Since ancient times, people have been interested in numbers. They counted the number of days in a year, the number of stars in the sky, the amount of grain collected, the costs of building roads and buildings, and so on. It is no exaggeration to say that numbers are at the heart of human activity of absolutely any nature. In order to perform a mathematical calculation, you must have an appropriate system and be able to use it. This article will focus on the unary number system.
Concept of number system
By this concept is meant a combination of symbols, the rules for compiling numbers from them and performing mathematical operations. That is, using the number system, you can perform various calculations and get the result of solving the problem in the form of a number.
An important role in various number systems is played by the way numbers are represented. In the general case, it is customary to distinguish positional and non-positional representations. In the first case, the value of a digit depends on the position in which it is located, in the second case, the value of a digit in a number does not differ from that if the digit independently formed a number.
For example, our number system is positional, so in the number "22" - the first number "2" characterizes tens, the same number "2", but already standing in the second position, defines units. An example of a non-positional number system is Latin numbers, so the number "XVIII" should be interpreted as the sum: X + V + I + I + I = 18. In this system, only the contribution to the total number of each digit changes depending on the number in front of her, but her very meaning does not change. For example, XI = X + I = 11, but IX = X - I = 9, here the characters "X" and "I" characterize the numbers 10 and 1, respectively.
Unary number system
It is understood as such a way of representing numbers, which is based on only one digit. Thus, this is the simplest number system that can exist. It is called unary (from the Latin word unum - “one”) because it is based on a single digit. For example, we will denote it by the symbol "|".
To represent a certain number of any elements of N in a unary number system, it is enough to write N corresponding characters in a row ("|"). For example, the number 5 is written like this: |||||.
Ways to represent a number in a unary system
From the above example, it becomes obvious that if you increase the number of elements, it will be necessary to write a lot of “sticks” for their presentation, which is extremely inconvenient. Therefore, people have come up with various ways to simplify writing and reading numbers in the number system in question.
One of the popular methods is the representation of "fives", that is, 5 elements are grouped in a certain way using "sticks". So, in Brazil and France, this numerical grouping is a square with a diagonal: "|" - this is the number 1, "L" (two "sticks") - the number 2, "U" (three "sticks") - 3, closing the "U" on top, get a square (number 4), finally, "|", put on the diagonal of the square, will display the number 5.
Historical reference
No known ancient civilization used this primitive system to perform calculations, however, the following fact was precisely established: the unary number system was the basis of almost all numerical representations in antiquity. We give the following examples:
- The ancient Egyptians used it to count from 1 to 10, then they added a new symbol for dozens and continued the count, “folding sticks”. Reaching hundreds, they again entered a new corresponding symbol, and so on.
- The Roman numeral system was also formed from unary. The reliability of this fact is confirmed by the first three numbers: I, II, III.
- The history of the unary number system is also present in eastern civilizations. So, for counting in China, Japan and Korea, as well as in the Roman system, the unary way of writing is used first, and then new characters are added.
Examples of using the system in question
Despite all its simplicity, the unary system is currently used in some mathematical operations. As a rule, it turns out to be useful and easy to use for cases when the finite number of elements does not matter, and you need to keep the score one by one, adding or subtracting the element. So examples of unary number system are the following:
- Simple finger counting.
- Counting the number of visitors to an institution over a certain period of time.
- Counting the number of votes during the election.
- Children in the 1st grade are taught counting and the simplest mathematical operations precisely using the unary system (on colored sticks).
- The unary number system in computer science is used to solve some problems, for example, problems of P-complexity. To do this, it is important to represent the number in a unary way, since it is easier to decompose it into components, each of which is processed in parallel by a computer processor.
Advantages and disadvantages of the unary system
The main advantage has already been named, it is to use just one character ("|") to represent any number of elements. In addition, using the unary number system, it is easy to perform addition and subtraction.
The disadvantages of its use are more significant than the advantages. So, there is no zero in it, which is a huge obstacle to the development of mathematics. It is extremely inconvenient to represent large numbers in a unary system, and operations with them, such as multiplication and division, are extremely complicated.
These reasons explain the fact that the system in question is used only for small numbers, and only for simple mathematical operations.