HEXADECIMAL NOTATION and TWO'S COMPLEMENT ARITHMETIC

Hexadecimal

Hexadecimal notation (hex) is based on radix 16. This is particularly convenient for expressing binary values as 4 bits are represented by one hexadecimal digit. Hex digits range from 0 to 15 (decimal) which is 0 to F (hex). Hex numbers above 9 use the letters A through F as follows: 
	     decimal	   hexadecimal	       binary
		9		9		1001
		10		A		1010
		11		B		1011
		12		C		1100
		13		D		1101
		14		E		1110
		15		F		1111
Therefore a byte of data can be expressed as two hex digits. 
	 Example:			10001010

	 Split into two 4-bit fields	1000 1010 = 8A (hexadecimal)

Representation of negative numbers

We may sometimes wish to use a number, representation that enables us to have positive and negative numbers, rather than be confined to positive numbers only. An 8-bit byte can represent 256 (2 to the 8th power) different quantities and the usual way of interpreting these would be as the numbers 0 to 255. If we require our range to include negative numbers as well, we might decide that a byte can represent any number in the range -128 to +127 (256 values, including zero). Remember that the exact interpretation of a byte is entirely in the hands of the programmer, so we could allocate the 256 codes to the 256 numbers in any way we wish.

However it makes sense to allocate them in such a way that arithmetic operations remain as simple as possible and do not involve extra decoding and encoding steps.

The most general way of representing negative integers in computing and microprocessing is the "Two's Complement" (TC) notation. Using TC notation, positive and negative numbers may be treated exactly alike during calculations and interpreted as positive or negative only on input/output. Particularly important is the necessity for the sum of a positive and a negative number to yield the correct result. I will show below how the TC of a number is calculated and that the above condition is satisfied.

RULE for finding the TC of a number

                a)     Invert the number (0 becomes 1; 1 becomes 0)
                b)     Add 1

So if 90 (decimal) is 01011010 in binary (5A in hex) then -90 in TC is found as follows: 

                a)     Invert:        10100101      (A5 in hex)
                b)     Add 1:         10100110      (A6 in hex)
We must prove that this is correct, which we do by adding 90 to -90: 
                                      01011010       90
                                      10100110      -90
				      --------      ---
                       Carry ->     1 00000000        0
				      ========      ===
The Carry is ignored; it has "spilled" over the end of the register.

What is 'zero' in Two's Complement? 

                       00000000              0
                a)     11111111
                b)  1  00000000             -0 = 0 as required
Note that in TC notation, a negative number always has its most significant bit set (equal to 1).

Subtraction in Two's Complement

To subtract two numbers the simplest way is to take the TC of the second (subtrahend) and add it to the first:

We want to know 25H - 17H (numbers in hex). 

         Take the TC of 17H:          00010111 = 17
                       a)             11101000
                       b)             11101001 = E9

         The TC of 17H is E9H.

         Now add them together:       00100101 = 25
                                      11101001 = -17

                     Carry ->    1    00001110 = OEH (Correct, try it)
What if we wanted 17H - 25H? 
         Take the TC of 25H:          -25H = DB

         Now add 17H:                 11011011 = DB
                                      00010111 = 17 +

                     Carry ->    0    11110010 = F2
This is obviously a negative number (MSB is set). To find out which number this is the negative of, we find the TC: 
                                      11110010 = F2

                       a)             00001101
                       b)             00001110 = 0E
So F2H is -0EH which is what we should expect.

On any of these calculations, the answer may have been outside the range of -128 to +127. In this case, the answer will appear negative when it should be positive and vice-versa. The ALU can detect this condition, known as "overflow" and set its overflow flag accordingly. This flag may then be tested in the program and appropriate action taken. 

         EXAMPLE               25H + 60H           (=85H)

                               00100101              25H
                               01100000             +60H
                               --------             ----
                 Carry -> 0    10000101              85H, but MSB indicates negative no.
			       ========		    ====
                          0    11000000    <-   carrys from previous bit
                          ^    ^
The OVERFLOW flag is set when the Carry out of the MSB (into CARRY) is different from the Carry out of the next most significant bit. In other words, it is the exclusive-OR of the carrys arrowed above.
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