Numbering system . Set of rules and conventions that allow the representation of all natural numbers , in principle, from a limited collection of basic symbols .
Summary
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- 1 Types
- 2 Base
- 1 Positional Number Systems
- 2 Binary System or Dyadic System
- 3 Decimal System
- 4 Octal System
- 3 Hexadecimal system
- 4 Conversion of numbers
- 1 Decimal-Binary
- 2 Decimal-Octal
- 3 Decimal-Hexadecimal
- 4 Octal and hexadecimal numbers
- 5 Conversion to decimal system
- 1 From binary to decimal
- 2 From octal to decimal
- 3 From hexadecimal to decimal
- 6 Binary sum
- 7 Binary subtraction
- 8 Sources
Types
- Positional:It is the one in which the value of the number changes according to the position the figure occupies within the number . Examples of them are: binary , decimal , hexadecimal , octal systems , etc.
- Non-positional:It is one in which the value of the number does not depend on the position it occupies within the number. This indicates that there are two types of figure values. An example of this is the Roman numerals .
Base
It is equal to the number of different digits that the system has, from which any number can be represented.
- Decimal System: 10 digits: (0,1,2,3,4,5,6,7,8,9)
- Binary System: 2 digits: (0,1)
Positional Number Systems
In the decimal number system the base or root is said to be 10 because it uses 10 digits, and the coefficients are multiplied by powers of 10.
The binary system only has two possible values that are 0 and 1, in which each coefficient AJ is multiplied by 2J , as an example we have the development of the binary number 11010.11 which will be represented by the following way:
1 * 24 + 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20 + 1 * 2 – 1 + 1 * 2 – 2 16 + 8 + 0 + 2 + 0 + 0.5 + 0.25 = 26.75 Therefore we have that a number in a base system ( r ) has coefficients multiplied by powers of ( r ) and would be represented as follows: an * rn + an * rn +. . . + a2 * r2 + a1 * r1 + a0 * r0 + a – 1 * r – 1 +. . . + a – m * r – m
Binary System or Dyadic System
In the binary system the base is 2 and only two digits are required, 0 and 1 therefore to represent a number. The figures 0 and 1 have the same meaning as in the decimal system, but differ in terms of their position.
In the binary system the individual digit represents the coefficient of the powers of two (2) instead of those of ten (10), as it happens in the decimal system. The value of any number expressed in the binary number system is equal to the sum of the terms that result from multiplying each of the digits that make up the number in question by the powers of 2 that corresponds according to the position that said digit occupies within the number.
An illustrative example is the decimal number 19, which is written in binary representation as 10011 since:
10011 = 1 x 24 + 0 x 23 + 0 x 22 + 1 x 21 + 1 x 20
10011 = 16 + 0 + 0 + 2+ 1 = 19
Decimal System
It is the system commonly used, base 10 and has ten digits, from 0 to 9. The value of any number expressed in this system is equal to the sum of the terms that result from multiplying each of the digits that make up the number in question. by the power of 10 that corresponds according to the position that said digit occupies within the number.
To write a number greater than 9, a meaning is assigned to the position of each digit in the whole number.
An example of this is number 1264:
1264 = 1 x 10 3 + 2 x 10 2 + 6 x 10 1 + 4 x 10 0
The adoption, use and dissemination of this base, as expressed by the Russian mathematician Nikolai Luzin, is due to the zoomorphic structure of the human being who has 10 fingers on both hands. Firstly, a bijection of the set of fingers with the objects to count. No object, nothing, or zero. Up to 9 objects, but if there were 10 objects, both hands collided, which meant a new result: what has been called a dozen. A dialectical leap was made, ten objects form a new unit, out of the tens. Then ten tens, the second dialectical jump, the hundreds. Then 10 hundreds, a thousand. Having organized a result you had, to say 4 units, 2 tens, 7 hundreds, 3 thousands. To simplify or compact, the figures were used: mcdu; in this example, 3724 turns out. A great finding was the importance of the position and the relative and absolute values of a figure. The origin of the figures or figures or digits that are used occurs in the Indian culture; but the Arabs contribute the symbol of zero and take it to Europe, and later it reaches America, so it is worth calling the Indo-Arabic figures for this bunch: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Octal system
In the same way as the decimal system , the octal system needs eight digits to be able to express or represent any number . The base of this system is 8. It consists of the digits 0,1,2,3,4,5,6,7.
Hexadecimal system
This system needs 16 digits as a base to express or represent any number , the first ten digits of this system coincide with those of the decimal number system and the remaining six digits are taken as the first six letters (capitals) of the alphabet : A, B, C, D, E, F, that is: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, F.
Combinations of digits will appear for the first time, one of which may be a letter but which for the purposes of the hexadecimal system is considered a number. As in the two previous cases, the value of any number expressed in the hexadecimal number system is equal to the sum of the terms that result from multiplying each of the digits that make up the number in question by the corresponding power of 16 according to the position that said digit occupies within the number.
Conversion of numbers
This method consists of repeatedly dividing a number by the base of the desired system, until finding a quotient such that it is not divisible by the divisor or the base. Then the latter quotient is taken and the remains, from right to left, thus forming the number in the requested system.
Decimal-Binary
- A binary number (x) can be converted to a decimal by summing the powers whose value is one.
Example: (1010.011) 2 = 1 * 2³ + 0 * 2² + 1 * 2¹ + 0 * 2º + 0 * 2 ־¹ + 1 * 2 ־² + 1 * 2 ־³ = 8 + 0 + 2 + 0 + 0 + 0.25 + 0.125 = 10,375
- For the numbers expressed in base ( r) it could be converted to a decimal by multiplying each coefficient by the corresponding power of r and adding.
Example :
(630.4) 8 = 6 * 82 + 3 * 81 + 0 * 80 + 4 * 8-1 = 384 + 24 + 0.5 = 408.5
- When you want to convert from decimal to binary or any other system with base r, itis more convenient if the number is separated into an integer part and a fractional part, and the conversion of each part is done separately.
- To convert any decimal integer to any base system r,division is done by r instead of 2.
- To convert a decimal fraction to binary, the following system is similar to the one we use for integers, however, multiplication is used instead of division, and integers are accumulated instead of residuals.
- When you want to convert a decimal fraction to a number expressed in base r, the procedure is similar, the multiplication is done with r instead of 2 and the coefficients are found with the integers.
- When you want to do the conversion of a decimal numberfrom an integer part and a fractional part, the conversion is done separately and then the two answers are combined.
Decimal-Octal
For this, an operation similar to the previous conversion is performed. The number in question is divided by the base of the system to be converted.
If we take the number 243 as an example, when dividing it by 8 which is the base of the octal system , it turns out to be 30, with a first remainder of 3; When dividing 30 by 8, it turns out to be 24, so the rest is 6 and the last quotient is 3. Taking from right to left is 363. So 243 in the decimal system is 3638 in the octal system 243 / 8 = 30 30/8 = 3 (3 6 remainders) 3 – last quotient.
Decimal-Hexadecimal
An illustrative example of this conversion is repeated with number 243. If you want to convert it to hexadecimal, you should proceed similarly to the previous conversions.
The number 243 is divided by the base 16. In doing so, we obtain a quotient of 15 with a remainder of 3, so that it is taken from right to left: last quotient (15) and the remainder that is 3. In hexadecimal the 15 is represented by the letter F, completing the result. The number 243 in decimal corresponds to F316 in hexadecimal.
Octal and hexadecimal numbers
Conversions between binary, octal, and hexadecimal code is very important in digital comparisons, as each octal digit corresponds to three binary digits and each hexadecimal digit corresponds to four binary digits.
(10110001101011.111100000110) 2 -> (26153.7406) 8
When you want to convert a binary number to hexadecimal, the process is similar except that the binary number is divided into groups of 4.
(10110001101011.11110010) 2 -> (2C6B.F2) 16
The conversion to hexadecimal in binary is done with an inverse procedure to the previous one, that is; each octal digit becomes its three-digit binary equivalent and each hexadecimal digit becomes its four-digit binary equivalent.
Conversion to decimal system
For this, the successive power multiplication method is used.
From binary to decimal
Convert the number 111100112 to decimal. The binary number contains eight digits, so a sum is made of each digit multiplied by 2 raised to the corresponding power starting with 0, 1, 2 … n, up to the last digit.
This operation is performed from right to left.
In this case it would be: 1×20 + 1×21 + 0x22 + 0x23 + 1×24 + 1×25 + 1×26 + 1×27 = 1 + 2 + 0 + 0 + 16 + 32 + 64 + 128 = 243 So the number 111100112 equals 243 in the decimal system.
From octal to decimal
It is done in a similar way to the previous one but taking into account that the base of the system is now eight (8).
322258 = 5×80 + 2×81 + 2×82 + 2×83 + 3×84 = 5 + 16 + 128 + 1024 + 12288 = 5 + 16 + 128 + 1024 + 12288 = 13461
From hexadecimal to decimal
This operation is performed from right to left. Each digit is taken, multiplied by the base 8 raised first to 0, one, ect. The value of F in the decimal system is 15.
We begin: FF16 = F x 160 + F x 161 = 15 x 160 + 15 x 161 = 15 x 1 + 15 x 16 = 15 + 240 = 255
So the base 16 FF number is equivalent to the base 10 255.
Binary sum
All operations are done through the binary sum (fundamental operation).
Sum table
| TO | B | Sum | Haulage |
| 0 | 0 | 0 | 0 |
| 0 | one | one | 0 |
| one | 0 | one | 0 |
| one | one | 0 | one |
As observed in the table, there is only carry when the value of one (1) is given to the two variables.
Binary subtraction
The subtraction consists of a negated sum (that is, its complement). If you want to subtract A – B , for example, the steps to follow would be:
- We take the number Aas it is.
- Number B
- A sum of both values is made.
- 1 is added to the result.
- The carry is removed.
