The Roman number system - beautiful, but difficult? Presentation on the topic "Roman number system".

16.10.2019

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 It was superseded by simpler ones that made arithmetic much simpler and easier.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, people had no need to write 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 bits 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, solemn 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.

>> Roman numeral system

§ 4.3. Roman number system

An example of a non-positional number system that has survived to this day is number systems, 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:

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. Built on this basis tables, which is convenient to use to denote 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 system). Then, for numbers in the thousands, hundreds, tens and units places, select the appropriate ones from the table.

In order to write down numbers greater than 3999, special rules are used, but familiarization with 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 were easy to counterfeit).

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

Bosova L. L. Computer Science: Textbook for 6th grade / L. L. Bosova. - 3rd ed., rev. and additional - M.: BINOM. Laboratory of Knowledge, 2005. - 208 pp.: ill.

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Note 1

This system refers to a non-positional number system that uses letters of the Latin alphabet to write numbers.

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:

Figure 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 today. 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.

Scientists continue to argue about the origin of other Roman numerals. 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

To easily remember the letter designations of 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: $CCCLXXXII$. 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 didn’t 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, a special instrument was used - the 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.

Note 1

This system refers to a non-positional number system that uses letters of the Latin alphabet to write numbers.

Number designation

Figure 1.

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.

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.

Note 2

To easily remember the letter designations of 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

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

Figure 2.

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 $XL$).

$D - CCLXIII = CCCCLXXXXVIIIII - CCLXIII = CCXXXVII$

Multiplication.

With multiplication the situation was much more complicated.

Let’s say you needed to multiply $126$ by $37$ (the Romans didn’t 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.

Using the Roman numeral system

Page 1

The Roman number system is an example of a system with a very complex way of writing numbers and cumbersome rules for performing arithmetic operations.

The Roman number system is inconvenient to use and is currently almost never used.

The Roman number system is not positional, since the value of a number does not depend on the position of the digit in a series of numbers.

The Roman number system, common in the Middle Ages in Europe, turned out to be inconvenient for arithmetic operations and fell into oblivion. We began to carry out the necessary calculations quickly and easily, completely forgetting about the art of counting in the Roman number system. So should we regret that the routine art of integration is also becoming a thing of the past? Isn’t it better to direct your knowledge, skills, ingenuity and imagination to tasks that are still waiting to be solved?

In the Roman number system, the meaning of a digit does not depend on its position in the number record.

An example of a non-positional system is the Roman number system, which has survived to this day.

So, for example, in the Roman number system, the number XXX contains the same symbol X in all digits, which means 10 units regardless of its position in the image of the number.

A more complex non-positional number system is the Roman number system. This system uses the principles of not only addition, but also subtraction. If a figure with a smaller quantitative equivalent is located to the right of a figure with a larger quantitative equivalent, then their quantitative equivalents are added; if on the left, then they are subtracted.

One of the varieties of non-positional systems has survived to this day - the Roman number system.

In positional number systems, the meaning of each digit depends and changes on its position in the number notation. The Roman number system is non-positional, in which the meaning of a digit does not depend on its location in the number.

In the Roman number system, each numerical sign in the recording of any number has the same meaning, i.e. the meaning of a number sign does not depend on its location in the number notation. Thus, the Roman number system is not a positional number system.

Number systems are divided into positional and non-positional. For example, the decimal number system is positional, and the Roman number system is non-positional.

A non-positional number system is a system in which the quantitative equivalent of a digit does not depend on its location in the number record. An example of a non-positional number system based on the principle of addition and subtraction is the well-known Roman number system, which has almost no practical application and is not considered further.