More than two thousand years ago, Roman numeration appeared, that is, in ancient Rome, numbers were written using the letters of the Latin alphabet.
I - 1; V - 5; X - 10; L-50; C - 100; D - 500; M - 1000 - these letters are called Roman numerals, and writing a number in Roman numerals is called writing a number in Roman numeration.
Addition and subtraction are used to write numbers in Roman numerals.
We agreed that in cases where addition is implied in the notation of a number, put the smaller number after the larger one, and when subtraction is implied in the notation of the number, put the smaller number (subtracted) before the larger one (decreased).
An example of writing Roman numbers
VI = 5 + 1 IV = 5 − 1But writing large numbers in this way is quite difficult, so now Roman numeration is used to write relatively small numbers - chapter numbers in books, centuries, etc.
Note that in the entry of the number 555 the number 5 is used three times, however, the number is read - “five hundred and fifty five”.
Just as writing numbers in Roman numerals means addition and subtraction, writing numbers in Arabic numerals means addition and multiplication:
555 = 500 + 50 + 5 = 5 ⋅ 100 + 5 ⋅ 10 + 5
Writing a number in this form is called the sum of bit terms.
This means that the significance of a digit depends on its place in the notation of the number, that is, on its position.
In such cases, the number is said to be written positional way.
What appeared before Roman or Arabic numeration?
In our usual system of writing numbers, 10 digits are used.
The account in it goes in tens, hundreds (10 tens), thousands (10 hundreds), etc.
Therefore, our counting system is called decimal, or decimal number system.
The numbers we use are called Arabic numerals. It was invented in 400 AD in India. In 800 AD Arabic numbering was borrowed by the Arabs, and in 1200 Arabic numbering began to be used in Europe. In Russia, Arabic numeration began to be used under Peter I.
Roman numbering originated in ancient Rome between 900 and 800 BC. Thus, Roman numbering arose earlier than Arabic.
Tasks for Roman numeration
Example #1. Determine the number written in Roman numerals: MMDCCCXXII.
Solution:
Recall that I - 1; V - 5; X - 10; L-50; C - 100; D - 500; M - 1000.
It is known that when writing numbers in Roman numerals, addition and subtraction are used. We agreed that in cases where addition is implied in the notation of a number, put the smaller number after the larger one, and when subtraction is implied in the notation of the number, put the smaller number (subtracted) before the larger one (decreased).
So MMDCCCXXII = 1000 + 1000 + 500 + 100 + 100 + 100 + 10 + 10 + 1 + 1 = 2822.
Answer: MMDCCCXXII = 2822.
Example #2. Determine the number written in Roman numerals: XXIX.
Solution:
XXIX = 10 + 10 + 9 = 29.
Answer: XXIX = 29.
Example #3. Enter the smallest five-digit number.
Solution:
It is known: to write down the smallest five-digit number, you need to use only the number 1 in the entry - once - and the number 0 - four times.
We get the number 10000.
Answer: The smallest five-digit number is 10,000.
Example #4. Enter the smallest eleven digit number.
Answer: 10,000,000,000
Example #5. Write down the number in words: 79 402 720 (write the number in lowercase letters, without any punctuation marks).
Answer: seventy-nine million four hundred two thousand seven hundred and twenty.
Example #6. Compare the numbers if the individual digits in them are replaced by asterisks: 27∗∗∗ and 28∗∗∗.
Solution:
Analyzing these numbers, in which individual digits are replaced by asterisks:
27∗∗∗ and 28∗∗∗ - we notice that both numbers are five-digit, in the highest digit of tens of thousands - the same digits, and in the digit of thousands of the first number the digit is less than the second, which means that the first number is less than the second, i.e. e. 27∗∗∗< 28∗∗∗.
Answer: 27∗∗∗< 28∗∗∗
Example #7. Write down the number that is 90 less than the largest four-digit number.
Solution
The largest four-digit number is 9999, and the number that is 90 less than the largest four-digit number is 9999 - 90 = 9909.
Answer: 9909.
Example #8. AT farming 3 hectares are occupied by the estate and buildings, under crops - 380 hectares, under haymaking - 310 hectares, under forest - 40 hectares and under pasture - 110 hectares. How much land does the farmer have in total?
Solution
To determine the entire area of land in use by the farmer, you need to add up the areas occupied by the estate and buildings, crops, haymaking, forest and pasture. We get:
3 + 380 + 310 + 40 + 110 = 843 ha
Answer: 843 ha.
Example #9. Write the number 2458 as the sum of the bit terms in two ways.
Example: 348 = 300 + 40 + 8 = 3 ⋅ 100 + 4 ⋅ 10 + 8.
Solution
Analyzing the sample given in the task for writing a number as a sum of digit terms, we apply it to a given four-digit number 2458.
Note that his senior digit is units of thousands, so the entry will be as follows: 2458 = 2000 + 400 + 50 + 8 = 2 ⋅ 1000 + 4 ⋅ 100 + 5 ⋅ 10 + 8.
Answer: 2458 = 2000 + 400 + 50 + 8 = 2 ⋅ 1000 + 4 ⋅ 100 + 5 ⋅ 10 + 8.
Example #10. Write a number instead of ∗ so that you get the correct equality: 750000:∗=75000.
Solution:
In order for the equality 750000: ∗ = 75000 to be true, instead of ∗ we write the number 10, because as a result we get a number consisting of the same digits as the dividend, only shifted one digit to the right, i.e. the number has decreased 10 times.
Answer: the number is 10.
Example #11. Identify all three-digit numbers that only use the digits 1 and/or 5.
Solution:
To determine all three-digit numbers, in the record of which only the numbers 1 and 5 are used, let's start reasoning like this:
in the first place (in the hundreds place) this number can have the number 1 or the number 5, i.e. we have
1∗∗ or 5∗∗
In the second place (in the place of tens) in each of these two cases, there can also be one of the digits - 1 or 5.
In the third place (in the category of units) in each of the four cases already received, there can also be one of the numbers - 1 or 5.
Continuing similar reasoning and sorting through all possible options we get
Thus, eight numbers can be formed:
111;115;151;155;511;515;551;555.
Answer: 111;115;151;155;511;515;551;555
Example #12. What is the position in which the number 7 is in the number 7 890 214. Continue the sentence: "The number is in the category __________."
dozens
hundreds
units million
units thousand
Solution:
It is known that the significance of a digit depends on its place in the notation of the number, that is, on its position.
Recall the table of ranks and the name of the classes.
Table of ranks and classes
We all use Roman numerals - we mark the numbers of centuries or months of the year with them. Roman numerals are on watch dials, including those on the chimes of the Spasskaya Tower. We use them, but we don't know much about them.
How are Roman numerals arranged?
The Roman counting system in its modern version consists of the following basic signs:
I 1
V 5
X 10
L 50
C 100
D500
M 1000
To remember numbers that are unusual for us using the Arabic system, there are several special mnemonic phrases in Russian and English:
We Give Juicy Lemons, Enough for Everyone Ix
We Advise Only Well-Brought-Up Individuals
I Value Xylophones Like Cows Dig Milk
The system of arrangement of these numbers relative to each other is as follows: numbers up to three inclusive are formed by adding units (II, III), - the fourfold repetition of any number is prohibited. To form numbers greater than three, the larger and smaller digits are added or subtracted, to subtract, the smaller digit is placed before the larger one, to add - after, (4 \u003d IV), the same logic works with other numbers (90 \u003d XC). The arrangement of thousands, hundreds, tens and units is the same as we are used to.
It is important that any digit should not repeat more than three times, so the longest number up to a thousand is 888 = DCCCLXXXVIII (500+100+100+100+50+10+10+10+5+1+1+1).
Alternatives
The ban on the fourth use of the same number in a row began to appear only in the 19th century. Therefore, in ancient texts one can see variants IIII and VIIII instead of IV and IX, and even IIIII or XXXXXX instead of V and LX. The remains of this writing can be seen on the clock, where four is often marked with exactly four units. In old books, there are also frequent cases of double subtractions - XIIX or IIXX instead of the standard XVIII in our days.
Also in the Middle Ages, a new Roman numeral appeared - zero, which was denoted by the letter N (from the Latin nulla, zero). Large numbers were marked with special characters: 1000 - ↀ (or C|Ɔ), 5000 - ↁ (or |Ɔ), 10000 - ↂ (or CC|ƆƆ). Millions are obtained by double underlining the standard digits. Fractions were also written in Roman numerals: ounces were marked with the help of icons - 1/12, half was marked with the symbol S, and everything that was more than 6/12 was added: S = 10\12. Another option is S::.
Origin
On the this moment there is no unified theory of the origin of Roman numerals. One of the most popular hypotheses is that the Etruscan-Roman numerals originated from a counting system that uses notches instead of numbers.
Thus, the number "I" is not the Latin or more ancient letter "i", but a notch that resembles the shape of this letter. Every fifth notch was marked with a bevel - V, and the tenth was crossed out - X. The number 10 in this account looked like this: IIIIΛIIIIX.
It is thanks to such a record of numbers in a row that we owe a special system for adding Roman numerals: over time, the record of the number 8 (IIIIΛIII) could be reduced to ΛIII, which convincingly demonstrates how the Roman counting system got its specifics. Gradually, the notches turned into graphic symbols I, V and X, and gained independence. Later they began to be identified with Roman letters - as they were outwardly similar to them.
An alternative theory belongs to Alfred Cooper, who suggested considering the Roman counting system from the point of view of physiology. Cooper believes that I, II, III, IIII is a graphical representation of the number of fingers of the right hand thrown out by the trader when naming the price. V - this is a set aside thumb, forming together with the palm a figure similar to the letter V.
That is why Roman numerals sum up not only units, but also add them to fives - VI, VII, etc. - this is the thumb and other exposed fingers of the hand. The number 10 was expressed using the crossing of hands or fingers, hence the symbol X. Another option is that the number V was simply doubled, getting X. Large numbers were transmitted using the left palm, which counted tens. So gradually the signs of the ancient finger count became pictograms, which then began to be identified with the letters of the Latin alphabet.
Modern application
Today in Russia, Roman numerals are needed, first of all, to record the number of the century or millennium. It is convenient to put Roman numerals next to Arabic ones - if you write a century in Roman numerals, and then a year in Arabic, then your eyes will not ripple from the abundance of identical signs. Roman numerals are somewhat archaic. With their help, they also traditionally indicate the serial number of the monarch (Peter I), the number of the volume of a multi-volume edition, and sometimes the chapter of the book. Roman numerals are also used in antique watch dials. Important numbers, such as the year of the Olympiad or the number of a scientific law, can also be recorded using Roman numerals: World War II, Euclid's fifth postulate.
AT different countries Roman numerals are used a little differently: in the USSR it was customary to use them to indicate the month of the year (1.XI.65). In the West, Roman numerals often write the number of the year in movie credits or on building facades.
In a part of Europe, especially in Lithuania, one can often find Roman numerals designating the days of the week (I - Monday, and so on). In the Netherlands, Roman numerals sometimes represent floors. And in Italy, they mark 100-meter sections of the path, marking, at the same time, with Arabic numerals each kilometer.
In Russia, when writing by hand, it is customary to underline Roman numerals from below and from above at the same time. However, often in other countries, an underscore from above meant an increase in the case of a number by a factor of 1000 (or 10,000 times with a double underscore).
There is a common misconception that modern Western clothing sizes have something to do with Roman numerals. In fact, the designations XXL, S, M, L, etc. have no connection with them: these are abbreviations of the English words eXtra (very), Small (small), Large (large).
Roman numerals- numerals used by the ancient Romans in their non-positional number system.
Natural numbers are written by repeating these digits. At the same time, if a large number is in front of a smaller one, then they are added (the principle of addition), if the smaller one is in front of the larger one, then the smaller one is subtracted from the larger one (the principle of subtraction). The last rule applies only to avoid the fourfold repetition of the same figure.
Roman numerals appeared around 500 BC with the Etruscans.
NumbersTo fix the alphabetic designations of numbers in descending order, there is a mnemonic rule:
M s D arim FROM face-to-face L imony, X vatite V sem I X.
Respectively M, D, C, L, X, V, I
To correctly write large numbers in Roman numerals, you must first write down the number of thousands, then hundreds, then tens, and finally units.
There is a "shortcut" for writing large numbers, such as 1999. It is not recommended, but is sometimes used for simplicity. The difference is that to reduce a digit, any digit can be written to the left of it:
- 999. Thousand (M), subtract 1 (I), 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), we get 950 (LM). Consequence: 1950 - MLM instead of MCML
It was only in the 19th century that the number “four” was written universally as “IV”, before that the record “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 1390. Watch dials have traditionally used "IIII" instead of "IV" in most cases, mainly for aesthetic reasons: this spelling provides visual symmetry with the numbers "VIII" on the opposite side, and the reversed "IV" is more difficult to read than "IIII".
Application of Roman Numerals
In Russian, Roman numerals are used in the following cases:
- Century or millennium number: XIX century, II millennium BC. e.
- The serial number of the monarch: Charles V, Catherine II.
- Volume number in a multi-volume book (sometimes numbers of book parts, sections or chapters).
- In some editions - page numbers with the preface to the book, so as not to correct references inside the main text when changing the preface.
- Antique watch dial markings.
- Other important events or list items, such as: V postulate of Euclid, II World War, XXII Congress of the CPSU, etc.
In other languages, the scope of Roman numerals may have some peculiarities, for example, in Western countries, Roman numerals sometimes record the year number.
Roman Numerals and Unicode
The Unicode standard defines characters to represent Roman numerals as part of Numeric forms(English) Number Forms), in the area of characters with codes from U+2160 to U+2188. For example, MCMLXXXVIII can be represented in the form ⅯⅭⅯⅬⅩⅩⅩⅧ . This range includes both lowercase and uppercase digits from 1 (Ⅰ or I) to 12 (Ⅻ or XII), including combined glyphs for compound numbers such as 8 (Ⅷ or VIII), mainly for compatibility with East Asian character sets in industry standards such as JIS X 0213 where these characters are defined. Combined glyphs are used to represent numbers that were previously made up of single characters (e.g. Ⅻ instead of its representation as Ⅹ and Ⅱ). In addition, glyphs exist for archaic 1000, 5000, 10000, big reversed C (Ɔ), late 6 (ↅ, similar to the Greek stigma: Ϛ), early 50 (ↆ, similar to to the down arrow ↓⫝⊥ ), 50,000, and 100,000. It should be noted that the small back c, ↄ is not included in Roman numeral characters, but is included in the Unicode standard as the uppercase Claudian letter Ↄ .
| The code | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Meaning | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 50 | 100 | 500 | 1 000 |
| U+2160 | Ⅰ
2160 | Ⅱ
2161 | Ⅲ
2162 | Ⅳ
2163 | Ⅴ
2164 | Ⅵ
2165 | Ⅶ
2166 | Ⅷ
2167 | Ⅸ
2168 | Ⅹ
2169 | Ⅺ
216A | Ⅻ
216B | Ⅼ
216C | Ⅽ
216D | Ⅾ
216E | Ⅿ
216F |
| U+2170 | ⅰ
2170 | ⅱ
2171 | ⅲ
2172 | ⅳ
2173 | ⅴ
2174 | ⅵ
2175 | ⅶ
2176 | ⅷ
2177 | ⅸ
2178 | ⅹ
2179 | ⅺ
217A | ⅻ
217B | ⅼ
217C | ⅽ
217D | ⅾ
217E | ⅿ
217F |
| Meaning | 1 000 | 5 000 | 10 000 | - | - | 6 | 50 | 50 000 | 100 000 | |||||||
| U+2160! U+2180 | ↀ
2180 | ↁ
2181 | ↂ
2182 | Ↄ | ↄ | ↄ | ↄ | ↄ | ↄ | |||||||
Characters in the range U+2160-217F are present only for compatibility with other standards that define those characters. In everyday life, ordinary letters of the Latin alphabet are used. The display of such characters requires software, which supports the Unicode standard, and a font containing glyphs corresponding to these characters.
They are written by repeating these numbers. At the same time, if a large number is in front of a smaller one, then they are added (the principle of addition), if the smaller one is in front of a larger one, then the smaller one is subtracted from the larger one (the principle of subtraction). The last rule applies only to avoid the fourfold repetition of the same figure.
Roman numerals appeared 500 BC from the Etruscans (see the Etruscan alphabet), who could borrow some of the numbers from the proto-Celts.
Roman notation for numbers is now better known than any other ancient number system. This is explained not so much by some special merits of the Roman system, but by the enormous influence that the Roman Empire enjoyed in the relatively recent past. The Etruscans conquered Rome in the 7th century. BC e., were influenced by Eastern Mediterranean cultures. This partly explains the similarity of the basic principles of the Roman and Attic number systems. Both systems were decimal, although the number five played a special role in both number systems. Both systems used repeated characters when writing numbers.
The old Roman symbols for the numbers 1, 5, 10, 100, and 1000 were, respectively, the symbols I, V, X, Θ(or ⊕ , or ⊗ ) and Φ (or ↀ , or CIƆ). Although much has been written about the original meaning of these symbols, there is still no satisfactory explanation for them. According to one common theory, the Roman numeral V depicts an open hand with four fingers pressed together and the thumb extended; the symbol X, according to the same theory, depicts two crossed hands or a double digit V. The symbols for the numbers 100 and 1000 probably originate from the Greek letters Θ and φ. It is not known if later designations originated C and M from old Roman symbols, or they are acrophonically related to the initial letters of the Latin words meaning 100 (centum) and 1000 (mille). It is believed that the Roman symbol for the number 500, the letter D, arose from half of the old symbol for 1000. Apart from the fact that most Roman symbols were most likely not acrophonic and that the intermediate symbols for the numbers 50 and 500 were not combinations of symbols for the numbers 5 and 10 or 5 and 100, then the rest of the Roman the number system resembled the attic. The Romans often used the principle of subtraction, so sometimes they used IX instead of VIIII, and XC instead of LXXXX; comparatively later, the symbol IV instead of IIII.
In general, the Romans were not inclined to do mathematics, so they did not feel much need for large numbers. However, they occasionally used the symbol to represent 10,000 CCIƆƆ, and for the number 100000 - the symbol CCCIƆƆƆ. The halves of these symbols were sometimes used to represent the numbers 5000 ( IƆƆ) and 50000 ( IƆƆƆ).
The Romans avoided fractions as stubbornly as large numbers. In practical measurement problems, they did not use fractions, subdividing the unit of measure usually into 12 parts, so that the result of the measurement is presented as a composite number, the sum of multiples of various units, as is done today when length is expressed in yards, feet and inches. English words "ounce" ( ounce) and "inch" ( inch) come from the Latin word lat. uncia ( ounce), denoting one twelfth of the basic unit of length.
To correctly write large numbers in Roman numerals, you must first write down the number of thousands, then hundreds, then tens, and finally units.
There is no zero in the Roman numeral system, but zero was previously used as nulla (no), nihil (nothing) and N (the first letter of these words).
In this case, some of the numbers (I, X, C, M) may be repeated, but no more than three times in a row; thus, they can be used to write any integer no more than 3999(MMMCMXCIX). AT early periods there were signs to indicate larger numbers - 5000, 10,000, 50,000 and 100,000 [ ] (then the maximum number according to the mentioned rule is 399,999). When writing numbers in the Roman numeral system, the smaller digit may be to the right of the larger one; in this case it is added to it. For example, the number 283 in Roman is written as CCLXXXIII, that is, 100+100+50+30+3=283. Here, the number representing a hundred is repeated twice, and the numbers representing ten and one, respectively, are repeated three times.
Example: number 1988. One thousand M, nine hundred CM, eight tens LXXX, eight units VIII. Let's write them together: MCMLXXXVIII.
Quite often, to highlight numbers in the text, a line was drawn over them: LXIV. Sometimes the 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 typographic typesetting due to technical complexity). For other authors, the overline could indicate an increase in the value of the figure by 1000 times: V = 5000.
It was only in the 19th century that the number “four” was written down as “IV” everywhere, before that the record “IIII” was most often used. However, the entry "IV" can already be found in the documents of the manuscript "Forme of Cury", dating back to 1390. Watch dials have traditionally used "IIII" instead of "IV" in most cases, mainly for aesthetic reasons: this spelling provides visual symmetry with the numbers "VIII" on the opposite side, and the reversed "IV" is more difficult to read than "IIII". There is also a version that IV was not written on the dial because IV is the first letters of the name of the god Jupiter (IVPITER).
The smaller number can be written to the left of the larger one, then it should be subtracted from the larger one. In this case, only numbers denoting 1 or powers of 10 can be subtracted, and only the nearest two numbers in the number series to the subtracted (that is, the subtracted, multiplied by 5 or 10) can act as a minuend. Repetitions of a smaller number are not allowed. Thus, there is only six options using the "rule of subtraction":
For example, the number 94 will be XCIV \u003d 100 - 10 + 5 - 1 \u003d 94 - the so-called "subtraction rule" (appeared in the era of late antiquity, and before that the Romans wrote the number 4 as IIII, and the number 40 as XXXX).
It should be noted that other methods of "subtraction" are not allowed; thus, the number 99 should be written as XCIX, but not as IC. However, nowadays, in some cases, a simplified notation of Roman numbers is also used: for example, in Microsoft Excel, when converting Arabic numerals to Roman using the “ROMAN ()” function, you can use several types of representation of numbers, from classical to highly simplified (for example, the number 499 can be written as CDXCIX, LDVLIV, XDIX, VDIV, or ID). The simplification is that to reduce any digit, any other digit can be written to the left of it:
Cases of such notation of numbers (usually years) are often found in the credits of US television series. For example, for the year 1998: IIMM instead of MCMXCVIII.
Roman numerals can also be used to write large numbers. To do this, a line is placed above those numbers that represent thousands, and a double line is placed above the numbers that represent millions. For example, the number 123123 would look like this:
A similar format was used in medical certificates in the 1970s and 1980s.With the transition to computer processing of information, date formats based on Roman numerals have practically fallen into disuse.
In other languages, the scope of Roman numerals may differ. In Western countries, the number of the year is often written in Roman numerals, for example, on the gables of buildings and in the credits of film and video products.
Displaying all of these characters requires software that supports the Unicode standard and a font that contains the corresponding glyphs for these characters (for example, the Universalia font).
To convert numbers written in Arabic numerals to Roman, special functions are used.
For example, in the English version of Microsoft Excel and in any version of OpenOffice.org Calc, there is a function for this ROMAN(argument; form), in the Russian version of Microsoft Excel this function is called ROMAN(number; shape). The optional argument "shape" can take values from 0 to 4, as well as "False" and "True". The absence of the argument "Form" or its equality to 0 or "True" gives the "classical" (strict) form of the transformation; a value of 4 or "False" gives the most simplified; values 1, 2, 3 give variants intermediate in rigor-simplification. Differences appear, for example, on the numbers 45, 49, 495, 499 (the first ones in the range are indicated).
Non-integer values of the "number" argument are rounded down to an integer; if after that the value is greater than 3999 or less than 0, then the function returns "#Value"; for a value of 0, an empty cell is returned.
string-join(for $num in (1999) return (("","M","MM","MMM")[($num idiv 1000) mod 10+1], ("","C", "CC","CCC","CD","D","DC","DCC","DCCC","CM")[($num idiv 100) mod 10+1], (""," X","XX","XXX","XL","L","LX","LXX","LXXX","XC")[($num idiv 10) mod 10+1], (" ","I","II","III","IV","V","VI","VII","VIII","IX")[$num mod 10+1]), "" ) /// The class is intended for converting Arabic numbers into Roman numbers and vice versa///
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