Numbers in the Roman numeral system. Great encyclopedia of oil and gas

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Roman number system

An example of a non-positional number system that has survived to this day is the number system used more than two and a half thousand years ago in Ancient Rome.

The Roman number system is based on the signs I (one finger) for the number 1, V (open palm) for the number 5, X (two folded palms) for 10, as well as special signs for the numbers 50, 100, 500 and 1000.

The notation for the last four numbers has undergone significant changes over time. Scientists suggest that initially the sign for the number 100 looked like a bunch of three lines like the Russian letter Zh, and for the number 50 it looked like the upper half of this letter, which was later transformed into the sign L:

To denote the numbers 100, 500 and 1000, the first letters of the corresponding Latin words began to be used (Centum - one hundred, Demimille - half a thousand, Mille - one thousand).

To write a number, the Romans used not only addition, but also subtraction of key numbers. The following rule was applied.

The value of each smaller sign placed to the left of the larger one is subtracted from the value of the larger sign.

For example, the entry IX represents the number 9, and the entry XI represents the number 11. The decimal number 28 is represented as follows:

XXVIII =10 + 10 + 5 + 1 + 1 + 1.

The decimal number 99 is represented as follows: XCIX = (-10 + 100) (- 1 + 10).

The fact that when writing new numbers, key numbers can not only be added, but also subtracted, has a significant drawback: writing in Roman numerals deprives the number of unique representation. Indeed, in accordance with the above rule, the number 1995 can be written, for example, in the following ways:

MCMXCV = 1000 + (1000 - 100) + (100 -10) + 5,
MDCCCCLXXXXV = 1000 + 500 + 100 + 100 + 100 + 100 + 50 + 10 + 10 + 10 + 10 + 5,
MVM = 1000 + (1000 - 5),
MDVD = 1000 + 500 + (500 - 5) and so on.

There are still no uniform rules for recording Roman numerals, but there are proposals to adopt an international standard for them.

Nowadays, it is proposed to write any of the Roman numerals in one number no more than three times in a row. Based on this, a table has been constructed that is convenient to use to designate numbers in Roman numerals:

This table allows you to write any integer from 1 to 3999. To do this, first write your number as usual (in decimal). Then, for numbers in the thousands, hundreds, tens and ones places, select the appropriate code groups from the table.

In order to write down numbers greater than 3999, special rules are used, but getting to know them is beyond the scope of our course.

Roman numerals have been used for a very long time. Even 200 years ago, in business papers, numbers had to be indicated by Roman numerals (it was believed that ordinary Arabic numerals easy to fake).

The Roman numeral system is used today mainly for naming significant dates, volumes, sections and chapters in books.

3.1. Basic concepts of number systems

3.2. Types of number systems

3.3. Rules for converting numbers from one number system to another

3.4. Illustrated support material

3.5. Testing

3.6. Control questions

Different peoples at different times used different number systems. Traces of ancient counting systems are still found today in the culture of many peoples. The division of an hour into 60 minutes and an angle into 360 degrees dates back to ancient Babylon. To Ancient Rome - the tradition of writing down the numbers I, II, III, etc. in Roman notation. To the Anglo-Saxons - counting by dozens: there are 12 months in a year, 12 inches in a foot, the day is divided into 2 periods of 12 hours.

According to modern data, developed numbering systems first appeared in ancient Egypt. To write numbers, the Egyptians used hieroglyphs one, ten, one hundred, thousand, etc. All other numbers were written using these hieroglyphs and the operation of addition. The disadvantages of this system are the inability to write large numbers and its cumbersome nature.

In the end, the most popular number system turned out to be the decimal system. The decimal number system came from India, where it appeared no later than the 6th century. n. e. There are only 10 numbers in it: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, but not only the number carries information, but also the position in which it stands. In the number 444, three identical digits indicate the number of units, tens, and hundreds. But in the number 400, the first digit indicates the number of hundreds; two 0s by themselves do not contribute to the number, but are needed only to indicate the position of the number 4.

3.1. Basic concepts of number systems

Notation is a set of rules and techniques for writing numbers using a set of digital characters. The number of digits required to write a number in the system is called number system base. The base of the system is written on the right side of the number in the subscript: ;;etc.

There are two types of number systems:

positional, when the value of each digit of a number is determined by its position in the number notation;

non-positional, when the value of a digit in a number does not depend on its place in the notation of the number.

An example of a non-positional number system is the Roman one: numbers IX, IV, XV, etc.

An example of a positional number system is the decimal system used every day.

Any integer in the positional system can be written in polynomial form:

where S is the base of the number system;

Digits of a number written in a given number system;

n is the number of digits of the number.

Example. Number will be written in polynomial form as follows :

3.2. Types of number systems

Roman number system is a non-positional system. It uses letters of the Latin alphabet to write numbers. In this case, the letter I always means one, the letter V means five, X means ten, L means fifty, C means one hundred, D means five hundred, M means a thousand, etc. For example, the number 264 is written as CCLXIV. When writing numbers in the Roman number system, the value of a number is the algebraic sum of the digits included in it. In this case, the digits in the number record are, as a rule, in descending order of their values, and it is not allowed to write more than three identical digits side by side. When a digit with a larger value is followed by a digit with a smaller value, its contribution to the value of the number as a whole is negative. Typical examples illustrating general rules records of numbers in the Roman numeral system are given in the table.

Table 2. Writing numbers in the Roman numeral system

The disadvantage of the Roman system is the lack of formal rules for recording numbers and, accordingly, arithmetic operations with multi-digit numbers. Due to its inconvenience and great complexity, the Roman number system is currently used where it is really convenient: in literature (chapter numbering), in the design of documents (passport series, securities, etc.), for decorative purposes on a watch dial and in a number of other cases.

Decimal number system- currently the most famous and used. The invention of the decimal number system is one of the main achievements of human thought. Without it, modern technology could hardly exist, much less arise. The reason why the decimal number system became generally accepted is not at all mathematical. People are used to counting in the decimal number system because they have 10 fingers on their hands.

The ancient image of decimal digits (Fig. 1) is not accidental: each digit represents a number by the number of angles in it. For example, 0 - no corners, 1 - one corner, 2 - two corners, etc. The writing of decimal numbers has undergone significant changes. The form we use was established in the 16th century.

The decimal system first appeared in India around the 6th century AD. Indian numbering used nine numeric characters and a zero to indicate an empty position. In early Indian manuscripts that have come down to us, numbers were written in reverse order - the most significant number was placed on the right. But it soon became a rule to place such a number on the left side. Particular importance was attached to the zero symbol, which was introduced for the positional notation system. Indian numbering, including zero, has survived to this day. In Europe, Hindu methods of decimal arithmetic became widespread at the beginning of the 13th century. thanks to the work of the Italian mathematician Leonardo of Pisa (Fibonacci). Europeans borrowed the Indian number system from the Arabs, calling it Arabic. This historical misnomer continues to this day.

The decimal system uses ten digits—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—as well as the symbols “+” and “–” to indicate the sign of a number, and a comma or period to separate the integer and decimal parts. numbers.

IN computers used binary number system, its base is the number 2. To write numbers in this system, only two digits are used - 0 and 1. Contrary to popular misconception, the binary number system was invented not by computer design engineers, but by mathematicians and philosophers long before the advent of computers, back in the 17th century. XIX centuries. The first published discussion of the binary number system is by the Spanish priest Juan Caramuel Lobkowitz (1670). General attention to this system was attracted by an article by the German mathematician Gottfried Wilhelm Leibniz, published in 1703. It explained the binary operations of addition, subtraction, multiplication and division. Leibniz did not recommend the use of this system for practical calculations, but emphasized its importance for theoretical research. Over time, the binary number system becomes well known and develops.

The choice of the binary system for use in computing is explained by the fact that electronic elements- triggers that make up computer chips can only be in two operating states.

Using the binary coding system, you can capture any data and knowledge. This is easy to understand if we recall the principle of encoding and transmitting information using Morse code. A telegraph operator, using only two symbols of this alphabet - dots and dashes, can transmit almost any text.

The binary system is convenient for a computer, but inconvenient for a person: the numbers are long and difficult to write and remember. Of course, you can convert the number to the decimal system and write it in this form, and then, when you need to convert it back, but all these translations are labor-intensive. Therefore, number systems related to binary are used - octal and hexadecimal. To write numbers in these systems, 8 and 16 digits are required, respectively. In hexadecimal, the first 10 digits are common, and then capitals are used letters. Hexadecimal digit A corresponds to the decimal number 10, hexadecimal B to the decimal number 11, etc. The use of these systems is explained by the fact that the transition to writing a number in any of these systems from its binary notation is very simple. Below is a table of correspondence between numbers written in different systems.

Table 3. Correspondence of numbers written in different number systems

Number designation

The designation of numbers in Ancient Rome was reminiscent of the first method of Greek numbering. The Romans adopted special notations not only for the numbers $1$, $10$, $100$ and $1000$, but also for the numbers $5$, $50$ and $500$. Roman numerals looked like this:

Picture 1.

The seven numbers presented in the table were called nodal and with their help it was possible to write down any multi-digit numbers. Initially, the writing of Roman numerals was somewhat different from the numbers we are used to using nowadays. Their appearance has undergone slight changes over time.

Scientists are still debating the origin of Roman numerals. There are several views on this problem. If you take a closer look at the numbers $1$, $5$ and $10$, you can see what they look like:

$I$ sign – on a stick;

$V$ sign - on an open hand;

$X$ – on two crossed arms.

But there is another explanation for this fact.

Initially, the numbers from $1$ to $9$ were represented by the corresponding number of vertical sticks. To depict a ten, they did the following: after drawing $9$ of sticks, the tenth one was crossed out. In order not to write many sticks, they crossed out one. This is how the image of the $X$ sign appeared. The image of the sign $V$ (the number $5$) was obtained by cutting the sign $X$ (the number $10$) in half. In turn, the Etruscan people, neighboring the Romans, who were conquered by the Roman Empire, used the lower part of the symbol $X$ to write the number $5$, and the Romans themselves used the upper part.

When indicating the number $100$, the stick was crossed out twice or the image of a circle with a dot inside was used. Apparently $50$ was represented by half of this sign.

Disputes between scientists about the origin of other Roman numerals continue. Most likely, the designations $C$ and $M$ are associated with the Roman names for hundreds and thousands. The Romans called a thousand "mille"(word "mile" once denoted a path of a thousand steps).

Note 2

For easy memorization letter designations numbers in descending order use the mnemonic rule:

$M$y $D$arim $C$full $L$imons, $X$vat $V$sem $I$х

Which corresponds to $M, D, C, L, X, V, I$.

Rules for writing numbers

When designating numbers, the Romans wrote down such a number of them that their sum reached the required number. For example, they wrote the number $8$ as $VIII$, and the number $382$ as: $CCCLXII$. When writing this number, you can note that large numbers are written first, and only then small ones.

However, sometimes the Romans did the opposite, i.e. the smaller number was placed in front of the larger one, which meant that it was necessary to subtract rather than add.

Example 1

For example, the number $4$ was designated $IV$ (minus one is five), and the number $9 was designated IX$ (minus one is ten). The entry $XC$ meant $90$ (minus one hundred). A digit with a larger value could be preceded by only one digit of a smaller value ($IV$ is a correct notation of a number, $IIV$ is an incorrect notation).

If two identical numbers stood next to each other, their values ​​were added together. For example: $CC – 200$, $XX – 20$. Moreover, the same number could not be written more than three times in a row.

In any number, the same digits $V$, $L$, $D$ could not be used separately from each other more than once ($DC$ and $DL$ are the correct notation of numbers, $VV$ is an incorrect notation of the number) .

Another rule is that if a digit of a larger value is preceded by a digit of a smaller value, then the latter can only be represented by one of the digits $I$, $X$, $C$ ($IX$ is the correct notation of the number, $VX $ is an invalid entry).

If a digit of a larger value is preceded by a digit of a smaller value, then after the larger digit in this pair there may be a digit that has a value less than that of the smaller digit of the pair ($CDX$ is a correct number entry, $CDC$ is an incorrect entry ).

If a digit was mentioned in a number as a smaller digit before a larger one, then it could not be used again (read from left to right) in that number, except in situations where it acted as a larger digit following a smaller one ($CDXC$ - correct number entry, $CDCC$ is an incorrect entry).

In the case when a digit with a larger value was followed by a digit with a smaller one, its contribution to the value of the number as a whole was negative. Examples that illustrate the general rules for writing numbers in the Roman numeral system are given in the table:

Figure 2.

The largest number that the Romans could designate was $100,000$. Therefore, usually in the names of large sums of money the words “hundreds of thousands” were omitted. The entry meant $10$ thousand hundreds, i.e. million.

We have given several rules for writing numbers that were used in the Roman number system. So, if you are now traveling somewhere in Europe and notice on an ancient building an inscription in Roman numerals $MDCCCXLIV$, you can easily determine that it was built in $1844$.

Rules for performing arithmetic operations with numbers

Addition and subtraction.

Adding two Roman numerals is quite simple. For example:

$XIX + XXVI = XXXV$

Addition is performed in the following sequence:

a) $IX + VI = XV$ ($I$ after $V$ “destroys” $I$ before $X$);

b) $X + XX = XXX$ (when adding another $X$, we get $XXXX$, or $XL$).

The difficulty of subtracting Roman numerals is approximately the same. For example, to subtract the number $263$ from $500$, the minuend must first be decomposed into smaller components, and then reduce the repeating signs in the minuend and subtrahend:

$D - CCLXIII = CCCCLXXXXVIIIII - CCLXIII = CCXXXVII$

Multiplication.

With multiplication the situation was much more complicated.

Let's say you needed to multiply $126$ by $37$ (the Romans did not have action signs; the names of actions were written in words).

$CXXVI \cdot XXXVII$

We had to multiply the multiplicand by each digit of the multiplier separately, and then add all the products.

This technique for performing multiplication is similar to multiplying polynomials.

Division.

Doing division was very difficult in the Roman number system. For this purpose it was used special tool– abacus (ancient abacus). Only highly educated people knew how and could work with him.

Using the Roman numeral system

Although Roman numbering was not entirely convenient, it spread throughout ecumene- this is what the ancient Greeks called the inhabited world they knew. The Romans are conquerors, they enslaved and subjugated many countries, which led to the growth of their empire. They collected huge taxes from enslaved peoples, and to do this they needed to use numbers. Therefore, the inhabitants of these countries had to learn Roman numbering while cursing their enslavers. And even after the collapse of the Roman Empire, this inconvenient numbering continued to be used in the business papers of Western Europe. It is inconvenient because it is difficult to perform arithmetic operations with multi-digit numbers in this system. Still, Roman numbering was used in Italy until the 13th century, and in other Western European countries until the 16th century.

Disadvantage of the Roman system notation is that it lacks formal rules for writing numbers and, accordingly, rules for arithmetic operations with multi-digit numbers. Due to the fact that the system is not entirely convenient and complex, currently we use it only where it is really convenient: for numbering chapters and volumes in literature, for determining centuries and serial numbers of monarchs in history, when registering securities, for marking the watch dial and in a number of other cases.

The Roman number system was widespread in Europe in the Middle Ages, however, due to the fact that it turned out to be inconvenient to use, it is practically not used today. It was superseded by simpler ones that made arithmetic much simpler and easier.

The Roman system is based on ten, as well as their halves. In the past, man had no need to record large and long numbers, so the set of basic numbers initially ended with a thousand. The numbers are written from left to right, and their sum indicates the given number.

The main difference is that the Roman number system is non-positional. This means that the location of a digit in a number notation does not indicate its meaning. The Roman numeral "1" is written as "I". Now let’s put the two units together and look at their meaning: “II” is exactly the Roman numeral 2, while “11” is written in Roman numeral as “XI”. In addition to one, other basic numbers in it are five, ten, fifty, one hundred, five hundred and a thousand, which are designated V, X, L, C, D and M, respectively.

In the decimal system we use today, in the number 1756, the first digit refers to the number of thousands, the second to hundreds, the third to tens, and the fourth represents the number of ones. That is why it is called a positional system, and calculations using it are carried out by adding the corresponding digits to each other. The Roman one is structured completely differently: in it, the value of an integer digit does not depend on its order in the notation of the number. In order, for example, to translate the number 168, you need to take into account that all the numbers in it are obtained from basic symbols: if the number on the left is greater than the number on the right, then these numbers are subtracted, otherwise they are added. Thus, 168 will be written there as CLXVIII (C-100, LX - 60, VIII - 8). As you can see, the Roman number system offers a rather cumbersome notation of numbers, which makes adding and subtracting large numbers extremely inconvenient, not to mention performing division and multiplication operations on them. The Roman system also has another significant drawback, namely the absence of a zero. Therefore, in our time, it is used exclusively to designate chapters in books, numbering centuries, and special dates, where there is no need to carry out arithmetic operations.

In everyday life, it is much easier to use the decimal system, the meaning of the numbers in which corresponds to the number of angles in each of them. It first appeared in the 6th century in India, and the symbols in it were finally established only by the 16th century. Indian numerals, called Arabic numerals, came to Europe thanks to the work of the famous mathematician Fibonacci. To separate the integer and fractional parts in the Arabic system, a comma or period is used. But in computers it is most often used, which spread in Europe thanks to the work of Leibniz, which is due to the fact that computer technology uses triggers that can only be in two working positions.

Numbers

To fix in memory the letter designations of numbers in descending order, there is a mnemonic rule:

M s D arim WITH face-to-face L imons, X vatit V seven I X.

Respectively M, D, C, L, X, V, I

Examples

To correctly write large numbers in Roman numerals, you must first write the number of thousands, then hundreds, then tens, and finally units.

Example: number 1988. One thousand M, nine hundred CM, eighty LXXX, eight VIII. Let's write them down together: MCMLXXXVIII.

Quite often, to highlight numbers in the text, a line was drawn over them: LXIV. Sometimes a line was drawn both above and below: XXXII- in particular, it is customary to highlight Roman numerals in Russian handwritten text (this is not used in typesetting due to technical complexity). For other authors, the overbar could indicate an increase in the value of the figure by 1000 times: V M = 6000.

There is a "shortcut" for writing large numbers such as 1999. It is not recommended, but is sometimes used to simplify things. The difference is that to reduce a digit, any digit can be written to the left of it:

• 999. Thousand M, subtract 1 (I), we get 999 (IM) instead of CMXCIX. Consequence: 1999 - MIM instead of MCMXCIX
• 95. One hundred C, subtract 5 (V), get 95 (VC) instead of XCV
• 1950: thousand M, subtract 50 (L), get 950 (LM). Consequence: 1950 - MLM instead of MCML

It was only in the 19th century that the number “four” was written down as “IV”; before that, the number “IIII” was most often used. However, the entry “IV” can be found already in the documents of the “Forme of Cury” manuscript, dating back to the year. Watch dials have traditionally used "IIII" instead of "IV" in most cases, mainly for aesthetic reasons: this spelling provides visual symmetry with the "VIII" numerals on the opposite side, and an inverted "IV" is more difficult to read than "IIII".

Application

In Russian, Roman numerals are used in the following cases.

• Century or millennium number: XIX century, II millennium BC. e.
• Serial number of the monarch: Charles V, Catherine II.
• The volume number in a multi-volume book (sometimes the numbers of parts of the book, sections or chapters).
• In some publications - the numbers of sheets with the preface to the book, so as not to correct the links within the main text when the preface is changed.
• Antique-style markings on watch dials.
• Other important events or list items, for example: V postulate of Euclid, World War II, XXII Congress of the CPSU, etc.

In other languages, the scope of application of Roman numerals may have specific features; for example, in Western countries, the year number is sometimes written in Roman numerals.

Extension

Roman numerals provide the ability to write numbers from 1 to 3999 (MMMCMXCIX). To solve this problem, [ Who?] extended Roman numerals .

Unicode

The Unicode standard defines characters to represent Roman numerals as part of Number forms(English) Number Forms), in the area of ​​characters with codes U+2160 to U+2188. For example, MCMLXXXVIII can be represented in the form ⅯⅭⅯⅬⅩⅩⅩⅧ . This range includes both lowercase and uppercase numerals from 1 (Ⅰ or I) to 12 (Ⅻ or XII), including combination glyphs for composite numbers such as 8 (Ⅷ or VIII), primarily for compatibility with East Asian character sets in industry standards such as JIS X 0213 where these characters are defined. Combination glyphs are used to represent numbers that were previously composed of individual characters (for example, Ⅻ instead of its representation as Ⅹ and Ⅱ). In addition to this, glyphs exist for the archaic forms of writing the numbers 1000, 5000, 10,000, major reverse C (Ɔ), the late form of writing 6 (ↅ, similar to the Greek stigma: Ϛ), the early form of writing the number 50 (ↆ, similar to the downward-pointing arrow ↓⫝⊥ ), 50,000, and 100,000. It should be noted that the backsmall small c, ↄ is not included in Roman numeral characters, but is included in the Unicode standard as the Claudian capital Ↄ.

Characters in the range U+2160-217F are present only for compatibility with other standards that define these characters. In everyday life, ordinary letters of the Latin alphabet are used. Displaying such symbols requires software, which supports the Unicode standard, and