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• Examples of the functions are,
			AND
			source A N7:0
			source B N7:1
			dest. N7:2
			OR
			source A N7:0
			source B N7:1
			dest. N7:3
			XOR
			source A N7:0
			source B N7:1
			dest. N7:4
			NOT
			source A N7:0
			dest. N7:5
	addr.	data (binary)
	N7:0	0011010111011011
	N7:1	1010010011101010
	after N7:2	1010010011001010
	N7:3	1011010111111011
	N7:4	1001000100110001
	N7:5	1100101000100100

24.2.3 Boolean Operations

• In most discrete systems the inputs and outputs (I/O) are either on or off. This is a binary state that will be represented with,

1 = on

0 = off

•Because there are many inputs and outputs, these can be grouped (for convenience) into binary numbers.

•Consider an application of binary numbers. There are three motors M1, M2 and M3

100 = Motor 1 is the only one on

111 = All three motors are on

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in total there are 2n or 23 possible combinations of motors on.

• The most common Binary operations are,

Name	Example		Result
AND	0010	* 1010	0010
OR	0010	+ 1010	1010
NOT	0010		1101
EOR	0010 eor 1010		1000
NAND	0010	* 1010	1101
shift left	111000		110001	(other results are possible)
shift right	111000		011100	(other results are possible)
etc.

*There are many other operations in addition to this list.

** These operations and the others can be useful for some complicated operations

24.2.4 Binary Mathematics

•These include standard logic forms such as,

-and/or/add, etc.

-compliments

•Negative numbers are a particular problem with binary numbers. As a result there are two common numbering systems use,

-signed binary - the most significant bit (MSB) of the binary number is used to indicate positive/negative

-2s compliment - negative numbers are represented by complimenting the binary number and then adding 1.

•Signed binary numbers are easy to understand, but much harder to work with when doing calculations.

•An example of 2s compliments are given below,

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	decimal	binary byte	METHOD FOR MAKING A NEGATIVE NUMBER
			1. write the binary number for the positive
	2	00000010
	1	00000001	for -30 we write 30 = 00011110
	0	00000000
	-1	11111111	2. Invert (compliment) the number
	-2	11111110	00011110 becomes 11100001
			3. Add 1
			11100001 + 00000001 = 11100010

• When adding 2s compliment numbers, additional operations are not needed to deal with negative numbers. Consider the examples below,

00000001 = 1 + 11111111 = -1

C+00000000 = 0

ignore the carry bits

	00000001 = 1		00000010 = 2
+	11111110 = -2	+	11111111 = -1
	11111111 = -1	C+00000001 = -1

Note: Normally the carry bit is ignored during the operation, but some additional logic is required to make sure that the number has not ‘overflowed’ and moved outside of the range of the numbers. Here the 2s compliment byte can have values from -128 to 127.

24.2.5 BCD (Binary Coded Decimal)

• Each digit is encoded in 4 bits
1263	BCD
0001 0010 0110 0011	decimal

Note: this example shows four digits in two bytes. The hex values would also be 1263.

• This numbering system makes poor use of the digits, but is easier to convert to/from base 10