Convert the Hexadecimal Number Dead Beef to Binary

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3.4 The Hexadecimal Numbering Arrangement

A large problem with the binary system is verbosity. To correspond the value 20210 requires 8 binary digits. The decimal version requires merely three decimal digits and, thus, represents numbers much more compactly than does the binary numbering system. This fact was not lost on the engineers who designed binary computer systems. When dealing with big values, binary numbers apace become likewise unwieldy. Unfortunately, the figurer thinks in binary, so most of the time it is user-friendly to utilise the binary numbering system. Although we can convert between decimal and binary, the conversion is non a trivial task. The hexadecimal (base 16) numbering system solves these problems. Hexadecimal numbers offer the two features we're looking for: they're very compact, and it's unproblematic to catechumen them to binary and vice versa. Because of this, most computer systems engineers apply the hexadecimal numbering organization. Since the radix (base of operations) of a hexadecimal number is sixteen, each hexadecimal digit to the left of the hexadecimal betoken represents some value times a successive power of 16. For example, the number 123416 is equal to:

1 * 163   +   2 * 162   +   3 * xvi1   +   iv * xvi0                            

or

4096 + 512 + 48 + 4 = 466010.             

Each hexadecimal digit can represent ane of sixteen values between 0 and 1510. Since at that place are only ten decimal digits, we need to invent half dozen additional digits to represent the values in the range 10x through 1510. Rather than create new symbols for these digits, we'll employ the letters A through F. The following are all examples of valid hexadecimal numbers:

123416   Dead16   Beefxvi   0AFBxvi   FEED16   DEAF16

Since nosotros'll often demand to enter hexadecimal numbers into the computer system, nosotros'll demand a different machinery for representing hexadecimal numbers. Afterward all, on most computer systems you cannot enter a subscript to denote the radix of the associated value. We'll prefer the following conventions:

• All hexadecimal values begin with a "$" character, e.g., $123A4.
• All binary values begin with a percent sign ("%").
• Decimal numbers do not take a prefix character.
• If the radix is clear from the context, this text may drop the leading "$" or "%" character.

Examples of valid hexadecimal numbers:

$1234  $DEAD  $Beefiness  $AFB  $FEED  $Deaf

As you lot can see, hexadecimal numbers are compact and easy to read. In addition, you can hands convert betwixt hexadecimal and binary. Consider the post-obit table:

Table 4: Binary/Hex Conversion

Binary	Hexadecimal
%0000	$0
%0001	$ane
%0010	$2
%0011	$3
%0100	$4
%0101	$five
%0110	$6
%0111	$7
%1000	$8
%1001	$ix
%1010	$A
%1011	$B
%1100	$C
%1101	$D
%1110	$Eastward
%1111	$F

This table provides all the information you'll ever demand to convert whatsoever hexadecimal number into a binary number or vice versa.

To convert a hexadecimal number into a binary number, just substitute the corresponding four bits for each hexadecimal digit in the number. For instance, to convert $ABCD into a binary value, simply convert each hexadecimal digit co-ordinate to the table above:

0 A B C D Hexadecimal

0000 1010 1011 1100 1101 Binary

To convert a binary number into hexadecimal format is almost as like shooting fish in a barrel. The start footstep is to pad the binary number with zeros to brand sure that at that place is a multiple of four bits in the number. For example, given the binary number 1011001010, the offset step would be to add two bits to the left of the number so that it contains 12 bits. The converted binary value is 001011001010. The adjacent step is to separate the binary value into groups of 4 bits, due east.m., 0010_1100_1010. Finally, await upwardly these binary values in the table higher up and substitute the advisable hexadecimal digits, i.due east., $2CA. Contrast this with the difficulty of conversion betwixt decimal and binary or decimal and hexadecimal!

Since converting betwixt hexadecimal and binary is an operation you will need to perform over and over over again, you should take a few minutes and memorize the table above. Even if you lot have a reckoner that volition do the conversion for yous, you'll notice manual conversion to be a lot faster and more convenient when converting between binary and hex.

iii.five Arithmetics Operations on Binary and Hexadecimal Numbers

There are several operations nosotros can perform on binary and hexadecimal numbers. For example, we can add, subtract, multiply, divide, and perform other arithmetic operations. Although you needn't become an adept at it, you should be able to, in a compression, perform these operations manually using a piece of paper and a pencil. Having merely said that you should be able to perform these operations manually, the correct way to perform such arithmetic operations is to have a calculator that does them for you lot. At that place are several such calculators on the market; the following tabular array lists some of the manufacturers who produce such devices:

Some manufacturers of Hexadecimal Calculators (circa 2002):

• Casio
• Hewlett-Packard
• Abrupt
• Texas Instruments

This list is past no means exhaustive. Other computer manufacturers probably produce these devices besides. The Hewlett-Packard devices are arguably the best of the bunch . However, they are more expensive than the others. Sharp and Casio produce units which sell for well under $50. If you lot plan on doing any assembly language programming at all, owning one of these calculators is essential.

To understand why you should spend the money on a estimator, consider the following arithmetics trouble:

  $ix   + $ane   ----             

Y'all're probably tempted to write in the answer "$10" as the solution to this problem. But that is non correct! The right answer is 10, which is "$A", not sixteen which is "$10". A like trouble exists with the arithmetic problem:

 $10   - $1   ----             

You're probably tempted to answer "$9" even though the true answer is "$F". Think, this problem is asking "what is the departure between sixteen and one?" The answer, of class, is fifteen which is "$F".

Even if the two issues to a higher place don't bother y'all, in a stressful state of affairs your encephalon will switch dorsum into decimal style while you lot're thinking about something else and y'all'll produce the wrong result. Moral of the story - if yous must do an arithmetic computation using hexadecimal numbers past hand, take your time and be conscientious about it. Either that, or convert the numbers to decimal, perform the operation in decimal, and catechumen them back to hexadecimal.

three.6 A Notation About Numbers vs. Representation

Many people confuse numbers and their representation. A common question showtime associates language students have is "I've got a binary number in the EAX register, how practise I convert that to a hexadecimal number in the EAX register?" The reply is "yous don't." Although a strong statement could exist fabricated that numbers in retentivity or in registers are represented in binary, it'due south best to view values in memory or in a annals as abstract numeric quantities. Strings of symbols like 128, $fourscore, or %1000_0000 are non different numbers; they are simply different representations for the aforementioned abstract quantity that we oft refer to every bit "one hundred twenty-eight." Within the figurer, a number is a number regardless of representation; the only time representation matters is when y'all input or output the value in a homo readable form.

Human readable forms of numeric quantities are always strings of characters. To impress the value 128 in man readable form, yous must convert the numeric value 128 to the three-character sequence `1' followed past `ii' followed past `viii'. This would provide the decimal representation of the numeric quantity. If you prefer, yous could catechumen the numeric value 128 to the three character sequence "$80". It's the same number, just we've converted it to a different sequence of characters considering (presumably) we wanted to view the number using hexadecimal representation rather than decimal. Likewise, if we desire to run across the number in binary, then we must convert this numeric value to a string containing a one followed past seven zeros.

By default, HLA displays all byte, word, and dword variables using the hexadecimal numbering system when you lot employ the stdout.put routine. Likewise, HLA'south stdout.put routine volition display all register values in hex. Consider the following plan that converts values input as decimal numbers to their hexadecimal equivalents:

              plan ConvertToHex;   #include( "stdlib.hhf" );   static        value: int32;          begin ConvertToHex;           stdout.put( "Input a decimal value:" );           stdin.get( value );           mov( value, eax );        stdout.put( "The value ", value, " converted to hex is $", eax, nl );          stop ConvertToHex;      Programme 3.xi	 Decimal to Hexadecimal Conversion Program          

In a similar fashion, the default input base of operations is also hexadecimal for registers and byte, give-and-take, or dword variables. The post-obit programme is the converse of the one higher up- it inputs a hexadecimal value and outputs it as decimal:

              program ConvertToDecimal;   #include( "stdlib.hhf" );   static        value: int32;      begin ConvertToDecimal;           stdout.put( "Input a hexadecimal value: " );          stdin.get( ebx );  mov( ebx, value );        stdout.put( "The value $", ebx, " converted to decimal is ", value, nl );          end ConvertToDecimal;      Plan 3.12	 Hexadecimal to Decimal Conversion Plan          

Just because the HLA stdout.put routine chooses decimal as the default output base for int8, int16, and int32 variables doesn't hateful that these variables hold "decimal" numbers. Remember, memory and registers concord numeric values, not hexadecimal or decimal values. The stdout.put routine converts these numeric values to strings and prints the resulting strings. The choice of hexadecimal vs. decimal output was a design choice in the HLA language, goose egg more than. Yous could very easily alter HLA so that it outputs registers and byte, word, or dword variables as decimal values rather than as hexadecimal. If yous demand to print the value of a register or byte, word, or dword variable as a decimal value, simply call i of the putiX routines to do this. The stdout.puti8 routine will output its parameter every bit an eight-bit signed integer. Whatsoever eight-fleck parameter will piece of work. So you could pass an eight-bit register, an int8 variable, or a byte variable as the parameter to stdout.puti8 and the effect volition always be decimal. The stdout.puti16 and stdout.puti32 provide the same capabilities for xvi-bit and 32-chip objects. The following program demonstrates the decimal conversion program (Program three.12 above) using simply the EAX register (i.e., it does non use the variable iValue ):

              program ConvertToDecimal2;   #include( "stdlib.hhf" );   brainstorm ConvertToDecimal2;               stdout.put( "Input a hexadecimal value: " );          stdin.get( ebx );         stdout.put( "The value $", ebx, " converted to decimal is " );        stdout.puti32( ebx );         stdout.newln();          end ConvertToDecimal2;      Programme 3.13	 Variable-less Hexadecimal to Decimal Converter          

Annotation that HLA'south stdin.go routine uses the same default base for input every bit stdout.put uses for output. That is, if you attempt to read an int8, int16, or int32 variable, the default input base is decimal. If you attempt to read a register or byte, word, or dword variable, the default input base is hexadecimal. If y'all want to alter the default input base to decimal when reading a register or a byte, word, or dword variable, then y'all tin can use stdin.geti8, stdin.geti16, or stdin.geti32.

If yous want to go in the opposite management, that is you want to input or output an int8, int16, or int32 variable as a hexadecimal value, you tin can call the stdout.putb, stdout.putw, stdout.putd, stdin.getb, stdin.getw, or stdin.getd routines. The stdout.putb, stdout.putw, and stdout.putd routines write eight-flake, sixteen-chip, or 32-bit objects every bit hexadecimal values. The stdin.getb, stdin.getw, and stdin.getd routines read eight-flake, 16-bit, and 32-bit values respectively; they return their results in the AL, AX, or EAX registers. The following program demonstrates the use of a few of these routines:

              plan HexIO;      #include( "stdlib.hhf" );      static       i32: int32;      brainstorm HexIO;          stdout.put( "Enter a hexadecimal value: " );       stdin.getd();       mov( eax, i32 );       stdout.put( "The value you entered was $" );       stdout.putd( i32 );       stdout.newln();          end HexIO;      Programme 3.14	 Sit-in of stdin.getd and stdout.putd          

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