_B_I_N___D_E_C___H_E_X(1)                       rrdtool                      _B_I_N___D_E_C___H_E_X(1)

NNAAMMEE
     bin_dec_hex - How to use binary, decimal, and hexadecimal notation.

DDEESSCCRRIIPPTTIIOONN
     Most  people use the decimal numbering system. This system uses ten symbols
     to represent numbers. When those ten symbols are used up,  they  start  all
     over  again  and  increment  the  position to the left. The digit 0 is only
     shown if it is the only symbol in the sequence, or if it is not  the  first
     one.

     If this sounds cryptic to you, this is what I've just said in numbers:

          0
          1
          2
          3
          4
          5
          6
          7
          8
          9
         10
         11
         12
         13

     and so on.

     Each  time the digit nine is incremented, it is reset to 0 and the position
     before (to the left) is incremented (from 0 to 1). Then  number  9  can  be
     seen as "00009" and when we should increment 9, we reset it to zero and in‐
     crement  the digit just before the 9 so the number becomes "00010". Leading
     zeros we don't write except if it is the only  digit  (number  0).  And  of
     course,  we  write  zeros  if they occur anywhere inside or at the end of a
     number:

      "00010" -> " 0010" -> " 010" -> "  10", but not "  1 ".

     This was pretty basic, you already knew this. Why did  I  tell  it?   Well,
     computers  usually  do not represent numbers with 10 different digits. They
     only use two different symbols, namely "0" and "1". Apply the same rules to
     this set of digits and you get the binary numbering system:

          0
          1
         10
         11
        100
        101
        110
        111
       1000
       1001
       1010
       1011
       1100
       1101

     and so on.

     If you count the number of rows, you'll see that these are again 14 differ‐
     ent numbers. The numbers are the same and mean the same  as  in  the  first
     list,  we just used a different representation. This means that you have to
     know the representation used, or as it is called the  numbering  system  or
     base.  Normally, if we do not explicitly specify the numbering system used,
     we implicitly use the decimal system. If we want to use any other numbering
     system, we'll have to make that clear. There are a few widely adopted meth‐
     ods  to  do  so.  One  common form is to write 1010(2) which means that you
     wrote down a number in its binary representation. It is the number ten.  If
     you would write 1010 without specifying the base, the number is interpreted
     as one thousand and ten using base 10.

     In  books,  another  form is common. It uses subscripts (little characters,
     more or less in between two rows). You can leave  out  the  parentheses  in
     that case and write down the number in normal characters followed by a lit‐
     tle two just behind it.

     As the numbering system used is also called the base, we talk of the number
     1100 base 2, the number 12 base 10.

     Within  the binary system, it is common to write leading zeros. The numbers
     are written down in series of four, eight or sixteen depending on the  con‐
     text.

     We  can  use the binary form when talking to computers (...programming...),
     but the numbers will have large representations. The number  65'535  (often
     in  the  decimal  system a ' is used to separate blocks of three digits for
     readability) would be written down as 1111111111111111(2) which is 16 times
     the digit 1.  This is difficult and prone to errors. Therefore, we  usually
     would  use  another base, called hexadecimal. It uses 16 different symbols.
     First the symbols from the decimal system are used, thereafter we  continue
     with  alphabetic  characters. We get 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C,
     D, E and F. This system is chosen because the hexadecimal form can be  con‐
     verted into the binary system very easily (and back).

     There  is yet another system in use, called the octal system. This was more
     common in the old days, but is not used very often anymore.  As  you  might
     find  it  in use sometimes, you should get used to it and we'll show it be‐
     low. It's the same story as with the other representations, but with  eight
     different symbols.

      Binary      (2)
      Octal       (8)
      Decimal     (10)
      Hexadecimal (16)

      (2)    (8) (10) (16)
      00000   0    0    0
      00001   1    1    1
      00010   2    2    2
      00011   3    3    3
      00100   4    4    4
      00101   5    5    5
      00110   6    6    6
      00111   7    7    7
      01000  10    8    8
      01001  11    9    9
      01010  12   10    A
      01011  13   11    B
      01100  14   12    C
      01101  15   13    D
      01110  16   14    E
      01111  17   15    F
      10000  20   16   10
      10001  21   17   11
      10010  22   18   12
      10011  23   19   13
      10100  24   20   14
      10101  25   21   15

     Most computers used nowadays are using bytes of eight bits. This means that
     they  store  eight  bits at a time. You can see why the octal system is not
     the most practical for that: You'd need three digits to represent the eight
     bits and this means that you'd have to use one complete digit to  represent
     only  two bits (2+3+3=8). This is a waste. For hexadecimal digits, you need
     only two digits which are used completely:

      (2)      (8)  (10) (16)
      11111111 377  255   FF

     You can see why binary and hexadecimal can be converted quickly:  For  each
     hexadecimal digit there are exactly four binary digits.  Take a binary num‐
     ber:  take  four digits from the right and make a hexadecimal digit from it
     (see the table above). Repeat this until there are no more digits. And  the
     other way around: Take a hexadecimal number. For each digit, write down its
     binary equivalent.

     Computers  (or  rather  the parsers running on them) would have a hard time
     converting a number like 1234(16). Therefore hexadecimal numbers are speci‐
     fied with a prefix. This prefix depends on the language you're writing  in.
     Some  of  the prefixes are "0x" for C, "$" for Pascal, "#" for HTML.  It is
     common to assume that if a number starts with a zero, it is octal. It  does
     not matter what is used as long as you know what it is. I will use "0x" for
     hexadecimal,  "%"  for binary and "0" for octal.  The following numbers are
     all the same, just their representation (base) is different:  021  0x11  17
     %00010001

     To  do  arithmetics  and conversions you need to understand one more thing.
     It is something you already know but perhaps you do not "see" it yet:

     If you write down 1234, (no prefix, so it is decimal) you are talking about
     the number one thousand, two hundred and thirty four. In sort of a formula:

      1 * 1000 = 1000
      2 *  100 =  200
      3 *   10 =   30
      4 *    1 =    4

     This can also be written as:

      1 * 10^3
      2 * 10^2
      3 * 10^1
      4 * 10^0

     where ^ means "to the power of".

     We are using the base 10, and the positions 0,1,2 and  3.   The  right-most
     position should NOT be multiplied with 10. The second from the right should
     be multiplied one time with 10. The third from the right is multiplied with
     10 two times. This continues for whatever positions are used.

     It is the same in all other representations:

     0x1234 will be

      1 * 16^3
      2 * 16^2
      3 * 16^1
      4 * 16^0

     01234 would be

      1 * 8^3
      2 * 8^2
      3 * 8^1
      4 * 8^0

     This  example  can not be done for binary as that system only uses two sym‐
     bols. Another example:

     %1010 would be

      1 * 2^3
      0 * 2^2
      1 * 2^1
      0 * 2^0

     It would have been easier to convert it to its hexadecimal  form  and  just
     translate  %1010  into  0xA. After a while you get used to it. You will not
     need to do any calculations anymore, but just know that 0xA means 10.

     To convert a decimal number into a  hexadecimal  you  could  use  the  next
     method.  It will take some time to be able to do the estimates, but it will
     be easier when you use the system more frequently. We'll look  at  yet  an‐
     other way afterwards.

     First you need to know how many positions will be used in the other system.
     To  do  so,  you  need  to  know the maximum numbers you'll be using. Well,
     that's not as hard as it looks. In decimal, the maximum number that you can
     form with two digits is "99". The maximum for three: "999". The next number
     would need an extra position. Reverse this idea and you will see  that  the
     number can be found by taking 10^3 (10*10*10 is 1000) minus 1 or 10^2 minus
     one.

     This can be done for hexadecimal as well:

      16^4 = 0x10000 = 65536
      16^3 =  0x1000 =  4096
      16^2 =   0x100 =   256
      16^1 =    0x10 =    16

     If  a  number is smaller than 65'536 it will fit in four positions.  If the
     number is bigger than 4'095, you must use position 4.  How many  times  you
     can  subtract  4'096  from the number without going below zero is the first
     digit you write down. This will always be a number from 1  to  15  (0x1  to
     0xF). Do the same for the other positions.

     Let's try with 41'029. It is smaller than 16^4 but bigger than 16^3-1. This
     means that we have to use four positions.  We can subtract 16^3 from 41'029
     ten  times without going below zero.  The left-most digit will therefore be
     "A", so we have 0xA????.  The number is reduced  to  41'029  -  10*4'096  =
     41'029-40'960  =  69.   69 is smaller than 16^3 but not bigger than 16^2-1.
     The second digit is therefore "0" and we now have 0xA0??.   69  is  smaller
     than 16^2 and bigger than 16^1-1. We can subtract 16^1 (which is just plain
     16) four times and write down "4" to get 0xA04?.  Subtract 64 from 69 (69 -
     4*16) and the last digit is 5 --> 0xA045.

     The  other  method  builds  up  the number from the right. Let's try 41'029
     again.  Divide by 16 and do not use fractions (only whole numbers).

      41'029 / 16 is 2'564 with a remainder of 5. Write down 5.
      2'564 / 16 is 160 with a remainder of 4. Write the 4 before the 5.
      160 / 16 is 10 with no remainder. Prepend 45 with 0.
      10 / 16 is below one. End here and prepend 0xA. End up with 0xA045.

     Which method to use is up to you. Use whatever works for you.  I  use  them
     both without being able to tell what method I use in each case, it just de‐
     pends  on  the number, I think. Fact is, some numbers will occur frequently
     while programming. If the number is close to one I am familiar with, then I
     will use the first method (like 32'770 which is into 32'768 + 2 and I  just
     know that it is 0x8000 + 0x2 = 0x8002).

     For binary the same approach can be used. The base is 2 and not 16, and the
     number  of positions will grow rapidly. Using the second method has the ad‐
     vantage that you can see very easily if you should write down a zero  or  a
     one:  if you divide by two the remainder will be zero if it is an even num‐
     ber and one if it is an odd number:

      41029 / 2 = 20514 remainder 1
      20514 / 2 = 10257 remainder 0
      10257 / 2 =  5128 remainder 1
       5128 / 2 =  2564 remainder 0
       2564 / 2 =  1282 remainder 0
       1282 / 2 =   641 remainder 0
        641 / 2 =   320 remainder 1
        320 / 2 =   160 remainder 0
        160 / 2 =    80 remainder 0
         80 / 2 =    40 remainder 0
         40 / 2 =    20 remainder 0
         20 / 2 =    10 remainder 0
         10 / 2 =     5 remainder 0
          5 / 2 =     2 remainder 1
          2 / 2 =     1 remainder 0
          1 / 2 below 0 remainder 1

     Write down the results from right to left: %1010000001000101

     Group by four:

      %1010000001000101
      %101000000100 0101
      %10100000 0100 0101
      %1010 0000 0100 0101

     Convert into hexadecimal: 0xA045

     Group %1010000001000101 by three and convert into octal:

      %1010000001000101
      %1010000001000 101
      %1010000001 000 101
      %1010000 001 000 101
      %1010 000 001 000 101
      %1 010 000 001 000 101
      %001 010 000 001 000 101
         1   2   0   1   0   5 --> 0120105

      So: %1010000001000101 = 0120105 = 0xA045 = 41029
      Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029(10)
      Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029

     At first while adding numbers, you'll convert them to  their  decimal  form
     and  then  back  into their original form after doing the addition.  If you
     use the other numbering system often, you will see that you'll be  able  to
     do arithmetics directly in the base that is used.  In any representation it
     is  the same, add the numbers on the right, write down the right-most digit
     from the result, remember the other digits and use them in the next  round.
     Continue with the second digit from the right and so on:

         %1010 + %0111 --> 10 + 7 --> 17 --> %00010001

     will become

         %1010
         %0111 +
          ||||
          |||+-- add 0 + 1, result is 1, nothing to remember
          ||+--- add 1 + 1, result is %10, write down 0 and remember 1
          |+---- add 0 + 1 + 1(remembered), result = 0, remember 1
          +----- add 1 + 0 + 1(remembered), result = 0, remember 1
                 nothing to add, 1 remembered, result = 1
      --------
        %10001 is the result, I like to write it as %00010001

     For low values, try to do the calculations yourself, then check them with a
     calculator. The more you do the calculations yourself, the more you'll find
     that you didn't make mistakes. In the end, you'll do calculi in other bases
     as easily as you do them in decimal.

     When  the numbers get bigger, you'll have to realize that a computer is not
     called a computer just to have a nice name. There are many different calcu‐
     lators available, use them. For Unix you could use "bc" which is short  for
     Binary  Calculator.  It  calculates  not  only in decimal, but in all bases
     you'll ever want to use (among them Binary).

     For  people  on  Windows:  Start  the  calculator  (start->programs->acces‐
     sories->calculator) and if necessary click view->scientific. You now have a
     scientific calculator and can compute in binary or hexadecimal.

AAUUTTHHOORR
     I  hope  you  enjoyed  the examples and their descriptions. If you do, help
     other people by pointing them to this document when they are  asking  basic
     questions.  They will not only get their answer, but at the same time learn
     a whole lot more.

     Alex van den Bogaerdt  <alex@vandenbogaerdt.nl>

1.11.0                             2026-08-08                     _B_I_N___D_E_C___H_E_X(1)
