pun intended) on zeros and ones?What Is "Hexadecimal" ?
and
"Hexa... What?" Well, that's hexa- (six) plus decimal (ten), or sixteen.
It stands for the Base 16 Number System. Programmers often refer to it as simply, Hex (which by itself should mean
only six, such as the number of sides that a "Hex wrench" has, but it almost always means Hexadecimal if the subject is computer
software rather than hardware).
You may already know that Binary (or Base 2) is what computers
actually use to carry out calculations. Octal is Base 8, and Decimal, of course, is Base 10. And just as we
have ten digits, 0 through 9, in the Decimal system, there are sixteen digits we could use for Hexadecimal (0-9
and six more).
The Basics of Hex
The following tables show how the first six letters of the Latin alphabet (which are the same for most European languages; including English) are used as symbols in Hex so we can have a total of 16 Hexadecimal digits. Though it's now standard practice to use the capitals A through F (or lower-case forms if their meaning is unambiguous) to symbolize the Tenth through Sixteenth digits of the Hexadecimal Number System. Remember, this is only a convention. Any other set of 6 symbols (such as letters from Arabic, Korean or Japanese) could function in the same way; just as we could use some other symbols for the Decimal digits 0-9.
Hexadecimal Decimal | Hexadecimal Decimal
------------- ----------- | ----------- -------
0 through 9 0 through 9 |
| 1D (1 x 16) + 13 = 29
A ( 9 + 1 = ) 10 | 1E " + 14 = 30
B ( 9 + 2 = ) 11 | 1F " + 15 = 31
C ( 9 + 3 = ) 12 | 20 (2 x 16) + 0 = 32
D ( 9 + 4 = ) 13 | 21 " + 1 = 33
E ( 9 + 5 = ) 14 | 22 " + 2 = 34
F ( 9 + 6 = ) 15 | 23 " + 3 = 35
10 (1 x 16) + 0 = 16 | 24 " + 4 = 36
11 (1 x 16) + 1 = 17 | 25 " + 5 = 37
12 " + 2 = 18 | 26 " + 6 = 38
13 " + 3 = 19 | 27 " + 7 = 39
14 " + 4 = 20 | 28 " + 8 = 40
15 " + 5 = 21 | 29 " + 9 = 41
16 " + 6 = 22 | 2A " + 10 = 42
17 " + 7 = 23 | 2B " + 11 = 43
18 " + 8 = 24 | 2C " + 12 = 44
19 " + 9 = 25 | 2D " + 13 = 45
1A " + 10 = 26 | 2E " + 14 = 46
1B " + 11 = 27 | 2F " + 15 = 47
1C " + 12 = 28 | 30 (3 x 16) + 0 = 48
|
Just as the location of the digits in a Decimal number represent the amount of ones, tens, hundreds, thousands, etc. in that number; which are the powers of 10. In a similar way, the digits in a Hex number stand for the ones, sixteens (16 to the power of 1), how many 256s (16 to the power of 2), the amount of 4096s (16 to the power of 3), etc. in its numbers.
The following table includes some important(*) Hex numbers in the PC industry; such as 200h (a small h, or capital H, at the end of a number means it's a Hexadecimal number; to prevent it from being mistaken as a Decimal number).
Hexadecimal Decimal Hexadecimal Decimal
----------- ------- ----------- -------
30 (3x16) + 0 = 48 | 100 (1x256) = 256
40 (4x16) + 0 = 64 |(*) 200 (2x256) = 512
50 (5x16) + 0 = 80 | 400 (4x256) = 1024
64 (6x16) + 4 = 100 | 500 (5x256) = 1280
(*) 7F (7x16) +15 = 127 | 800 (8x256) = 2048
80 (8x16) + 0 = 128 | A00 (10x256) = 2560
90 (9x16) + 0 = 144 | FFF 4095
9F (9x16) +15 = 159 | 1000 (1x4096) = 4096
A0 (10x16) + 0 = 160 | 4000 (4x4096) = 16,384
B0 (11x16) + 0 = 176 | 5000 (5x4096) = 20,480
F0 (15x16) + 0 = 240 | 8000 (8x4096) = 32,768
(*) FF (15x16) +15 = 255 |(*) FFFF 65,535
|
FFFFh = (15x4096) + (15x256) + (15x16) + 15
= 65,535 ; which is the largest value that can be represented within sixteen Binary digits. 200h is the Hex equivalent for 512, as
in: bytes per sector on both floppy and hard disks. Multiples of these sectors are sometimes called clusters.
The format for a Windows™ 2000, XP, 7 or later OS partition is usually NTFS; which often has 4 KiB clusters (or 8
"sectors per cluster"). Files are always stored on your computer (whether under the older FAT or NTFS file
systems) by dividing them up into clusters. The last cluster is almost never completely filled in by a file (and that unused part
of the last cluster is often called the file's slack space). Just before FAT32 was implemented, many people had hard drives that
used 32 KiB clusters; with lots of wasted space on them. The space you save by using NTFS with a Windows™ OS file system
rather than FAT32 can be huge since the size of new drives back then had already reached triple digits of Gigabytes; but
can easily be a Terabyte or two or even 3 today!
| So, what does a Base 16 number system have
to do with a machine that's based ( pun intended) on zeros and ones? |
Do any of those numbers above look familiar to you, specifically the number 1024 and its multiples? It's more than just a coincidence that this is equal to the term we call a Binary Kilobyte (1024 bytes, or 2 to the power of 10 or 1 KiB). The 2 Kib, 4 Kib and 16 Kib multiples of this Binary number should now be evident in the Table above. 10,000 Hex (FFFFh + 1) is equivalent to 64 Binary KB (65,536 bytes, or 2 to the power of 16) which is the largest size a true .COM program* can have.
After laying some groundwork from the past, we'll return to these observations and answer that important question in the box above! Hopefully you'll enjoy the following commentary on highlights from the history of computing.
_____________________
*Perhaps you've heard of the file COMMAND.COM (it's the kernel for the MS-DOS 7.1
Operating System on a Windows™ 98 Boot Disk). Well it's really an .EXE program! Microsoft®
simply felt it had to retain the same name for this file, most likely for compatibility reasons; it hasn't been an actual .COM
file since Windows™ 95 was released. A true .COM program can never be more than 64 KiB in size, and does
not begin with the letters "MZ" (which
you'll find at the beginning of this COMMAND.COM file and all other .EXE files!).
A Quick Look at the Amazing History of Computers
Over half a century ago, hot, room-sized machines built with vacuum tubes and relays were already using the same concepts we still use today to carry out instructions on numbers stored as "bits" (Binary digits) in structures known as registers. Today, even our hand-held programmable calculators could blow away those old computers in speed if not in results (number of digits). And the registers have shrunk from the size of handfuls of light bulbs to a space so tiny you need an electron microscope to see them; yet they still continue to add, subtract and compare bits by the same logical methods already known long before those now archaic machines could have been built.
Programs were often stored and fed into those first computers from patterns of
holes punched into a roll of paper tape (Punched Tape). This idea came from
an earlier device called the teletype (or teletypewriter or
teleprinter), which used a seven-bit code we now call ASCII to send
alphanumeric data over wires. Essentially this same bit code or expansions upon it, such as Unicode, are still used to store
data on a PC. Designed for communications, the original code included control characters to signal the start of, success, failure,
and end of transmissions. The ASCII code 7, was used to ring a real bell at a distant station, thus it is often abbreviated as
"bell" in an ASCII chart. Programs now use it to beep a computer's speaker!
Some of the 33 control codes used decades ago, such as ASCII 8 (a Backspace), 10 [0Ah] (a Line Feed), and
13 [0Dh] (a Carriage-Return) are still fairly close to their original meanings, while others were so "device dependent"
that modern day equivalents no longer exist.
Seven bits gives us a total of 128 different codes in the
standard ASCII character set. That's 64 + 32 + 16 + 8 + 4 + 2 + 1 or 127, plus the zero. At a much later
date, companies such as IBM® and Apple® introduced their own Extended ASCII character sets; which were eventually followed
by Unicode.
The revolution in electronics which made home computing possible, came about when integrated circuit companies, such as Motorola®,
Intel®, Rockwell® and Zilog®, began mass producing 8-bit microprocessors in the early Seventies. The Motorola® MC6800 series made the loveable little Apple™ computers an instant
success, and the Zilog® Z-80™ made the Tandy®
(Radio Shack) TRS-80 ™ a very common product. Finally, in 1979,
Intel's 4.77 MHz 8088 chip (a 16-bit CPU with only an 8-bit bus)
found a home inside the original IBM® Personal Computer™ [Note: The Intel 8086, with a full 16-bit bus, had already been produced in 1978, but IBM
considered it too expensive to use for their first PC project.] In 1984, IBM introduced it's AT (Advanced
Technology) series based on Intel's 80286 CPU. That computer (made by a
business rather than hobby company), may have been what convinced many managers and CEOs that microcomputers would soon become the
future of computing in the business world. (Well, that and the fact that further technological advances allowed PCs
to be "Networked" together through server machines! Once they became widely used in large businesses, the
PCs tended to cause data which had only been stored in the tape libraries of main-frame computers to become
distributed among "workstations" as they're now called; rather than those locations simply being "dumb"
terminals that had to access what some still thought of as a "real computer" in order to do something useful.)
Instead of making its own Disk Operating System, IBM® paid a little software company to put together a DOS for their new IBM Personal Computers. But, IBM did not pay for exclusive rights to any future uses of this DOS. And when that little company (called Microsoft®) made some changes and started selling MS-DOS™ to anyone, it wasn't long until a flood of "AT clones" and "IBM - compatibles" spread across the globe! Microsoft and Intel® made a fortune selling the essential software and hardware that made up a PC, but IBM simply acted like its usual "big blue" self over the losses in home and small business sales. I'm sure that IBM now wishes they had done things differently, but perhaps that's what they deserved for making a calculated decision to promote Microsoft's DOS instead of an 8086 version of CP/M by Gary Kildall (which later became Digital Research DOS). And in spite of what you may read elsewhere, Gary never 'lost out' to Bill Gates due to any personal differences or missed meetings! It was simply a business decision by IBM; which appears to have been done on purpose since IBM's DOS could only be used on their machines, whereas Gary's OS was almost machine independent. It's a little known fact (these days) that IBM actually licensed both DOS and CP/M-86 for their PC, but you had to special order CP/M-86 rather than having it come at a much lower price with most PCs as IBM's DOS did; that was clearly a business decision on the part of IBM!
Bits, Bytes, Characters and Words
Data in these new micro computers was contained in 8-bit units
called "bytes" (a combination of the word bit and to bite; a 4-bit unit is often humorously called a
"nibble" ). IBM® later added an extended-ASCII character set for their PCs,
which doubled the size from 128 to 256 codes. Inside the CPU (the processor) itself, there were also registers of 16-bit
lengths. The machine code which ran the CPU could have instructions of this size too, or even larger, arriving at the CPU in a number
of successive bytes. It was around this time, that the term "word" took on its usual meaning of two eight-bit
bytes. So, how did they manipulate such large binary numbers without making too many mistakes?
Why Octal and Hexadecimal are used On Computers
Long before this, programmers regularly used a convenient method to handle large binary numbers in either 3-bit or 4-bit groupings. Most people find it quite easy to see what decimal numbers are represented by only three bits; for example, 101 is 5, 011 is 3, etc. If we use the 3-bits at a time method to convert the binary number 111000101000 into groups of 3-bit decimal digits, we obtain: 7 0 5 0. "If you needed to turn this into an Octal (Base 8) number, what more would you need to do?" Nothing! The digits 7050 are the Octal equivalent of this Binary number; it's almost too simple. But to convert either of them into Decimal is a lot more difficult for most of us; yet it's quite often unnecessary for a programmer to do so.
Not long after computers needed to be programmed, the actual machine code
for the computer instructions was replaced by mnemonics (first m is silent when pronouncing this word; it essentially
means abbreviations). Three or four letters (at first), were used in what was called the Assembly Language for a particular CPU;
thus, today we have the x86 Assembly code (in various forms) for Intel-based computers. It was also possible to use more than one base
numeration for writing data in these Assembly languages, so programmers made sure their Assemblers [the programs which convert Assembly
language into a particular machine code] could understand Octal as well as Binary numbers.
Well, what about Hexadecimal? The same Binary number above, grouped into four bits at a time, will give us the
Hex numerals, E28. Now we're finally getting to the bottom of that important
question above; and the main reason I had wanted to write this essay.
The maximum number a four-bit Binary can represent is 15. Exactly the same as a single Hex digit. And if you sum up the next four higher bits in a Binary number having eight bits, for example, the ones in 11110000, you'll get 16 + 32 + 64 + 128 = 240, or simply 15 times 16. And the next four bits beyond that will give you 15 times 256, or 3840.
Let's compare the differences between grouping the Binary digits in sets of fours (for Hexadecimal) and threes (for Octal) in a more graphic manner :
Binary: | 2048 1024 512 256 | 128 64 32 16 | 8 4 2 1 | = 1111 1111 1111 HEX : | 256's | 16's | 1's | = F F F ------------------------------------------------------------------------- Computing (15 x 256) + (15 x 16 ) + 15 Decimal 3840 + 240 + 15 = 4 0 9 5 equivalent ========================================================================= Binary: |2048 1024 512 | 256 128 64 | 32 16 8 | 4 2 1 | = 111 111 111 111 OCTAL: | 512's | 64's | 8's | 1's | = 7 7 7 7 ------------------------------------------------------------------------- Computing ( 7 x 512 ) + ( 7 x 64 ) + (7 x 8) + 7 Decimal 3584 + 448 + 56 + 7 = 4 0 9 5 eqivalent ========================================================================= F F F h = 7 7 7 7 Octal = 4 0 9 5 (Decimal). |
This shows that it's almost as easy to extract a Hexadecimal number from a Binary
as it was to obtain its Octal eqivalent. We simply group the Binary bits by four digits instead of three.
Conclusion
There are two important aspects to the beauty of using Hexadecimal with computers: First, it can represent 16-bit words in only four Hex digits, or 8-bit bytes in just two; thus, by using a numeration with more symbols, it is both easier to work with (saving paper and screen space) and makes it possible to understand some of the vast streams of data inside a computer merely by looking at the Hex output. This is why programs such as DEBUG, use only Hexadecimal to display the actual Binary bytes of a Memory Dump rather than a huge number of ones and zeros!
The second aspect is closely related: Whenever it is necessary to convert the Hex representation back into the actual Binary bits, the process is simple enough to be done in your own mind. For example, FAD7 hex is 1111101011010111 (F=1111, A=1010, D=1101, 7=0111) in Binary. The reason one might wish to do this is in order to work with "logical" (AND, OR or XOR) or "bit-oriented" instructions (Bit tests, etc.) which may make it easier (at times) for a programmer to comprehend.
For example, if you wanted to logically AND the hex number FAD7 with D37E, you might have a difficult time without first changing these numbers into Binary. If you jot them out in Binary on scratch paper, the task will be much easier:
FAD7(hex) = 1 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1
D37E(hex) = 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0
------------------------------------
ANDing gives: 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0
the Answer: D 2 5 6
(in hex)
|
One final bit of advice: Remember that memory, stored
programs, or codes always begin counting their locations or offsets with a ZERO (0). So, to arrive at a total you must add a one (1) to whatever number you
see in the tables above: Thus, the amount of total memory locations in FF hex or the total length in bytes of a program that ends at offset FFh is 256 (inclusive), not just 255. In this case, an old phrase you
may have heard, "nothing counts," is quite literally true as a zero stands for the first (1st) location you start counting from.
However, any given 8-bit byte of memory, can never hold or count up to anything larger than FF hex, or 255, since a zero in a memory register
represents the idea of nothing being in there (a true nil or null). In this case, the zero in a data
register does not really count, because it is being used as a symbol to convey the idea that there is not anything
in there yet; it's only a void or emptiness!
Along with the advances in Microcomputers, "the Internet" has experienced
many changes as well. A code still used by "Web browsers" today had been invented to transfer information from servers to terminals in
a way that made the Internet a much more effective tool for research. That code is called "Hyper-Text Markup Language" (or HTML)
and it soon included a method which could theoretically reproduce background and text with a total of 16 million, 777 thousand, 216 different
colors.
The hardware available today has already advanced way beyond that limit. The main reason was a push to display pictures
"in living color;" now a common reality. And one of the first things a new computer user should always do is make sure their display
can be set to what's called "24-bit" or "True Color" (for those 16-million plus possible colors). As a matter of fact, every
video card today has the capacity to reproduce what's called "32-bit" color. But all those extra bits are not used for
increasing the number of colors! Why? Well, since the human eye is only
capable of distinguishing something like 7 million or so different colors, that would be a real waste of technology! But
you'll have to look for another page about video cards if you want to know more; our task here is to simply explain the use color with HTML code.
Soon there were two very typical lines of HTML code being used to create background colors:
<body bgcolor="#FFFFFF"> for an all white background and <body bgcolor="#CCCCCC"> for a light-gray backgroundAs a matter of fact, some browsers display an all-white background as their default (when no color is given) whereas others use a light-gray color to display such backgrounds. And "#000000" is used to make pages with an all-black background. The three PRIMARY colors (RED, GREEN and BLUE) were each assigned TWO HEXADECIMAL DIGITS within this six-digit hexadecimal number : Each of the component colors (R-G-B) can, therefore, contribute a maximum of 256 different shades (zero through 255) to the final appearance of the background or text color. Here's a list which shows how easy it is to make six basic colors. The lines shown here contain the HTML "tag" pairs on either side of the TEXT color they create. (For those who may have a problem seeing one or more of the colors, the name also appears at the end of each line in parentheses):
HTML TAG Names
<font color="#FF0000"> RED </font> (RED)
<font color="#00FF00"> GREEN </font> (GREEN) LIME
<font color="#0000FF"> BLUE </font> (BLUE)
<font color="#FFFF00"> YELLOW </font> (YELLOW)
<font color="#FF00FF"> VIOLET </font> (VIOLET) FUCHSIA
<font color="#00FFFF"> CYAN </font> (CYAN) AQUA
You can now see that YELLOW really is a combination of Red and Green. When you
combine Red and Blue it makes Violet (FUCHSIA), and displaying Green and Blue together causes the eye to see CYAN (AQUA). Other colors are formed
by limiting the intensity of one or more of the three Primary Colors. E.g., cutting the values for the Green and Blue of Cyan in half gives you
the default color for the MS-Windows™ 98 Desktop called TEAL (color = "#007F7F" ).
[Note: Although you'll often see the word "Green" as the name for the Primary Color
represented by "#00FF00" above, when it came time to assign the names of colors to HTML "tags" for Netscape's browser, they
ended up calling this color "Lime" and assigned the name Green to half the full value: either "#007F00" (or
"#008000").]
For some reason when the first web browsers (including Netscape) were made, even though video cards
used what was called VGA and could reproduce 256 different colors, some of those colors were not compatible with the web browsers! Later
browsers, such as Microsoft's series, could handle all 256 colors. Rather than have their webpages possibly look bad for some people, many
commercial HTML authors still use a subset of the 256 colors which are called "Browser Safe" or just "Safety" colors.
There are FOUR different webpages here which contain SAMPLES of all 216 colors in
this "safety palette." Each consists of a large table that shows either the Hexadecimal Code or the R-G-B Notation for each color on a
page with either a Black or a White background:
Hexadecimal - White Background
Hexadecimal - Black Background
R-G-B Notation - White Background
and R-G-B Notation - Black Background
[ Here's
the link to that FREE program I promised you. It does HEX and DECIMAL conversions,
shows Hexadecimal and R-G-B Color Notations, allows you to pick a 24-bit "True
Color" from a visual display and also performs logical operations and
conversions on a maximum of 32-bit binaries: B
I T S (Version 1.7)
]
Copyright ©1996, 1997, 1999, 2000, 2002, 2003, 2004, 2020 by Daniel B. Sedory < You can email us here >
Revised on November 7th, 2003. Updated: 25 OCT 2004; 31 MAY 2011.
Last Update: November 1, 2020.
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