Number systems - what is it? Even without knowing the answer to this question, each of us involuntarily uses number systems in our lives and does not suspect it. That's right, plural! That is, not one, but several. Before giving examples of non-positional number systems, let's understand this issue, let's talk about positional systems too.
Invoice Needed
Since ancient times, people had a need for counting, that is, they intuitively realized that they needed to somehow express a quantitative vision of things and events. The brain suggested that it was necessary to use objects for counting. Fingers have always been the most convenient, 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 (way) of representing quantitative values and quantities using certain signs (symbols or letters).
It is necessary to understand what is positional and non-positional in counting 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 the systems starts from zero. Modern computer technologies are based on the use of binary codes - the binary positional number system.
Decimal number system
Positionality is the presence of significant positions to varying degrees, on which the signs of the number are located. This can best be demonstrated using the example of the decimal number system. After all, we are accustomed to using it 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 own position (place). The seven takes the position reserved for single values (units), the two - tens, and the three - hundreds. Since the number is three-digit, therefore, there are only three positions in it.
Based on the above, thisa three-digit decimal number can be described as follows: three hundreds, two tens and seven units. Moreover, the significance (importance) of positions is counted from left to right, from a weak position (one) to a stronger one (hundreds).
We feel very comfortable in the decimal positional number system. We have ten fingers on our hands, and the same on our feet. Five plus five - so, thanks to the fingers, we easily imagine a dozen from childhood. That is why it is easy for children to learn the multiplication tables for five and ten. And it’s also so easy to learn how to count banknotes, which are most often multiples (that is, divided 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 counting system, our brain is used to doing some calculations. Until now, mankind has been using six and duodecimal number systems. That is, in such a system there are only six characters (in hexadecimal): 0, 1, 2, 3, 4, 5. In duodecimal there are twelve of them: 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 count time in sixes, don't we? One hour is sixty minutes (six tens), 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 duodecimal series. But we are so used to it that we do not even think about it when counting time.
Non-positional number systems. Unary
It is necessary to define what it is - a non-positional number system. This is such a sign system in which there are no positions for the signs of a number, or the principle of "reading" a number does not depend on the position. It also has its own rules for writing or calculating.
Let's give examples of non-positional number systems. Let's go back to antiquity. People needed an account and came up with the simplest invention - knots. The non-positional number system is nodular. 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, there were as many knots on the rope as many bags of rice were bought (as an example). But it could also be notches on a wooden stick, on a stone slab, etc. Such a number system became known as nodular. She has a second name - unary, or single ("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 it (the position) is only one! Oddly enough, in some parts of the Earth, the unary non-positional number system is still in use.
Also, non-positional number systems include:
- Roman (letters are used to write numbers - Latin characters);
- ancient Egyptian (similar to Roman, symbols were also used);
- alphabetic (letters of the alphabet were used);
- Babylonian (cuneiform - used direct andinverted "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 counting system. Two hundred years ago, Roman numerals were used to denote amounts in business documents (thus counterfeiting was avoided).
Roman numeration is an example of a non-positional number system, we know it now. Also, the Roman system is actively used, but not for mathematical calculations, but for narrowly focused actions. For example, with the help of 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 decorate watch dials. And also Roman numeration is an example of a non-positional number system.
The Romans denoted numbers with Latin letters. Moreover, they wrote down the numbers according to certain rules. There is a list of key symbols in the Roman numeral system, with the help of which all numbers were written without exception.
| Number (decimal) | Roman numeral (letter of the Latin alphabet) |
| 1 | I |
| 5 | V |
| 10 | X |
| 50 | L |
| 100 | C |
| 500 | D |
| 1000 | M |
Rules for composing numbers
The required number was obtained by adding signs (Latin letters) and calculating their sum. Let's consider how signs are written symbolically in the Roman system and how they should be "read". Let's list the main laws of number formation in the Roman non-positional number system.
- The 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, you need to add, then you get the number six - VI.
- It is necessary to add two identical signs next to each other. For example: SS is 200 (C is 100), or XX is 20.
- If the first sign of a number is less than the second, then the third character in this row can be a character whose value is even less than the first. To avoid confusion, here is an example: CDX - 410 (in decimal).
- Some large numbers can be represented in different ways, which is one of the disadvantages of the Roman counting 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.
Arithmetic tricks
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 are not easyfor modern people. We 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 steps:
- First - take and add the smaller fractions of numbers: IX + VI=XV (I after V and I before X "destroy" each other).
- Second - add large fractions of two numbers: X + XX=XXX.
Subtraction is somewhat more complicated. The number to be reduced must be divided into its constituent elements, and then the duplicated characters to be reduced in the number to be reduced and to be subtracted. Subtract 263 from 500:
D - CCLXIII=CCCCLXXXXVIIIII - CCLXIII=CCXXXVII.
Roman numeral multiplication. By the way, it is necessary to mention that the Romans did not have signs of arithmetic operations, they simply denoted them with words.
The multiple number had to be multiplied by each individual symbol of the multiplier, resulting in several products that had to be added. This is how polynomials are multiplied.
As for division, this process in the Roman numeral system was and remains the most difficult. The ancient Roman abacus was used here. To work with him, people were specially trained (and not every person managed to master such a science).
On the disadvantages of non-positional systems
As mentioned above, non-positional number systems have their drawbacks, inconveniences in use. Unary is simple enough for simple counting, but for arithmetic and complex calculations, it is notgood enough.
In Roman there are no uniform rules for the formation of large numbers and confusion arises, and it is also very difficult to make calculations in it. Also, the largest number the ancient Romans could write down with their method was 100,000.
