Numeral radixes
Decimal, Hexadecimal and Octal

Numeral radixes

All of us we are accustomed since youngs to use decimal numbers to express quantities. This nomenclature that seems so logical to us may not seems it to a Roman of the classic Rome. For them each symbol that they put to express a number always represented the same value:
I      1
II     2
III    3
IV     4
V      5
If one pays attention all the I signs always represents the value 1 (one) wherever they be placed, like the V sign always represents our 5 (five). Nevertheless that does not take place in our decimal system. When we write the decimal symbol 1 we are not always talking about value 1 (I in Roman numbers). For example:
  1    I
 10    X
100    C
In these cases our symbol 1 does not always has a value of 1 (I in Roman numbers). For example, in the second case, the symbol 1 represents the value 10 (ten, X in Roman) and in the third one, 1 represents the value 100 (one hundred, C).

Another example:

is not equivalent to 2+7+5, rather could be decomposed as 200+70+5:
+ 70
therefore, the first sign 2 is equivalent to 200 (2 x 100), the second sign, 7 is equivalent to 70 (7 x 10) whereas the last sign corresponds to value 5 (5 x 1).

All the previous can be mathematically represented in a very simple way. For example, to represent the value 182736 we can assume that each digit is the product of itself multiplied by 10 powered to its place as exponent, beginning from the right with 100, following with 101, 102, and so on:

Octal numbers (radix 8)

Like our "normal" numbers are radix 10 because we have 10 different digits (from the 0 to the 9):
the octals numbers include only the signs from the 0 to the 7:
and, therefore, its mathematical radix is 8. In C++ octal numbers have the peculiarity that they always they begin by a 0 digit. Let us see how we would write the first numbers in octal:
octal  decimal
-----  -------
  0       0   (zero)
 01       1   (one)
 02       2   (two)
 03       3   (three)
 04       4   (four)
 05       5   (five)
 06       6   (six)
 07       7   (seven)
010       8   (eight)
011       9   (nine)
012      10   (ten)
013      11   (eleven)
014      12   (twelve)
015      13   (thirteen)
016      14   (fourteen)
017      15   (fifteen)
020      16   (sixteen)
021      17   (seventeen)
Thus, for example, number 17 (seventeen, or XVII in Romans) it is expressed 021 as an octal number.

We can apply the scheme that we saw previously with the decimal numbers to the octal numbers simply by considering that its radix is 8. For example, taking the octal number 071263:

therefore the octal numbers 071263 it is expressed as 29363 in decimal numbers.

Hexadecimal numbers (radix 16)

Like decimal numbers have 10 different digits to be represented (0123456789) and octal numbers have 8 (01234567), hexadecimal numbers have 16: numbers from 0 to 9 and letters A, B, C, D, E and F that together they serve us to represent the 16 different symbols that we need:
hexadecimal  decimal
-----------  -------
      0         0   (zero)
    0x1         1   (one)
    0x2         2   (two)
    0x3         3   (three)
    0x4         4   (four)
    0x5         5   (five)
    0x6         6   (six)
    0x7         7   (seven)
    0x8         8   (eight)
    0x9         9   (nine)
    0xA        10   (ten)
    0xB        11   (eleven)
    0xC        12   (twelve)
    0xD        13   (thirteen)
    0xE        14   (fourteen)
    0xF        15   (fifteen)
   0x10        16   (sixteen)
   0x11        17   (seventeen)
Once again we can use the same method to translate a number from a base to another one:

Binary representations

Octal and hexadecimal numbers have a considerable advantage over our decimal numbers in the world of bits, and is that their bases (8 and 16) are perfect multiples of 2 (23 and 24) which allows us to make easier conversions to binary than with decimal numbers (whose base is 2x5). For example, suppose that we want to pass the following binary sequence to numbers of other bases:
In order to pass it to decimal we would need to conduct a mathematical operation similar to the one we have used previously to convert from hexadecimal or octal, which would give us the decimal number 212628.

Nevertheless to pass this sequence to octal it will only take to us some seconds and we can do it just seeing it: Since 8 is 23, we will separate the binary value in groups of 3 numbers:

110 011 111 010 010 100
and now we just have to translate to octal numberal radix each group separately:
110 011 111 010 010 100
 6   3   7   2   2   4
giving the number 637224 as result. This same process can be inversely performed to pass from octal to binary.

In order to conduct the operation with hexadecimal numbers we only have to perform the same process but separating the binary value in groups of 4 numbers (16 = 24):

11 0011 1110 1001 0100
3    3    E    9    4
Therefore, the binary expression 110011111010010100 can be represented in C++ as 212628 (decimal), as 0637224 (octal) or as 0x33e94 (hexadecimal).

The hexadecimal code is specially interesting in computer science since nowadays computers are based on bytes composed of 8 binary bits and therefore each byte matches with the rank that 2 hexadecimal numbers can represent. For that reason is the most used type when representing values translated from binary.

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