lec03zczc
TRANSCRIPT
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OperationsinNumberSystem&
BooleanAlgebraDr.A.Sahu
De t of Com . Sc. & En .
IndianInstituteofTechnologyGuwahati
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NumberSystem&Conversion
Decimal,
Binary,
Octal,
Hex Otherrepresentation: Signed,Complement
OperationsinNumberSystem
, u , u , v, o
Howtohandlerealnumberefficientl Float,Double
oo ean ge ra: a es eorem2
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Addition
Subtraction
Division
Modulus
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0nebit
0
+ 0=
01 + 0= 1
0 + 1= 1
1
+ 1=
01
(Carrybit)
u cons ercarry
1 1 0 1
= 1 0 1 1
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0nebit
0
0=
01 0= 1
0 1= 1 1 (Carrybit)
1
1=
0
Multibit (considercarry)
1 1 1 0
1
0 0
1
= 0 1 0 1
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BinarySubtraction Multibit (considercarry)
1 1 1 0
1
0 0
1
= 0 1 0 1
add2s
com lement=
(1001)+1=
0110+1=0111
Otherway (add2scomplement&discardcarry)
1 1 1 0
+ 0 1 1 1= 1 0 1 0 1
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,
Oct:Add,
Sub,
Multiplication,
Division
&
Mod
Rea yourse
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Floatingformat:
2.03455x103
7 .
Realnumbersmustbenormalizedusing
0.12n wherenisaninteger
andthemostsignificantdigitofthefractionisa 1 ALWAYS!
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complex
rational
5/8 real
rea 3
complex
2
3i
rational
inte er
Extremelylargeandsmallvalues:
distanceplutosun=5.91012 m
28 .
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5 8 =5/8
r ngsw exp c ec ma po n
2 4 7 . 0 9 Implicitpointatafixedposition
010011010110001011
Floatingpoint
fraction x base power
implicitpoint
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= 2 1 0 1 2. .
=4+1+.+
0.5
+0.25
=5.7510
0.6=0.10011001100110011001.....
.6x2=1+.2
.2x2=0+.4
=. .
.8x2=1+.6
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1
8
23
S E F
1 11 20+32
110101101011000101100
S E F
10110001011011001011010111010110
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,
Makesame
power,
operate,
normalize
. .
2x106 3X104 =2x106 0.03x106 =1.97x106
u , v
Dooperation,normalize
2.0x106 x3.0x103 =
2x3
x10
(6+3) =6.0x109
2x106 3x103 = 2/3x10(63) =0.666x103 =6.66x102
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BooleanAlgebra
Computer hardware using binary circuit
grea y s mp y es gn
Binary circuits: To have a conceptual
framework to manipulate the circuits
George Boole (1813-1864): developed a
ma ema ca s ruc ureTo deal with binary operations with just two
values.
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Element 0 : FALSE. Element 1 : TRUE.
+ operation OR,* operation AND and operation NOT.
OROR 00 11 ANDAND 00 11 NOTNOT
00 00 11 00 00 00 00 11
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11 11 11 11 00 11 11 00
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+ operation OR
X Y R=XORY=
0 0 0X
Y
R=X+Y
0 1 1
1 0 1
00 00 11 1 1 1
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11 11 11
1+
Y
=
1
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* operation AND
X Y R=XAND
Y
R=X*Y0 0 0
X
Y
R=X*Y
1 0 0ANDAND 00 11
1 1 100 00 00
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11 00 11
0*Y=0
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operation NOT
X R=X=
0 1XR=X
1 0
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BooleanAlgebraDefined
Boolean Algebra B : 5-tuple, +, , , ,
nary ,
u .
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BooleanAlgebraDefined Axiom #1: Closure
(a + b) and (a * b) are Boolean.
Axiom #2: Cardinality
if a is Boolean then a is Boolean
(a + b) = (b + a)
(a * b) = (b * a)
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BooleanAlgebraDefined x om : ssoc at ve : a an are oo ean
a + b + c = a + b + c
(a * b) * c = a * (b * c)
Axiom #6: Distributive
a * (b + c) = (a * b) + (a * c)
* *2nd oneisNotTrueforDecimalnumbersSystem
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B has identity to + and *
0 is identity element for + : a + = a
* * =
a + a' = 1
a * a' = 0
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Thanks
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