FROM

The Art of Assembly Language Programming

by Randall Hyde

 

1.3 - The Hexadecimal Numbering System
1.4 - Arithmetic Operations on Binary and Hexadecimal Numbers


1.3 The Hexadecimal Numbering System

A big problem with the binary system is verbosity. To represent the value 202 (decimal) requires eight binary digits. The decimal version requires only 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 large values, binary numbers quickly become too unwieldy. Unfortunately, the computer thinks in binary, so most of the time it is convenient to use the binary numbering system. Although we can convert between decimal and binary, the conversion is not 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 simple to convert them to binary and vice versa. Because of this, most binary computer systems today use the hexadecimal numbering system. Since the radix (base) of a hexadecimal number is 16, each hexadecimal digit to the left of the hexadecimal point represents some value times a successive power of 16. For example, the number 1234 (hexadecimal) is equal to: 
1 * 16**3   +   2 * 16**2   +   3 * 16**1   +   4 * 16**0

or 

4096 + 512 + 48 + 4 = 4660 (decimal).

Each hexadecimal digit can represent one of sixteen values between 0 and 15. Since there are only ten decimal digits, we need to invent six additional digits to represent the values in the range 10 through 15. Rather than create new symbols for these digits, we'll use the letters A through F. The following are all examples of valid hexadecimal numbers:

1234 DEAD BEEF 0AFB FEED DEAF

Since we'll often need to enter hexadecimal numbers into the computer system, we'll need a different mechanism for representing hexadecimal numbers. After all, on most computer systems you cannot enter a subscript to denote the radix of the associated value. We'll adopt the following conventions: 

		
Examples of valid hexadecimal numbers:
		
		1234h 0DEADh 0BEEFh 0AFBh 0FEEDh 0DEAFh
		
		As you can see, hexadecimal numbers are compact and easy to read. In addition, you can easily convert between hexadecimal and binary. Consider the following table:
		
		
			Binary/Hex Conversion			
				Binary				Hexadecimal			
			
				0000				0			
			
				0001				1			
			
				0010				2			
			
				0011				3			
			
				0100				4			
			
				0101				5			
			
				0110				6			
			
				0111				7			
			
				1000				8			
			
				1001				9			
			
				1010				A			
			
				1011				B			
			
				1100				C			
			
				1101				D			
			
				1110				E			
			
				1111				F			
		
		
		
		This table provides all the information you'll ever need to convert any hexadecimal number into a binary number or vice versa.
		
		To convert a hexadecimal number into a binary number, simply substitute the corresponding four bits for each hexadecimal digit in the number. For example, to convert 0ABCDh into a binary value, simply convert each hexadecimal digit according 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 easy. The first step is to pad the binary number with zeros to make sure that there is a multiple of four bits in the number. For example, given the binary number 1011001010, the first 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 next step is to separate the binary value into groups of four bits, e.g., 0010 1100 1010. Finally, look up these binary values in the table above and substitute the appropriate hexadecimal digits, e.g., 2CA. Contrast this with the difficulty of conversion between decimal and binary or decimal and hexadecimal!
		
		Since converting between hexadecimal and binary is an operation you will need to perform over and over again, you should take a few minutes and memorize the table above. Even if you have a calculator that will do the conversion for you, you'll find manual conversion to be a lot faster and more convenient when converting between binary and hex.
		
				
1.4 Arithmetic Operations on Binary and Hexadecimal Numbers
		
		
		There are several operations we 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 expert at it, you should be able to, in a pinch, perform these operations manually using a piece of paper and a pencil. Having just said that you should be able to perform these operations manually, the correct way to perform such arithmetic operations is to have a calculator which does them for you. There are several such calculators on the market; the following table lists some of the manufacturers who produce such devices:
		
		Manufacturers of Hexadecimal Calculators:				

		This list is, by no means, exhaustive. Other calculator manufacturers probably produce these devices as well. 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 plan on doing any assembly language programming at all, owning one of these calculators is essential.
		
		Another alternative to purchasing a hexadecimal calculator is to obtain a TSR (Terminate and Stay Resident) program such as SideKick which contains a built-in calculator. However, unless you already have one of these programs, or you need some of the other features they offer, such programs are not a particularly good value since they cost more than an actual calculator and are not as convenient to use.
		
		To understand why you should spend the money on a calculator, consider the following arithmetic problem:		
  9h+ 1h----
		
You're probably tempted to write in the answer "10h" as the solution to this problem. But that is not correct! The correct answer is ten, which is "0Ah", not sixteen which is "10h". A similar problem exists with the arithmetic problem:		
 10h- 1h----
		
You're probably tempted to answer "9h" even though the true answer is "0Fh". Remember, this problem is asking "what is the difference between sixteen and one?" The answer, of course, is fifteen which is "0Fh".
		
		Even if the two problems above don't bother you, in a stressful situation your brain will switch back into decimal mode while you're thinking about something else and you'll produce the incorrect result. Moral of the story - if you must do an arithmetic computation using hexadecimal numbers by hand, take your time and be careful about it. Either that, or convert the numbers to decimal, perform the operation in decimal, and convert them back to hexadecimal.
		
		You should never perform binary arithmetic computations. Since binary numbers usually contain long strings of bits, there is too much of an opportunity for you to make a mistake. Always convert binary numbers to hex, perform the operation in hex (preferably with a hex calculator) and convert the result back to binary, if necessary.