=head1 NAME
-bin_dec_hex - How Binary, Decimal, Hexadecimal
-
-=for html <div align="right"><a href="bin_dec_hex.pdf">PDF</a> version.</div>
+bin_dec_hex - How to use binary, decimal, and hexadecimal notation.
=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
+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 said in numbers:
+If this sounds cryptic to you, this is what I've just said in numbers:
0
1
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 ".
+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 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:
+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
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
+different 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 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 (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 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
+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 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).
+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 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.
+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)
(2) (8) (10) (16)
00000 0 0 0
00001 1 1 1
- 00010 2 2 2
+ 00010 2 2 2
00011 3 3 3
00100 4 4 4
00101 5 5 5
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
+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. 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, "$" for Pascal, "#" for HTML.
-It is common to assume that if a number starts with a zero, it is octal.
-It does not matter what is used as long as you know what it is.
-I will use "0x" for hexadecimal, "%" for binary and "0" for octal.
-The following numbers are all the same, just the way they are written is
-different: 021 0x11 17 %00010001
+You can see why binary and hexadecimal can be converted quickly: For
+each hexadecimal digit there are exactly four binary digits. Take a
+binary number: take four digits from the right and make a hexadecimal
+digit from it (see the table above). Repeat this until there are no
+more digits. And the other way around: Take a hexadecimal number. For
+each digit, write down its binary equivalent.
+
+Computers (or rather the parsers running on them) would have a hard
+time converting a number like 1234(16). Therefore hexadecimal numbers
+are specified with a prefix. This prefix depends on the language
+you're writing in. Some of the prefixes are "0x" for C, "$" 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
+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, (so it is decimal) you are talking about the
-number one thousand, two hundred and thirty four. In sort of a formula:
+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 * 8^1
4 * 8^0
-This example can not be done for binary as that system can only use two
+This example can not be done for binary as that system only uses two
symbols. Another example:
%1010 would be
1 * 2^1
0 * 2^0
-It would have been more easy to convert it to its hexadecimal form and
+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.
+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.
+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 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.
+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 too:
+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 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
+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.
-Number is 41029. It is smaller than 16^4 but bigger than 16^3-1. This
+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 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.
+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 know 0xA0??.
+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?.
-Take 64 from 69 (69 - 4*16) and the last digit is 5 --> 0xA045.
+Subtract 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).
+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).
- 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.
+ 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. 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).
-
+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 simple if you should write down
+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
-was an even number and one if it was an odd number:
-
+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
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.
+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 rightmost digit from the result, remember the other
-digits and use them in the next round. Continue with the second digits
+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
--------
%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
+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 easy as you do in decimal.
+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
-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).
+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->programs->accessories->calculator)
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
+basic questions. They will not only get their answer, but at the same
time learn a whole lot more.
-Alex van den Bogaerdt E<lt>alex@ergens.op.het.netE<gt>
+Alex van den Bogaerdt E<lt>alex@vandenbogaerdt.nlE<gt>
projects / rrdtool.git / blobdiff