=head1 NAME

Binary Decimal Hexadecimal - How does it work

=for html <div align="right"><a href="bin_dec_hex.pdf">PDF</a> version.</div>

=head1 DESCRIPTION

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 just before this. 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 as crypto to you, this is what I've 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 should be incremented, it is reset to 0 and the
position before is incremented. 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". For zero's we write a 
space if it is not the only digit (so: number 0) and if it is the first
digit: "00010" -> " 0010" -> "  010" -> "   10". It is not "   1 ".

This was pretty basic, you already knew this. Why did I tell it ?
Well, computers do not represent numbers with 10 different digits. They
know of only two different symbols, being 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. It is
only 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 speak about the numbering system used, we're
using the decimal system. If we are talking about another numbering
system, we'll have to make that clear. There are a few wide-spread
methods to do so. One common form is to write 1010(2) which means that
you wrote down a number in the binary form. It is the number ten.
If you would write 1010 it means the number one thousand and ten.

In books, another form is most used. It uses subscript (little chars,
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
with a little two just behind it.

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

For the binary system, is is common to write leading zero's. 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 representations. The number 65535 would
be written down as 1111111111111111(2) which is 16 times the digit 1.
This is difficult and prone to errors. Therefore we normally would use
another base, called hexadecimal. It uses 16 different symbols. First
the symbols from the decimal system are used, thereafter we continue
with the 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 easy (and back).

There is yet another system in use, called the octal system. This was
more common in the old days but not anymore. You will find it in use
on some places so get used to it. The same story applies, but now with
only 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 preferred 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. Each time take four digits from the right and make
a hexadecimal digit from it (see the table above). Stop when there are
no more digits.
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 get 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 the way they are written 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, (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 can only use two
symbols. Another example:

%1010 would be

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

It would have been more easy 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 one you could use the next
method. It will take some time to be able to do the estimates but it will
be more and more easy when you use the system more frequent. Another way
is presented to you thereafter.

First you will need to know how many positions will be used in the other
system. To do so, you need to know the maximum numbers. Well, that's not
so 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 too:

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

If a number is smaller than 65536 it will thus fit in four positions.
If the number is bigger than 4095, you will need to use position 4.
How many times can you take 4096 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.

Number is 41029. 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 41029 ten times without going below zero.
The leftmost digit will be "A" so we have 0xA????.
The number is reduced to 41029 - 10*4096 = 41029-40960 = 69.
69 is smaller than 16^3 but not bigger than 16^2-1. The second digit
is therefore "0" and we know 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?.
Take 64 from 69 (69 - 4*16) and the last digit is 5 --> 0xA045.

The other method builds the number from the right. Take again 41029.
Divide by 16 and do not use fractions (only whole numbers).

 41029 / 16 is 2564 with a remainder of 5. Write down 5.
 2564 / 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. Personally
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 then
I will use the first method (like 32770, translate into 32768 + 2 and
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 simple if you should write down
a zero or a one: if you divide by two the remainder will be zero if it
was an even number and one if it was 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 in the base that is used.
In any representation it is the same, add the numbers on the right,
write down the rightmost digit from the result, remember the other
digits and use them in the next round. Continue with the second digits
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, check them with
a calculator. The more you do the calculations yourself, the more you
find that you didn't make mistakes. In the end, you'll do calculi in
other bases as easy as you do 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
called so as it is short for Binary Calculator. It calculates not only
in decimal, but in all bases you'll ever use (among them Binary).

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

=head1 AUTHOR

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@ergens.op.het.net>
