 5 – 1 = 4

IX = 10 – 1 = 9

XL = 50 – 10 = 40

XC = 100 – 10 = 90

4. The symbols, V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted. This means that

• The symbol I can be subtracted from V and X only.
• The symbol X can be subtracted from L, K and C only.
• The symbol C can be subtracted from D and K only.
• If a symbol numeral is placed between two larger numerals, it is always subtracted from the larger numeral immediately following it.

For example,

XIV = 10 + (5 – 1 ) = 10 + 4 = 14

XIX = 10 + ( 10 – 1 ) = 10 + 9 = 19

CXIV = 100 + 10 + ( 5 – 1 ) = 100 +10 +4 = 114

6. If a bar is placed over a numeral, it is multiplied by 1000.

For example,

$\overline{V}$ = 5000 and $\overline{L}$ = 50000

Let us now list different numbers according to the Roman numeral system.

Based on the cases above, we can define the six cases where subtraction is used to determine Roman numerals

The Six Cases of Subtraction

The following are the six cases of subtraction with the help of which we can determine and evaluate Roman numerals – 

1. I is placed before V

When I is placed before V, we will get IV = 5 – 1 = 4.

2. I is placed before X

When I is placed before X, we will get IX = 10 – 1 = 9.

3. X is placed before L

When X is placed before L, we will get XL = 50 – 10 = 40.

4. X is placed before C

When X is placed before C, we will get XC = 100 – 10 = 90.

5. C is placed before D

When C is placed before D, we will get CD = 500 – 100 = 400.

6. C is placed before M

When C is placed before M, we will get CM = 1000 – 100 = 900.

Roman Numerals from 1 to 100

The following table shows the Roman numerals from 1 to 100

Number	Roman Numeral	Number	Roman Numeral	Number	Roman Numeral	Number	Roman Numeral
1	I	26	XXVI	51	LI	76	LXXVI
2	II	27	XXVII	52	LII	77	LXXVII
3	III	28	XXVIII	53	LIII	78	LXXVIII
4	IV	29	XXIX	54	LIV	79	LXXIX
5	V	30	XXX	55	LV	80	LXXX
6	VI	31	XXXI	56	LVI	81	LXXXI
7	VII	32	XXXII	57	LVII	82	LXXXII
8	VIII	33	XXXIII	58	LVIII	83	LXXXIII
9	IX	34	XXXIV	59	LIX	84	LXXXIV
10	X	35	XXXV	60	LX	85	LXXXV
11	XI	36	XXXVI	61	LXI	86	LXXXVI
12	XII	37	XXXVII	62	LXII	87	LXXXVII
13	XIII	38	XXXVIII	63	LXIII	88	LXXXVIII
14	XIV	39	XXXIX	64	LXIV	89	LXXXIX
15	XV	40	XL	65	LXV	90	XC
16	XVI	41	XLI	66	LXVI	91	XCI
17	XVII	42	XLII	67	LXVII	92	XCII
18	XVIII	43	XLIII	68	LXVIII	93	XCIII
19	XIX	44	XLIV	69	LXIX	94	XCIV
20	XX	45	XLV	70	LXX	95	XCV
21	XXI	46	XLVI	71	LXXI	96	XCVI
22	XXII	47	XLVII	72	LXXII	97	XCVII
23	XXIII	48	XLVIII	73	LXXIII	98	XCVIII
24	XXXIV	49	XLIX	74	LXXIV	99	XCIX
25	XXV	50	L	75	LXXV	100	C

Roman Numbers from 100 to 1000

The following table shows the corresponding Roman numerals from numbers from 1 to 1000-

Number	Roman Numeral	Evaluation
100	C	100
200	CC	100 +100 = 200
300	CCC	100 + 100 + 100 = 300
400	CD	500 – 100 = 400
500	D	500
600	DC	500 + 100 = 600
700	DCC	500 + 100 + 100 = 700
800	DCCC	500 +100 +100 +100 = 800
900	DM	1000 – 100 = 900
1000	M	1000

Roman Letters

Similar to numerals in the Roman system, we have Roman letters as well. The roman letters are English alphabets but not all the alphabets are Roman. There are 23 roman alphabets out of 26 English alphabets, in which J, U and W are not considered as roman alphabets. Hence, the roman alphabets are:

A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, V, X, Y and Z. These roman letters are also called roman symbols.

For example, the year 2019 is written as MMXIX.

Limitations of Roman Numeral System

There were some limitations of the Roman numerals system which led to the replacement of this system by the Hindu-Arabic system that we use today. Some of these limitations were – 

1. Having no symbol for zero was one of the major limitations of the Roman numeral system. This hindered the ability to develop a universally understood, sophisticated math system, and made trading more difficult.
2. Another major limitation of this system of no rule is defined for calculating fractions. This was again a challenge for various day to day calculations that involved the use of fractional numbers. The Romans usually expressed fractions in terms of the uncia. An uncia originally meant 1/12 of the Roman measure of weight (English derived the word “ounce” from uncia).
3. Multiplication and division were fairly difficult operations in this system which even led the Roman to develop counting boards, similar to the abacus that we know today.

Despite all these limitations, there are areas where the Roma numerals are still in use today. Let us discuss the use of Roman Numerals in today’s age.

Where are Roman Numerals used today?

Although Roman numerals are not in use for regular mathematical operations, they still find their use in many other places. Some of the areas where Roman numerals are still used are – 

1. Roman numerals are often used in books, often to number the chapters. For instance, the bottom of the page of your textbook may Read, “ Textbook for Class VII “. Here VII is the Roman numeral used for the number 7. Moreover, pages in appendices or introductions are also numbered with Roman numerals.
2. Roman numerals can be seen on clocks and watches. Notice the way the numbers on the dials of many clocks and watches are written and you will be surprised to see that many of them make use of Roman numerals.
3. Annual sporting events, like the Summer and Winter Olympics and the Super Bowl, also mark the passage of years by using Roman numerals.
4. Many generations have a family name that has been passed down and includes a Roman numeral to signify the family member. For example, if a man’s name is John Philips and his father and grandfather were also named John that would make him John Philips III. Royal families also use this system.
5. Roman numerals are used in astronomy as well. For instance, they are used to count the number of moons of a planet.

Solved Examples

Example 1 Write 69 in roman numerals.

Solution We have been given the 69 and we are required to write it in Roman numerals. Following the rule stated above, we have,

69 = 60 + 9

= (50 + 10) + 9

= LX + IX

= LXIX

Hence 69 in Roman numerals will be written as LXIX

Example 2 Write 1969 in Roman Numerals 

Solution To write 1969 in Roman Numerals, we will have to break it up and write in expanded form, as we do in the number system. We will have,

1969 = 1000 + 900 + 60 + 9

Now, we will find the corresponding Roman numerals of all the numbers obtained in the expanded form.

We will have,

1000 in Roman numerals is equal to M

 900 in Roman numerals is equal to CM

60 in Roman numerals is equal to LX

9 in Roman numerals is equal to IX

So, we have,

1969 = 1000 + 900 + 60 + 9

Therefore,

1969 = MCMLXIX

Example 3 Find the value of the following Roman numeral MXXII – LXX – LII

Solution We have been given an expression in the form of Roman numerals and we need to find its value in numbers. To find the value, first, we will obtain the corresponding numbers for all the Roman numerals given to us. Let us find the values one by one. We have,

MXXII = 1000 + 10 +10 + 1 + 1 = 1022

LXX = 50 +10 +10 = 70

LII = 50 + 1 + 1 = 52

Now, that we have obtained the corresponding numbers for all the Roman numerals given to us, we will substitute these values in the expression that has been given to us. Therefore,

MXXII – LXX – LII = 1022 – 70 – 52 = 900.

Hence, the Roman expression MXXII – LXX – LII in numeric form is equal to 900.

Example 4 Write the Roman numerals for the following – 

a) 2359

b) 3786

Solution We have been given numbers in numeric form and we need to find the corresponding Roman numerals for them. Let us do them one by one.

a) 2359

First, let us write the given number in the expanded form. We have,

2359 = 2000 + 300 + 50 + 9

Now, we will find the respective Roman numeral for each of the number in the expanded form. We now have,

2000 in Roman numerals is equal to MM

300 in Roman numerals is equal to CCC

50 in Roman numerals is equal to L

9 in Roman numerals is equal to IX

Therefore, 

2359 = 2000 + 300 + 50 + 9 = MMCCCLIX

b) 3786

First, let us write the given number in the expanded form. We have,

3786 = 3000 + 700 + 80 + 6

Now, we will find the respective Roman numeral for each of the numbers in the expanded form. We now have,

3000 in Roman numerals is equal to MMM

700 in Roman numerals is equal to DCC

80 in Roman numerals is equal to LXXX

6 in Roman numerals is equal to VI

Therefore,

3786 = 3000 + 700 + 80 + 6 = MMMDCCLXXXVI

Key Facts and Summary

1. The Roman numeral system is one of the early numeral systems that are still in use today in many places.
2. There are 7 distinct symbols that form the entire Roman numeral system. These symbols are I, V, X, L, C, D and K.
3. Certain rules determine the writing of numbers in the Roman numeral system. These rules are
   • If a symbol is repeated, its value is added as many times as it occurs.
   • If a symbol of a smaller value is written to the right of a symbol of greater value, we add its value to the value of the greater symbol.
   • If a symbol of a smaller value is written to the left of a symbol of greater value, then its value is subtracted from the value of the greater symbol. 
   • The symbols, V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted. This means that
     1. The symbol I can be subtracted from V and X only.
     2. The symbol X can be subtracted from L, K and C only.
     3. The symbol C can be subtracted from D and K only.
   • If a symbol numeral is placed between two larger numerals, it is always subtracted from the larger numeral immediately following it.
   • If a bar is placed over a numeral, it is multiplied by 1000.
4. The roman letters are English alphabets but not all the alphabets are roman alphabets. There are 23 roman alphabets out of 26 English alphabets, in which J, U and W are not considered as roman alphabets.