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21.2 DATA VALUES

21.2.1 Binary

•Binary is best used in computers because signals are ON/OFF which is well suited to the two binary digits.

•Converting between number systems can be done by looking at digit magnitude.

26 = 64 25 = 32 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1

1 1 1 0 0 0 1

A binary to decimal conversion 64 32 16

1

113

•Conversion can also be done between systems by division.

•For division use remainders.

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e.g., convert 932 in decimal (base 10) to a binary number (base 2)

	932	=	466.0
for binary	--------
	2
	466	=	233.0
	--------
	2
	233	=	116.5
	--------
	2
	116	=		58.0
	--------
	2
	58	=		29.0
	-----
	2
	29	=		14.5
	-----
	2
	14		=	7.0
	-----
	2
		7
		2--	=	3.5
		3
		2--	=	1.5
		1
		2--	=	0.5

0

0

1

0

0

1

0 1110100100

1

1

1

multiply places after decimal by division base, in this case it is 2 because of the binary.

*This method works for other number bases also, the divisor and multipliers should be changed to the new number bases.

•Convert the following numbers to/from binary

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base 10

base 2

54,321

110000101101

• Binary bytes and words are shown below.

BYTE	WORD
MSB LSB	MSB	LSB
0110 1011	0110 1011 0100 0010
	most	least
	significant	significant
	byte	byte

21.2.2 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

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111 = All three motors are on

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

21.2.3 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,