Number systems - what is it? Without even knowing the answer to this question, each of us involuntarily in his life uses number systems and does not suspect about it. Exactly so, in the plural! That is, not one, but several. Before giving examples of non-positional number systems, let's look at this issue, let's talk about positional systems too.
Account requirement
Since ancient times, people had a need for counting, that is, they intuitively realized that it was necessary to somehow express a quantitative vision of things and events. The brain suggested that it was necessary to use objects for counting. The most comfortable fingers were always on the hands, and this is understandable, because they are always available (with rare exceptions).
So the ancient representatives of the human race had to bend their fingers in the literal sense - to indicate the number of killed mammoths, for example. Such elements of the account did not yet have names, but only a visual picture, a comparison.
Modern positional number systems
The number system is a method (method) for representing quantitative values and quantities by means of certain signs (symbols or letters).
You need to understand what positionalism and non-positioning are in the account before giving examples of non-positional number systems. There are many positional number systems. Now the following are used in various fields of knowledge: binary (includes only two significant elements: 0 and 1), hexadecimal (number of characters - 6), octal (characters - 8), duodecimal (twelve characters), hexadecimal (includes sixteen characters). Moreover, each row of characters in systems starts from scratch. Modern computer technology is based on the use of binary codes - a binary positional number system.
Decimal number system
Positionality is considered to be the presence, to varying degrees, of significant positions on which the signs of the number are located. This can best be demonstrated by decimal notation. After all, it’s what we used to use from childhood. There are ten signs in this system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Take the number 327. It has three signs: 3, 2, 7. Each of them is located in its position ( location). The seven occupies the position reserved for unit values (units), the two - tens, and the triple - hundreds. Since the number is three-digit, 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. Moreover, the importance (importance) of positions is measured from left to right, from a weak position (unit) to a stronger one (hundreds).
It is very convenient for us to feel in a decimal positional number system. We have ten fingers on our hands, and so are our feet. Five plus five - so, thanks to our fingers, since childhood we easily imagine a dozen. This is why it is easy for children to learn the multiplication table of five and ten. It’s also so simple to learn how to count money notes, which are most often multiple (that is, divide without a remainder) by five and ten.
Other positional number systems
To the surprise of many, it should be said that not only in the decimal system of counting is our brain used to doing certain calculations. Until now, humanity uses hexadecimal and duodecimal number systems. That is, in such a system there are only six characters (in hexadecimal): 0, 1, 2, 3, 4, 5. In twelve, there are twelve: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , A, B, where A is the number 10, B is the number 11 (since the sign must be one).
Judge for yourself. We think time is six, right? One hour - sixty minutes (six dozen), one day - this is twenty-four hours (two times twelve), a year - twelve months, and so on ... All time intervals easily fit into six- and duodecimal rows. But we are so used to it that we don’t even think about it when counting down the time.
Non-positional number systems. Unary
It is necessary to determine what it is - a non-positional number system. This is such a sign system in which there are no positions for the signs of the number, or the principle of "reading" the number does not depend on the position. It also has its own rules for writing or computing.
We give examples of non-positional number systems. Back to antiquity. People needed a bill and came up with the simplest invention - knots. Non-positional number system is nodular. One object (a bag of rice, a bull, a haystack , etc.) was counted, for example, when buying or selling and tied a knot on a rope.
As a result, on the rope it turned out as many nodules as many bags of rice were bought (as an example). But also it could be notches on a wooden stick, on a stone slab, etc. This number system began to be called nodular. She has a second name - unary, or singular ("uno" in Latin means "one").
It becomes obvious that this number system is non-positional. After all, what kind of positions can we talk about when she (position) is only one! Oddly enough, in some parts of the Earth, a unary non-positional number system is still in use.
Also non-positional number systems include:
- Roman (for writing numbers, letters are used - Latin characters);
- Ancient Egyptian (similar to Roman, symbols were also used);
- alphabetical (letters of the alphabet were used);
- Babylonian (cuneiform - used a straight and inverted "wedge");
- Greek (also referred to as alphabetical).
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 account system. Two hundred years ago, Roman numbers were used to indicate amounts in business documents (thus avoiding falsification).
Roman numbering is an example of a non-positional number system; it is known to us now. Also, the Roman system is actively used, but not for mathematical calculations, but for narrowly targeted actions. For example, using Roman numbers it is customary to designate historical dates, centuries, numbers of volumes, sections and chapters in book publications. Roman signs are often used to design watch dials. And Roman numeration is an example of a non-positional number system.
The Romans indicated the numbers in Latin letters. Moreover, they wrote down the numbers according to certain rules. There is a list of key characters in the Roman numeral system, with the help of them all numbers were recorded without exception.
Designations of numbers of the Roman numeral systemNumber (in decimal) | Roman number (letter of the Latin alphabet) |
| 1 | I |
| 5 | V |
| 10 | X |
| fifty | L |
| 100 | C |
| 500 | D |
| 1000 | M
|
Rules for compiling numbers
The required number was obtained by adding the characters (Latin letters) and calculating their sum. Consider how characters are written symbolically in the Roman system and how to "read" them. We list the basic laws of the formation of numbers in the Roman non-positional number system.
- Number four - IV, consists of two characters (I, V - one and five). It is obtained by subtracting the smaller sign from the larger one, if it is to the left. When the smaller sign is located on the right, it is necessary to add, then we get the number six - VI.
- It is necessary to fold two identical signs standing side by side. For example: SS is 200 (C - 100), or XX - 20.
- If the first character of the number is less than the second, then the third in this row may be a character whose value is even less than the first. In order not to get confused, we give an example: CDX - 410 (in decimal).
- Some large numbers can be represented in different ways, which is one of the minuses of the Roman system of counting. Here are some examples: MVM (Roman system) = 1000 + (1000 - 5) = 1995 (decimal system) or MDVD = 1000 + 500 + (500 - 5) = 1995. And there are more to come.
Arithmetic techniques
A non-positional number system is sometimes a complex set of rules for the formation of numbers, their processing (actions on them). Arithmetic operations in non-positional number systems is not an easy task for modern people. We do not envy the ancient Roman mathematicians!
An example of addition. Let's try to add two numbers: XIX + XXVI = XXXV, this task is performed in two actions:
- First, we take and add smaller fractions of numbers: IX + VI = XV (I after V and I before X “destroy” each other).
- Second - add up large fractions of two numbers: X + XX = XXX.
Subtraction is a bit more complicated. The decremented number needs to be divided into constituent elements, and after that, in the reducible and subtracted ones, reduce duplicate characters. Subtract 263 from the number 500:
D - CCLXIII = EGRLXXXXVIIIII - CCLXIII = CCXXXVII.
The multiplication of Roman numbers. By the way, it is necessary to mention that the Romans did not have signs of arithmetic operations, they simply designated them with words.
The multiplication number needed to be multiplied by each individual symbol of the multiplier, and several pieces were obtained that needed to be added. In this way, polynomials are multiplied.
As for division, this process in the Roman numeral system was and remains the most complex. Here the ancient Roman abacus was used - 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 mentioned above, in non-positional number systems there are disadvantages, inconveniences in use. Unary is simple enough for a simple calculation, but for arithmetic and complex calculations, it is not suitable at all.
In Roman there are no uniform rules for the formation of large numbers and there is confusion, and it is also very difficult to make calculations in it. In addition, the largest number that the ancient Romans could record using their method was 100,000.