BIN_DEC_HEX(1)               rrdtool               BIN_DEC_HEX(1)



NNAAMMEE
       bin_dec_hex - How to use binary, decimal, and hexadecimal
       notation.

DDEESSCCRRIIPPTTIIOONN
       Most people use the decimal numbering system. This system
       uses ten symbols to represent numbers. When those ten sym-
       bols 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 increment 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 sym-
       bols, 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 different numbers. The numbers are the same and
       mean the same as in the first list, we just used a differ-
       ent representation. This means that you have to know the
       representation used, or as it is called the numbering sys-
       tem 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 methods 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 (lit-
       tle 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 little 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 context.

       We can use the binary form when talking to computers
       (...programming...), but the numbers will have large rep-
       resentations. The number 65'535 (often in the decimal sys-
       tem 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, there-
       after 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 converted
       into the binary system very easily (and back).

       There is yet another system in use, called the octal sys-
       tem. 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 below. 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 number: take four digits
       from the right and make a hexadecimal digit from it (see
       the table above). Repeat this until there are no more dig-
       its. 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). There-
       fore hexadecimal numbers are specified 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 represenatation (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 per-
       haps 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 symbols. 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 another 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 tak-
       ing 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 posi-
       tions.  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 ub the number from the right.
       Let's try 41'029 again.  Divide by 16 and do not use frac-
       tions (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 depends 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 advantage 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 number 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 calcula-
       tions 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 calculators 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->pro-
       grams->accessories->calculator) and if necessary click
       view->scientific. You now have a scientific calculator and
       can compute in binary or hexadecimal.

AAUUTTHHOORR
       I hope you enjoyed the examples and their descriptions. If
       you do, help other people by pointing them to this docu-
       ment 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@ergens.op.het.net>



1.2.15                      2006-07-14             BIN_DEC_HEX(1)
