Number systems. Example of nonpositional number systems

Number systems - what is it? Even without knowing the answer to this question, each of us willy-nilly in our lives uses number systems and does not suspect about it. That's right, in the plural! That is not one, but several. Before we give examples of non-position number systems, let's look at this issue, let's talk about positional systems too.

Need for an account

Since ancient times, people had a need for an account, that is, they intuitively realized that it was necessary to express in some way a quantitative vision of things and events. The brain suggested that you need to use items for the account. The most convenient were always the fingers on their hands, and this is understandable, because they are always available (with rare exceptions).

So it was necessary for ancient representatives of the human race to bend fingers in the literal sense - to denote the number of killed mammoths, for example. The names of such elements of the account did not yet exist, but only a visual picture, a comparison.

Modern positional number systems

The number system is a method (method) for the presentation of quantitative values and quantities by means of certain signs (symbols or letters).

It is necessary to understand what is positional and non-positional in the account, before giving examples of non-position number systems. Positional number systems are many. Now they use the following in different fields of knowledge: binary (includes only two significant elements: 0 and 1), six-digit (number of characters - 6), octal (signs - 8), duodecimal (twelve characters), hexadecimal (includes sixteen characters). And every series of signs in systems starts from zero. Modern computer technology is based on the use of binary codes - binary position number system.

Decimal number system

Positivity is the presence in varying degrees of significant positions on which the signs of the number are located. This can best be demonstrated by the example of a decimal number system. After all, we used to use it from childhood. The signs in this system are ten: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Take the number 327. There are three signs: 3, 2, 7. Each of them is located in its position Place). Seven occupies a position reserved for unit values (units), two dozens, and a triple - hundreds. Since the number is three-valued, therefore, there are only three positions in it.

Based on the foregoing, such a three-digit decimal number can be described as follows: three hundred, two tens and seven units. And the importance (importance) of positions is counted from left to right, from weak position (unit) to stronger (hundreds).

We feel very comfortable in the decimal position system. We have ten fingers on our hands, as well. Five plus five - so, thanks to the fingers, we are from childhood easily imagine a dozen. That's why it's easy for children to learn the multiplication table by five and by ten. And it's so easy to learn to count money notes, which are often multiples (that is, they divide without a remainder) by five and ten.

Other positioning systems

To the surprise of many, it should be said that not only in the decimal system of account our brain is accustomed to make certain calculations. Until now, mankind has used the six-and-twelve-digit number systems. That is, in such a system there are only six characters (in a hexadecimal): 0, 1, 2, 3, 4, 5. In the twelfth order there are twelve: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , A, B, where A - denotes the number 10, B - the number 11 (since the sign must be one).

Judge for yourself. We think time is six, is not it? One hour is sixty minutes (six dozen), one day is twenty four hours (two times twelve), a year is twelve months and so on ... All time intervals easily fit into six- and twelve-row rows. But we are so used to it that we do not even think about counting time.

Non-position number systems. Unary

It is necessary to determine what it is - the non-positional number system. It is such a sign system, in which there are no positions for the signs of a number, or the principle of "reading" a number from a position does not depend. It also has its own rules for writing or calculating.

We give examples of non-position number systems. Let's return to antiquity. People needed an account and came up with the simplest invention - nodules. The non-positioning system is the nodal system. One item (a bag of rice, a bull, a haystack , etc.) was counted, for example, when buying or selling and tied a knot on a string.

As a result, on the rope turned out so many knots, how many bags of rice bought (as an example). But also it could be notches on a wooden stick, on a stone slab, etc. Such a numbering system has become known as the nodal system. It has a second name - unary, or single ("uno" means "one" in Latin).

It becomes obvious that this number system is non-positional. After all, what positions can there be when it (position) is only one! Strangely enough, in some parts of the Earth there is still a unary non-position number system in the course of the process.

Also to non-positioning systems are:

• Roman (for the writing of numbers the letters are used - Latin symbols);
• Ancient Egyptian (similar to the Roman, also used symbols);
• Alphabetic (letters of the alphabet were used);
• Babylonian (cuneiform - used a straight and inverted "wedge");
• Greek (also referred to as alphabetic).

Roman numeral system

The ancient Roman Empire, as well as its science, was very progressive. The Romans gave the world many useful inventions of science and art, including their system of accounts. Two hundred years ago, Roman numbers were used to refer to amounts in business documents (thus avoiding forgery).

Roman numeration is an example of a non-position number system, it is known to us now. Also the Roman system is actively used, but not for mathematical calculations, but for narrowly directed actions. For example, with the help of Roman numerals, it is customary to designate historical dates, ages, numbers of volumes, sections and chapters in book editions. Often use Roman signs to decorate the dials of watches. And also the Roman numeration is an example of a non-positioning number system.

The Romans denoted the numbers in Latin letters. And the numbers they wrote down by certain rules. There is a list of key symbols in the Roman numeral system, with the help of them all numbers were recorded without exception.

Rules for compiling numbers

The required number was obtained by adding the signs (Latin letters) and calculating their sum. Consider how the signs in the Roman system are symbolically written and how to "read" them. Let's enumerate the basic laws of number formation in the Roman non-position number system.

1. Number four - IV, consists of two signs (I, V - one and five). It is obtained by subtracting the smaller sign from the larger if it is to the left. When the smaller sign is located on the right, it is necessary to add, then the number six - VI will be obtained.
2. It is necessary to add two identical signs standing side by side. For example: CC is 200 (C-100), or XX-20.
3. If the first character of a number is less than the second, then the third in this series can be a symbol whose value is even less than the first. In order not to get confused, let's give an example: CDX-410 (in decimal).
4. Some large numbers can be represented in many ways, which is one of the drawbacks of the Roman account system. Here are some examples: MVM (Roman system) = 1000 + (1000 - 5) = 1995 (decimal system) or MDVD = 1000 + 500 + (500 - 5) = 1995. And that's not all the ways.

Methods of arithmetic

The non-positioning number system is sometimes a complex set of rules for the formation of numbers, their processing (actions on them). Arithmetic operations in non-position number systems are not easy for modern people. Do not envy the ancient Roman mathematicians!

Example of addition. Let's try to add two numbers: XIX + XXVI = XXXV, this task is performed in two actions:

1. First, we take and add the smaller parts of numbers: IX + VI = XV (I after V and I before X "destroy" each other).
2. Secondly, we add the large fractions of two numbers: X + XX = XXX.

Subtraction is somewhat more complicated. The decremented number needs to be broken up into composite elements, and after that, the duplicated symbols in the reduced and subtractable ones should be reduced. From the number 500 we subtract 263:

D - CCLXIII = CCCCLXXXXVIIIII - CCLXIII = CCXXXVII.

Multiplication of Roman numbers. By the way, it is necessary to mention that the Romans did not have signs of arithmetic operations, they simply denoted them with words.

Multiply multiplication was needed for each individual symbol of the multiplier, resulting in several works that needed to be added. In this way, multiplication of polynomials is performed.

As for division, this process in the Roman numeral system has been and remains the most complex. Here used ancient Roman abacus - the abacus. To work with him, people were specially trained (and not every person managed to master such a science).

On the disadvantages of non-position systems

As it was said above, in non-position number systems there are some disadvantages, inconveniences in use. Unary is simple enough for simple counting, but it is not suitable for arithmetic and complex calculations at all.

In Roman there are no uniform rules for the formation of large numbers and confusion arises, and it is very difficult to make calculations in it. In addition, the largest number that the ancient Romans could record with the help of their method was 100,000.