The most famous number systems. Number systems. Non-positional number systems. Alphabetic number systems

It all depends on the specific number system.

The decimal number system is obviously used almost everywhere.

Roman number system in modern world used most often when you want to indicate a number in order. For example, “10” means quantity (ten pieces), and the Roman “X” means “tenth”.

The binary number system is the most widely used in computers, since one digit of a binary number corresponds to one bit - the minimum unit of information in computer technology.

Also, the binary number system is traditionally used when indicating linear dimensions in inches, for example, 7 15 / 16 ″, 3 11 / 32 ″. The very first known use The binary number system belongs, perhaps, to the ancient Indian mathematician Pingala (approximately 2nd-5th centuries BC).

The hexadecimal number system is widely used in low-level programming as well as in computer documentation. In modern computers, the minimum unit of memory is an 8-bit byte, the values of which are conveniently written in two hexadecimal digits. This use began with the IBM/360 system, where all documentation used the hexadecimal number system.

Everything is interesting with the octal number system. It was used, for example, by some American Indians, since they believed that quantities should be counted not by the number of fingers, but by the number of spaces between the fingers.

In Europe in 1716, King Charles XII of Sweden asked Emmanuel Swedenborg to develop a 64-digit number system, to which Emmanuel Swedenborg noted that ordinary people not with such a high intelligence as the king would have difficulty understanding a number system with such a large base and proposed to use, therefore, the octal number system. It would be interesting to know why Charles XII chose this particular foundation.

Also, the octal number system is sometimes used in computers - apparently most often when determining permissions in Unix-like operating systems. Once upon a time there were computers that used 24 and 36 bit words. In such computers it was very convenient to use the octal number system, since all the bits of a word could be represented by a whole number of octal digits and there was no need to always add insignificant zero bits at the beginning. For example, a 36-bit word requires exactly 12 octal digits.

In our discrete mathematics course, we study the octal system because it is one of the systems into which we can directly convert from the binary number system, bypassing the decimal number system.

The sexagesimal number system is widely used in calculating minutes and seconds. The origins of the sexagesimal system are unclear. Perhaps it is related to the duodecimal number system (60 = 5 × 12, where 5 is the number of fingers on the hand). There is also a hypothesis by O. Neugebauer (1927) that after the Akkadian conquest of the Sumerian state, two monetary units simultaneously existed there for a long time: the shekel (shekel) and the mina, and their ratio was established as 1 mina = 60 shekels. Later, this division became customary and gave rise to a corresponding system for recording any numbers.

Is it possible to add zeros to the beginning of a number in the hexadecimal number system?

All rules for all positional number systems are the same. In the decimal number system, it is allowed to add insignificant zeros at the beginning, and after the decimal point at the end. In the same way, insignificant zeros can be added in any other positional number system.

What symbols are used to write a number in the 25-ary number system?

The hexadecimal number system is a fairly common number system. There is a standard for this number system - numbers greater than 9 are written in letters of the Latin alphabet from A to F.

All other positional number systems with a base greater than 10 are not common and there is no recording standard for them. But, by analogy, it would be convenient to also use letters of the Latin alphabet in these number systems.

In particular, in the 25-ary number system, the first 10 digits coincide with the numbers in the decimal number system - from 0 to 9, and the remaining 15 are encoded in letters of the Latin alphabet from A to O. The same rules apply to other positional number systems.

But what about a number system for which there are not enough letters of the Latin alphabet?

There is no universal standard in this area. Except in the case of more or less widely used number systems.

If you have to operate with such a number system, then either adhere to the rules that others have come up with (if anyone else uses such a number system), or come up with your own rules.

In practice, an example of such a number system with a large base is the 60-digit number system for counting seconds and minutes. We all know how time is recorded. For example, the entry “34:17”, meaning “34 minutes 17 seconds”, is actually a number written in sexagesimal with two digits.

How to correctly read numbers in number systems other than decimal?

In general, there is no standard for how to correctly read such numbers.

Strictly speaking, calling 20 8 the word “twenty” is not entirely correct, since everyone knows that “twenty” means “tens,” and in the octal number system this two means not the number of tens, but the number of eights. This number would probably be correctly read as "two zero", but this is not the standard.

When using the hexadecimal number system, the letters are pronounced as they are usually pronounced in the Latin alphabet: “A”, “Be”, “Tse”, “De”, “E”, “Ef”. The number 1E3.F 16 is usually pronounced like this: “one e three dot ef.”

However, if a number uses only decimal digits, the numbers are often read as if they were written in decimal notation. For example, “517.5 8” can be pronounced as “five hundred seventeen point five in octal notation.” It would probably be more accurate to say “five hundred seventeen point five eighths in the octal number system,” but in this case some may be confused about how to write “five eighths.”

Sometimes parts of a number are named according to different rules. For example, like this: “five hundred seventeen point five in the octal number system.” There seems to be no standard in this area yet either.

I think that the most important thing in pronouncing numbers is for others to understand what you mean.

How to remember the table of correspondence between binary numbers and octal and hexadecimal numbers?

You can remember this table only with experience - refer to it many times, and after a while you will know it by heart.

But you don't need to memorize this table! It’s so easy to determine the correspondence that I can’t even be sure whether I remember this table by heart or calculate it every time? To determine compliance, you only need to know a few very simple things:

One hexadecimal digit corresponds to 4 binary digits, and one octal digit corresponds to 3 binary digits. This is easy to remember, since 2 4 =16, and 2 3 =8.

You need to learn to mentally convert numbers from 0 to 7 from the octal number system to the decimal number system and vice versa. This is a very difficult operation, only prodigies can do it in their minds. If you are not a prodigy, you can simply remember that 0=0, 1=1, 2=2, 3=3, 4=4, 5=5, 6=6, and 7 is equal to 7.

You need to learn to mentally convert numbers from 0 to 15 from the decimal number system to hexadecimal. This is very simple, since the numbers from 0 to 9 coincide, and the numbers from 10 to 15 correspond to the letters of the Latin alphabet from A to F. You can count each time in your head (10 is A, 11 is B, 12 is C and etc.)

The hardest thing is to learn. But this skill alone covers a significant portion of the table.

Now you can easily convert any number from 0 to 15 from binary to decimal, and then to hexadecimal or octal. Or you can do the opposite.

To convert numbers, you need to be able to do long division. What if I don’t know how to do long division?

The theoretical material presented here assumes that you have some skills. If you do not already have these minimum skills, then in order to understand what is written here, it makes sense to first obtain these simple skills.

To understand all the theoretical material presented here, you will need:

Understand what a number is in principle.

Let's look at one of the most important topics in computer science -. IN school curriculum it is revealed rather “modestly,” most likely due to the lack of hours allocated to it. Knowledge on this topic, especially on translation of number systems, are a prerequisite for successful passing the Unified State Exam and admission to universities at the relevant faculties. Below we discuss in detail concepts such as positional and non-positional number systems, examples of these number systems are given, rules for translating integer decimal numbers, correct decimals and mixed decimal numbers to any other number system, conversion of numbers from any number system to decimal, conversion from octal and hexadecimal number systems to binary number system. There are a lot of problems on this topic in exams. The ability to solve them is one of the requirements for applicants. Coming soon: For each topic of the section, in addition to detailed theoretical material, almost all possible options will be presented tasks For self-study. In addition, you will have the opportunity to download ready-made ones from a file hosting service completely free of charge. detailed solutions to these tasks, illustrating various ways getting the correct answer.

positional number systems.

Non-positional number systems- number systems in which the quantitative value of a digit does not depend on its location in the number.

Non-positional number systems include, for example, Roman, where instead of numbers there are Latin letters.

I	1 (one)
V	5 (five)
X	10 (ten)
L	50 (fifty)
C	100 (one hundred)
D	500 (five hundred)
M	1000 (thousand)

Here the letter V stands for 5 regardless of its location. However, it is worth mentioning that although the Roman number system is a classic example of a non-positional number system, it is not completely non-positional, because The smaller number in front of the larger one is subtracted from it:

IL	49 (50-1=49)
VI	6 (5+1=6)
XXI	21 (10+10+1=21)
MI	1001 (1000+1=1001)

positional number systems.

Positional number systems- number systems in which the quantitative value of a digit depends on its location in the number.

For example, if we talk about the decimal number system, then in the number 700 the number 7 means “seven hundred”, but the same number in the number 71 means “seven tens”, and in the number 7020 - “seven thousand”.

Each positional number system has its own base. A natural number greater than or equal to two is chosen as the base. It is equal to the number of digits used in a given number system.

Binary- positional number system with base 2.
Quaternary- positional number system with base 4.
Five-fold- positional number system with base 5.
Octal- positional number system with base 8.
Hexadecimal- positional number system with base 16.

To successfully solve problems on the topic “Number systems”, the student must know by heart the correspondence of binary, decimal, octal and hexadecimal numbers up to 16 10:

10 s/s	2 s/s	8 s/s	16 s/s
0	0	0	0
1	1	1	1
2	10	2	2
3	11	3	3
4	100	4	4
5	101	5	5
6	110	6	6
7	111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10

It is useful to know how numbers are obtained in these number systems. You can guess that in octal, hexadecimal, ternary and others positional number systems everything happens in the same way as the decimal system we are used to:

One is added to the number and a new number is obtained. If the units place becomes equal to the base of the number system, we increase the number of tens by 1, etc.

This “transition of one” is what frightens most students. In fact, everything is quite simple. The transition occurs if the units digit becomes equal to number base, we increase the number of tens by 1. Many, remembering the good old decimal system, are instantly confused about the digits in this transition, because decimal and, for example, binary tens are different things.

Hence, resourceful students develop “their own methods” (surprisingly... working) when filling out, for example, truth tables, the first columns (variable values) of which are, in fact, filled with binary numbers in ascending order.

For example, let's look at getting numbers in octal system: We add 1 to the first number (0), we get 1. Then we add 1 to 1, we get 2, etc. to 7. If we add one to 7, we get a number equal to the base of the number system, i.e. 8. Then you need to increase the tens place by one (we get the octal ten - 10). Next, obviously, are the numbers 11, 12, 13, 14, 15, 16, 17, 20, ..., 27, 30, ..., 77, 100, 101...

Rules for converting from one number system to another.

1 Converting integer decimal numbers to any other number system.

The number must be divided by new number system base. The first remainder of the division is the first minor digit of the new number. If the quotient of the division is less than or equal to the new base, then it (the quotient) must be divided again by the new base. The division must be continued until we get a quotient less than the new base. This is the highest digit of the new number (you need to remember that, for example, in the hexadecimal system, after 9 there are letters, i.e. if the remainder is 11, you need to write it as B).

Example ("division by corner"): Let's convert the number 173 10 to the octal number system.

Thus, 173 10 =255 8

2 Converting regular decimal fractions to any other number system.

The number must be multiplied by the new number system base. The digit that has become the integer part is the highest digit of the fractional part of the new number. to obtain the next digit, the fractional part of the resulting product must again be multiplied by a new base of the number system until the transition to the whole part occurs. We continue multiplication until the fractional part equals zero, or until we reach the accuracy specified in the problem (“... calculate with an accuracy of, for example, two decimal places”).

Example: Let's convert the number 0.65625 10 to the octal number system.

Notation is a way of writing numbers. Usually, numbers are written using special characters - numbers (although not always). If you have never studied this question, then at least you should know two number systems - Arabic and Roman. The first uses the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and is a positional number system. And in the second - I, V, X, L, C, D, M and this is a non-positional number system.

In positional number systems, the quantity denoted by a digit in a number depends on its position, but in non-positional number systems it does not. For example:

11 - here the first unit denotes ten, and the second - 1.
II - here both units denote one.

345, 259, 521 - here the number 5 in the first case means 5, in the second - 50, and in the third - 500.

XXV, XVI, VII - here, wherever the number V is, it always means five units. In other words, the quantity denoted by the V sign does not depend on its position.

Addition, multiplication and other mathematical operations are easier to perform in positional number systems than in non-positional ones, because mathematical operations are carried out using simple algorithms (for example, multiplication by a column, comparison of two numbers).

Positional number systems are the most common in the world. In addition to the decimal system, which is familiar to everyone since childhood (which uses ten digits from 0 to 9), such number systems as binary (the numbers 0 and 1 are used), octal and hexadecimal are widely used in technology.

It should be noted the important role of zero. The “discovery” of this number in the history of mankind played a big role in the formation of positional number systems.

The base of a number system is the number of digits that is used to write numbers.

Place is the position of a digit in a number. The digit capacity of a number is the number of digits that make up the number (for example, 264 is a three-digit number, 00010101 is an eight-digit number). The digits are numbered from right to left (for example, in the number 598, eight occupies the first digit, and five occupies the third).

So, in the positional number system, numbers are written in such a way that each next (movement from right to left) digit is greater than the other by the power of the base of the number system. (come up with a diagram)

The same number (value) can be represented in different number systems. The representation of the number is different, but the meaning remains unchanged.

Binary number system

IN binary system numbering uses only two digits 0 and 1. In other words, two is the base of the binary number system. (Similarly, the decimal system has a base of 10.)

To learn to understand numbers in the binary number system, first consider how numbers are formed in the decimal number system familiar to us.

In the decimal number system we have ten digits (from 0 to 9). When the count reaches 9, a new digit (tens) is introduced, the ones are reset to zero and the count starts again. After 19, the tens digit increases by 1, and the ones are reset to zero again. And so on. When the tens reach 9, then the third digit appears - hundreds.

The binary number system is similar to the decimal number system, except that only two digits are involved in the formation of the number: 0 and 1. As soon as the digit reaches its limit (i.e., one), a new digit appears, and the old one is reset to zero.

Let's try to count in binary system:
0 is zero
1 is one (and that's the discharge limit)
10 is two
11 is three (and that's the limit again)
100 is four
101 - five
110 - six
111 - seven, etc.
Converting numbers from binary to decimal

It is not difficult to notice that in the binary number system, the lengths of numbers increase rapidly as the values increase. How to determine what this means: 10001001? Unaccustomed to this form of writing numbers, the human brain usually cannot understand how much it is. It would be nice to be able to convert binary numbers to decimal.

In the decimal number system, any number can be represented as a sum of units, tens, hundreds, etc. For example:

1476 = 1000 + 400 + 70 + 6

1476 = 1 * 103 + 4 * 102 + 7 * 101 + 6 * 100

Look at this entry carefully. Here the numbers 1, 4, 7 and 6 are a set of numbers that make up the number 1476. All these numbers are multiplied in turn by ten raised to one degree or another. Ten is the base of the decimal number system. The power to which ten is raised is the digit of the digit minus one.

Any binary number can be expanded in a similar way. Only the base here will be 2:

10001001 = 1*2 7 + 0*2 6 + 0*2 5 + 0*2 4 + 1*2 3 + 0*2 2 + 0*2 1 + 1*2 0

1*2 7 + 0*2 6 + 0*2 5 + 0*2 4 + 1*2 3 + 0*2 2 + 0*2 1 + 1*2 0 = 128 + 0 + 0 + 0 + 8 + 0 + 0 + 1 = 137

Those. The number 10001001 in base 2 is equal to the number 137 in base 10. You can write it like this:

10001001 2 = 13710
Why is the binary number system so common?

The fact is that the binary number system is the language of computer technology. Each number must be somehow represented on a physical medium. If this is a decimal system, then you will have to create a device that can have ten states. It's complicated. It is easier to produce a physical element that can only be in two states (for example, there is current or no current). This is one of the main reasons why so much attention is paid to the binary number system.
Converting a decimal number to binary

You may need to convert the decimal number to binary. One way is to divide by two and form a binary number from the remainder. For example, you need to get its binary notation from the number 77:

77 / 2 = 38 (1 remainder)
38 / 2 = 19 (0 remainder)
19 / 2 = 9 (1 remainder)
9 / 2 = 4 (1 remainder)
4 / 2 = 2 (0 remainder)
2 / 2 = 1 (0 remainder)
1 / 2 = 0 (1 remainder)

We collect the remainders together, starting from the end: 1001101. This is the number 77 in binary representation. Let's check:

1001101 = 1*2 6 + 0*2 5 + 0*2 4 + 1*2 3 + 1*2 2 + 0*2 1 + 1*2 0 = 64 + 0 + 0 + 8 + 4 + 0 + 1 = 77

Octal number system

So, modern “hardware understands” only the binary number system. However, it is difficult for a person to perceive long records of zeros and ones on the one hand, and on the other hand, converting numbers from binary to decimal system and back is quite time-consuming and labor-intensive. As a result, programmers often use other number systems: octal and hexadecimal. Both 8 and 16 are powers of two, and converting a binary number to them (as well as doing the inverse) is very easy.

The octal number system uses eight digits (from 0 to 7). Each digit corresponds to a set of three digits in the binary number system:

000 - 0
001 - 1
010 - 2
011 - 3
100 - 4
101 - 5
110 - 6
111 - 7

To convert a binary number to octal, it is enough to break it into triplets and replace them with their corresponding digits from the octal number system. You need to start dividing into triplets from the end, and replace the missing numbers at the beginning with zeros. For example:

1011101 = 1 011 101 = 001 011 101 = 1 3 5 = 135

That is, the number 1011101 in the binary number system is equal to the number 135 in the octal number system. Or 1011101 2 = 1358.

Reverse translation. Let's say you want to convert the number 1008 (don't be mistaken! 100 in octal is not 100 in decimal) into the binary number system.

100 8 = 1 0 0 = 001 000 000 = 001000000 = 10000002

Converting an octal number to a decimal number can be done using the already familiar scheme:

6728 = 6 * 8 2 + 7 * 8 1 + 2 * 8 0 = 6 * 64 + 56 + 2 = 384 + 56 + 2 = 44210
1008 = 1 * 8 2 + 0 * 8 1 + 0 * 8 0 = 6410

Hexadecimal number system

The hexadecimal number system, like the octal number system, is widely used in computer science because of the ease of converting binary numbers into it. Hexadecimal notation makes numbers more compact.

The hexadecimal number system uses numbers from 0 to 9 and the first six Latin letters - A (10), B (11), C (12), D (13), E (14), F (15).

When converting a binary number to hexadecimal, the first is divided into groups of four digits, starting from the end. If the number of digits is not divisible by an integer, then the first four is appended with zeros in front. Each four corresponds to a digit in the hexadecimal number system:

For example:
10001100101 = 0100 1100 0101 = 4 C 5 = 4C5

If necessary, the number 4C5 can be converted to the decimal number system as follows (C should be replaced with the number corresponding to this symbol in the decimal number system - this is 12):

4C5 = 4 * 162 + 12 * 161 + 5 * 160 = 4 * 256 + 192 + 5 = 1221

The maximum two-digit number that can be obtained using hexadecimal notation is FF.

FF = 15 * 161 + 15 * 160 = 240 + 15 = 255

255 is the maximum value of one byte, equal to 8 bits: 1111 1111 = FF. Therefore, using the hexadecimal number system it is very convenient to write down byte values briefly (using two digits). Attention! An 8-bit byte can have 256 states, but the maximum value is 255. Don’t forget about 0 - this is exactly the 256th state

Lecture 1. Number systems

1. The history of the emergence of number systems.

2. Positional and non-positional number systems.

3. Decimal number system, writing numbers in it.

4. Rank

A person constantly has to deal with numbers, so you need to be able to correctly name and write any number, and perform operations on numbers. As a rule, everyone copes with this successfully. The method of writing numbers that is currently used everywhere and is called the decimal number system helps here.

The study of this system begins in primary school, and, of course, the teacher needs certain knowledge in this area. He must know different ways of writing numbers, algorithms arithmetic operations and their rationale. The material in this lecture provides the minimum without which it is impossible to understand various methodological approaches to teaching. junior schoolchildren ways of writing numbers and performing operations on them.

History of the emergence of number systems.

The concept of number arose in ancient times. Then the need arose to name and write numbers. The language for naming, writing numbers and performing operations on them is called number system.

The simplest system records natural numbers requires only one number, such as "sticks" (or notches in wood, like primitive man, or a knot on a rope, like the American Indians), which represents a unit. By repeating this sign, you can write any number: each number n simply written n"sticks". In such a number system it is convenient to perform arithmetic operations. But this method of recording is very uneconomical and for large numbers inevitably leads to errors in counting.

Therefore, over time, other, more economical and convenient ways of writing numbers arose. Let's look at some of them.

IN Ancient Greece the so-called attic numbering. The numbers 1, 2, 3, 4 were indicated by dashes:

The number 5 was written with the sign G (the ancient form of the letter “pi”, with which the word “pente” - five) begins. The numbers 6, 7, 8, 9 were designated as follows:

The number 10 was denoted by Δ (the initial letter of the word “deca” is ten). The numbers 100, 1000 and 10,000 were designated H, X, M - the initial letters of the corresponding words.

Other numbers were written with various combinations of these signs.

In the third century BC, Attic numbering was supplanted by the so-called Ionian system. In it, the numbers 1 – 9 are indicated by the first nine letters of the alphabet: α (alpha), β (beta), γ (gamma), δ (delta), ε (epsilon), ς (wow) ζ (zeta),
η (eta), (theta).

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 – in the following nine letters: i(iota),
κ (kappa), λ (lambda), μ (mu), ν (nude), ξ (xi), ο (omicron), π (pi), With(cop).

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 are the last nine letters of the Greek alphabet.

In ancient times, Jews, Arabs and many other peoples of the Middle East had alphabetical numbering similar to the ancient Greek one. It is unknown which people it first appeared among.

IN Ancient Rome the “key” numbers were 1, 5, 10, 50, 100, 500 and 1000. They were designated by the letters I, V, X, L, C, D and M, respectively.

All integers (up to 5000) were written by repeating the above numbers. At the same time, if a larger number is in front of a smaller one, then they are added, but if the smaller one is in front of a larger one (in this case it cannot be repeated), then the smaller one is subtracted from the larger one: VI = 6, i.e. 5 + 1; IV = 4, i.e. 5 – 1;
XL = 40, i.e. 50 – 10; LX = 60, i.e. 50 + 10. The same number is placed no more than three times in a row: LXX = 70, LXXX = 80, the number 90 is written XC (not LXXXX).

For example: XXVIII = 28, XXXIX = 39, CCCXCVII = 397, MDCCCXVIII = 1818.

Performing arithmetic operations on multi-digit numbers in this notation is very difficult. However, Roman numbering has survived to this day. It is used to mark anniversaries, names of conferences, chapters in books, etc.

In ancient times, numbers were designated by letters in Rus'. To indicate that the sign is not a letter, but a number, a special sign called a “titlo” was placed above them. The first nine digits were written like this:

Tens are designated as follows:

Hundreds are designated as follows:

Thousands were designated by the same letters with “titles” as the first nine digits, but they had a “≠” sign on the left: ≠ A = 1000, ≠ B = 2000, ≠ E = 5000.

Tens of thousands were called " dark", they were designated by circling the unit signs:

10 000, = 20 000, = 80 000.

This is where the expression “Darkness to the people” comes from, i.e. there are a lot of people.

Hundreds of thousands were called " legions", they were designated by circling the unit signs with circles of dots:

100 000, = 200 000, = 800 000.

Millions were called " leodras" They were designated by circling the unit signs with circles of rays or commas:

1 000 000, = 2 000 000.

Tens of millions were called " crows"or "corvids" and they were designated by circling the unit signs with circles of crosses or placing the letter K on both sides:

Hundreds of millions were called " decks" The “deck” had a special designation - square brackets were placed above and below the letter:

Hieroglyphs of residents Ancient Babylon were made up of narrow vertical and horizontal wedges; these two icons were also used to record numbers. One vertical wedge meant one, and a horizontal one meant ten. In Ancient Babylon they counted in groups of 60 units. For example, the number 185 was represented as 3 times 60 and 5 more. Such a number was written using only two signs, one of which indicated how many times 60 were taken, and the other - how many units were taken.

There are many hypotheses about when and how the sexagesimal system arose among the Babylonians, but none has been proven yet. One of the hypotheses is that there was a mixture of two tribes, one of which used the sixfold system, and the other used the decimal system. The sexagesimal system arose as a compromise between these two systems. Another hypothesis is that the Babylonians considered the length of the year to be 360 days, which is naturally associated with the number 60.

The sexagesimal system, to some extent, has survived to this day, for example, in dividing the hour into 60 minutes, and the minute into 60 seconds, and in a similar system for measuring angles: 1 degree is equal to 60 minutes, 1 minute is 60 seconds.

Binary system Notation was used by some primitive tribes when counting; it was known to ancient Chinese mathematicians, but it was the great German mathematician Leibniz who truly developed and built the binary system, who saw in it the personification of a deep metaphysical truth.

The binary number system is used by some (local) cultures in Africa, Australia and South America.

To represent numbers in the binary number system, only two digits are required: 0 and 1. For this reason, the binary notation of a number is easy to represent using physical elements that have two different stable states. This is precisely what served as one of the important reasons for the widespread use of the binary system in modern electronic computers.

The most economical of all number systems is ternary. The binary system and the quaternary system, which is equivalent to it in terms of efficiency, are somewhat inferior in this regard to the ternary system, but are superior to all the main possible systems. If writing numbers from 1 to 10 in the decimal system requires 90 different states, and in the binary system - 60, then in the ternary system 57 states are sufficient.

The most common situation in which the need for ternary analysis manifests itself is, perhaps, weighing on a cup scale. Three different cases can arise here: either one of the cups will outweigh the other, or vice versa, or the cups will balance each other.

Quaternary number system used mainly by the Indian tribes of South America and the Yucca Indians of California, who count on the spaces between their fingers.

Five-fold number system was much more widespread than all the others. The Tamanacos Indians of South America use the same word for the number 5 as for “whole hand.” The word “six” in Tamanak means “one finger on the other hand,” seven means “two fingers on the other hand,” etc. for eight and nine. Ten is called "two hands". Wanting to name a number from 11 to 14, the Tamanakos extend both hands forward and count: “one on the leg, two on the leg,” etc. until they reach 15 - “the whole leg.” This is followed by “one on the other leg” (number 16), etc. to 19. The number 20 in Tamanak means “one Indian”, 21 means “one on the hand of another Indian”. "Two Indians" means 40, "three Indians" means 60.

The inhabitants of ancient Java and the Aztecs had a week of 5 days.

Some historians believe that the Roman numeral X (ten) was made up of two Roman 5s V (one of them inverted), and the number V in turn arose from a stylized image of a human hand.

Was widespread in ancient times duodecimal number system. Its origin is also connected with counting on fingers. Namely, since the four fingers of the hand (except the thumb) have a total of 12 phalanges, then along these phalanges, turning them over in turn with the thumb, they count from 1 to 12. Then 12 is taken as the unit of the next digit.

The main advantage of the duodecimal system is that its base is divisible by 2, 3 and 4. Proponents of the duodecimal system appeared in the 16th century. At a later time, these included: outstanding people, like Herbert Spencer, John Quincy Adams and George Bernard Shaw. There is even an American Duodecimal Society, which publishes two periodicals: the Duodecimal Bulletin and the Duodecimal System Manual. The society provides all “duodenums” with a special counting ruler, in which 12 is used as the base.

In oral speech, remnants of the duodecimal system have survived to this day: instead of saying “twelve,” some say “dozen.” The custom has been preserved of counting many items not by dozens, but by dozens, for example, cutlery in a service (a set for 12 people) or chairs in a furniture set.

The name of the third digit unit in the duodecimal number system is gross- is rare now, but in trade practice at the beginning of the 20th century it existed and, even a hundred years ago, it could be easily found. For example, in the poem “Plyushkin” written in 1928 by V.V. Mayakovsky, ridiculing the townspeople who buy up everything they need and don’t need, wrote:

Looking around

a scattering of goods,

The binary number system uses only two digits, 0 and 1. In other words, two is the base of the binary number system. (Similarly, the decimal system has a base of 10.)

To learn to understand numbers in the binary number system, first consider how numbers are formed in the decimal number system familiar to us.

Let's try to count in binary system:
0 is zero
1 is one (and this is the discharge limit)
10 is two
11 is three (and that's the limit again)
100 is four
101 – five
110 – six
111 – seven, etc.

Converting numbers from binary to decimal