copyright © 1997-2009 curt hill bitwise operators packing logicals with other bases as a bonus

Copyright © 1997-2009 Curt Hill

BitWise Operators

Packing Logicals with Other Bases as a Bonus


Aside• Before discussing bitwise operators

a more fundamental item needs to be introduced or reviewed

• Binary is the base 2 number system

• It is how things are represented in computers

• If an item cannot be represented in base two it cannot be stored on a computer


Binary• Base two number system• Each digit is 0 or 1 and multiplied

by a power of two• Hence a byte with just one bit

turned on is a power of two• All numbers on a computer are

represented in binary because of the hardware


8 bit example:

1 1 1 1 1 1 1 1

1 x 20 = 11 x 21 = 21 x 22 = 41 x 23 = 81 x 24 = 161 x 25 = 321 x 26 = 641 x 27 = 128


Other bases• Binary has some problems, the

largest of which is that it is very bulky

• Decimal from binary is not an easy conversion– It also obscures structure

• Most manufacturers use another power of two base to show things– Octal – base 8 = 23

– Hexadecimal – base 16 = 24


Octal• Base 8 has 8 digits, 0 – 7 and

every place is raised to a power of 8

• Conversion to and from binary is way too easy– Partition into groups of 3 bits from

right to left– Convert the three bits to octal

• Preferred by certain manufacturers such as DEC and Intel


Binary – Octal Conversion

Start with binary 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1

Partition (right to left) in groups of 3

0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1

Write digits 0 6 7 1 3 5

How easy is that?

Conversion of octal to binary is the reverse


Hexadecimal• Base 16 has 16 digits, 0 – 9, A-F and

every place is raised to a power of 16– A=10, B=11, C=12, D=13, E=14, F=15

• Conversion to and from binary is also easy– Partition into groups of 4 bits– Convert the four bits to hexadecimal

• Preferred by certain manufacturers such as IBM and Burroughs


Binary – Hexadecimal Conversion

Start with binary 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1

Partition (right to left) in groups of 4

0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1

Write digits 6 E 5 D

How easy is that?

Conversion of hexadecimal to binary is the reverse


C Notation• Since the C family of languages is

occasionally well designed they all provide a notation for octal and hexadecimal constants

• An octal constant must start with a zero and have only octal digits:– 031, 077123

• A hexadecimal constant starts with a zero and then an x:– 0x012cf8


Purpose• Bool values may contain one of

two values: – true or false

• Boolean variables take one byte each

• Each byte consists of 8 bits, each with a possible value of 0 or 1

• Bitwise operators allow us to use these bits like small arrays of bools


Packing• An integer is normally 4 bytes or

32 bits• An array of 4 bools occupies 4

bytes• In same space 4 booleans or 32

bits?• Allocation is always by bytes so we

cannot allocate 25 bits


Boolean operations• AND • OR• NOT• XOR


What is needed?• Declare these clusters of booleans• Initialize• Apply the operators• Obtain the results


Declaration• Any simple type may be used• The preference is for integral types• char gives 8 bits• short gives 16• int gives 32• double gives 64


Operators• Bitwise operators are the same as

the bool operators except one symbol instead of two– && is boolean AND– & is bit AND– | is OR– ^ is XOR


Example• int a, b, c;• c = a | b• This does 32 individual Ors in one

operation– Usually a very fast operation

• 10110111 ^ 00101101 gives 10011010


What about NOT?• To accomplish NOT use XOR with

all ones for one operand• 11111111 ^

01100111 gives 10011000

• Negation exampleint c;…c = c ^ 0xFFFFFFFF


Accessing results• Integers cannot be used as bools• There is no subscripting operation

for bits either• How are they accessed? • Using bit operations and constant

integers


Setting Values• To set values

– Add the power of two constants– c = 64 + 8 + 4 + 1;

• To check the presence of a bit– AND it with constant and check for

zero– c & 8 != 0

• true 8 was present• false 8 was not present


Shifting

• There are also shift operations• A left shift of one bit corresponds

to multiply by 2• A right shift of one bit corresponds

to divide by 2• Form is int << int

– Left int is one to be shifted– Right is number of bits to shift


Example Shift• c = a << 2;

– Same as multiply by 4 if a and c are not signed

• b = c >> a;– Shift the value from c to the right by a

bits and put result in b


In Theory and In Practice• The usual use of bits are for named

constants• An Event object contains an id

field, which has these flags among others– MOUSE_ENTER, MOUSE_DRAG,

MOUSE_EXIT, MOUSE_DOWN, MOUSE_UP, MOUSE_MOVE

– ACTION_EVENT– LIST_SELECT, LIST_DESELECT


Sets• Boolean algebra and set theory are

orthogonal– Any set operation or proof may be

converted into a boolean operation or proof or vice versa

• Thus an array of booleans may act like a set of small integers


Set Operations• The boolean AND is equivalent to

the set Intersection– A & B could be either intersection or

parallel And

• The boolean OR is the set Union• The membership function of a set

is merely determining if that integer is present

copyright © 1997-2009 curt hill bitwise operators packing logicals with other bases as a bonus

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