Numeration system

The set of methods of representation of numbers is known. Anyway the number is represented by the character or a character group (word) of some alphabet. Such characters are called digits.

Numeration systems

For representation of numbers also radix notations are used not position.

Not radix notations

As soon as people began to consider, they had a need for record of numbers. Finds archeologists on parking of primitive people demonstrate to what originally quantity of objects displayed equal quantity of any icons (labels): notches, hyphens, points. Later, for simplification of the account, these icons began to group on three or on five. Such recording system of numbers is called single (unary) as any number in it is formed by repetition of one sign symbolizing unit. Echoes of a single numeration system meet and today. So that to learn on what rate the cadet of military college studies, it is necessary to count what quantity of strips is sewed on its sleeve. Without realizing that, kids use a single numeration system, showing the age on fingers, and calculating sticks is used for training of pupils of the 1st class in the account. Let's consider different numeration systems.

A single system – not the most convenient writing method of numbers. Write thus large numbers tiresomely and records at the same time turn out very long. Eventually there were others, more convenient, numeration systems.

Ancient Egyptian decimal not radix notation. Approximately in the third millennium B.C. ancient Egyptians thought up the numerical system in which for designation of key numbers 1, 10, 100 special icons – hieroglyphs were, etc. used. All other numbers were formed from these key by means of addition operation. The numeration system of Ancient Egypt is decimal, but not position. In not radix notations the quantitative equivalent of each digit does not depend on its provision (the place, a position) in number designation. For example, to represent 3252 drew three flowers of a lotus (three thousand), two folded palm sheets (two hundred), five arcs (five tens) and two poles (two units). The value of number did not depend on in what order the signs making it were located: they could be written from top to down, from right to left or alternately.

Roman numeration system. The numeration system which was applied in Ancient Rome more than two and a half thousand years ago can serve as an example of not position system which remained up to now. Signs I (one finger) for number 1, V (the opened palm) for number 5, X (two put palms) for 10 were the cornerstone of the Roman numeration system, and began to apply the first letters of the matching Latin words to designation of numbers 100, 500 and 1000 (Centum – hundred, Demimille – a half thousands, Mille – one thousand). To write number, Romans decomposed it to the amount of thousands, polutysyach, hundreds, fifty, tens, pyatok, units. For example, the decimal number 28 is represented as follows:

XXVIII=10+10+5+1+1+1 (two tens, heels, three units).

For record of intermediate numbers Romans used not only addition, but also subtraction. At the same time the following rule was applied: each smaller sign delivered to the right of bigger increases to its value, and each smaller sign delivered to the left of bigger is subtracted from it. For example, – designates IX 9, XI – are designated by 11.

The decimal number 99 has the following idea:

XCIХ = –10+100–1+10.

Used the Roman digits very long. 200 years ago in official papers of number had to be designated by the Roman digits (was considered that it is easy to forge normal Arab digits). The Roman numeration system is used today, generally for the name of significant dates, volumes, sections and heads in books.

Alphabetic numeration systems. The alphabetic systems were more perfect not radix notations. Were among such numeration systems Greek, Slavic, Phoenician and others. In them number from 1 to 9, the whole number of tens (from 10 to 90) and the whole number of hundreds (from 100 to 900) were designated by alphabet letters. In an alphabetic numeration system of Ancient Greece of number 1, 2..., 9 were designated by the first nine letters of the Greek alphabet, etc. For designation of numbers 10, 20..., 90 the following 9 letters and were applied to designation of numbers 100, 200..., 900 – the last 9 letters.

At the Slavic people numerical values of letters were established as the Slavic alphabet which used at first the Glagolitic alphabet, and then Cyrillics.

In Russia Slavic numbering remained until the end of the 17th century. At Peter I so-called Arab numbering which we use and now prevailed. Slavic numbering remained only in prayer books.

Not radix notations have a number of essential shortcomings:

• There is a permanent requirement of introduction of new signs for record of large numbers.

• It is impossible to represent fractional and negative numbers.

• It is difficult to execute arithmetic transactions as there are no algorithms of their accomplishment.

Radix notations

In radix notations – the quantitative equivalent of each digit depends on its provision (position) in the code (record) of number. Nowadays we got used to use a decimal position system — numbers register using 10 digits. The most right digit designates units, more to the left — tens, is even more left — hundreds, etc.

For example: 1) six-denary (Ancient Babylon) – the first radix notation. Still at measurement of time the basis equal 60 is used (1 min. = 60s, 1 h = 60 min.); 2) a duodecimal system of notation (was widely adopted in the 19th century number 12 - "dozen": in day two dozens hours). The account not on fingers, and on joints of fingers. On each finger of a hand, except big, up to 3 joints – only 12; 3) now the most widespread radix notations are decimal, binary, octal and hexadecimal (it is widely used in low-level programming and in general in computer documentation as in modern computers the minimum unit of memory is the 8-bit byte which values it is convenient to write in two hexadecimal digits).

In any position system the number can be presented in the polynom form.

Let's show how present a decimal number in the polynom form:

Types of numeration systems

The most important that it is necessary to know about a numeration system – its type: additive or multiplicative. In the first type each digit has the value, and for reading of number it is necessary to put all values of the used digits:

XXXV = 10+10+10+5 = 35; CCXIX = 100+100+10–1+10 = 219;

In the second type each digit can have different values depending on the location in number:

(hieroglyphs one after another: 2, 1000, 4, 100, 2, 10, 5)

Here the hieroglyph "2" is twice used, and in each case it accepted different values "2000" and "20".

2´ 1000 + 4´ 100+2´ 10+5 = 2425

For an additive ("dobavitelny") system it is necessary to know all digits characters with their values (them there are about 4-5 tens), and a record order. For example, in Latin record if the smaller digit is written before bigger, then subtraction is made and if later, then addition (IV = (5–1) = 4; VI = (5+1) = 6).

For a multiplicative system it is necessary to know the image of digits and their value, and also a base radix. It is very easy to define the basis, it is only necessary to count the number of significant figures in a system. If it is simpler, then this number with which the second discharge at number begins. We, for example, use digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Them is exactly 10 therefore the basis of our numeration system too 10, and the numeration system is called "decimal". In the above-stated example digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used (auxiliary 10, 100, 1000, 10000, etc. it is not counted). The main digits here too 10, and a numeration system – decimal.

As it is possible to guess how many is numbers, as much maybe the bases of numeration systems. But only the most convenient bases of numeration systems are used. How do you think why the basis of the most common human numeration system 10? Yes, just because on hands at us 10 fingers. "But on one that to a hand only five fingers" – will tell some and will be right. History of mankind knows examples of quinary numeration systems. "And with legs – twenty fingers" – will tell others, and too will be absolutely right. Quite so Indians Maya considered. It is even visible on their digits.

The concept "dozen" is very interesting. All know that it is 12, but from where there was such number – very few people know. Look at the hands rather on one hand. How many phalanxes on all fingers of one hand, apart from big? Correctly, twelve. And the thumb is intended to note the counted phalanxes.

And if on other hand to lay off fingers the number of complete dozens, then we will receive all the known six-denary Babylon system.

In different civilizations calculated differently, but also now it is possible even in language, to find a remaining balance in names and images of digits absolutely of other numeration systems which were once used by these people.

So the French once had a dvadtsaterichny numeration system as 80 in French sounds as "four times twenty".

Romans, or their predecessors used once quinary system as V that other as the image of a palm with the set aside thumb, and X are two same hands.