the building blocks of number system

8/12/2019 The Building Blocks of Number System

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The Building Blocks:Binary Numbers, Boolean

Logic, and Gates

Chapter 4


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urpose o! Chapter

Learn ho" computers represent andstore in!ormation#

Learn "hy computers represent

in!ormation that "ay# Learn "hat the basic building de$ices in

a computer are, and ho" those de$icesare used to store in!ormation#

Learn ho" to build more comple%de$ices using the basic de$ices#


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&

'%ternal (epresentation o! )n!ormation

*hen "e communicate "ith each other, "eneed to represent the in!ormation in anunderstandable notation, e#g# *e use digits to represent numbers#

*e use letters to represent te%t# +ame applies "hen "e communicate "ith a

computer: *e enter te%t and numbers on the keyboard,

The computers displays te%t, images, and numberson the screen#

*e re!er to this as an e%ternal representation# But ho" do humanscomputers store the

in!ormation -internally./


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0umans:

Computers:

Te%t, numbers,

images, sounds

Te%t, numbers,

sounds

Te%t, numbers,images, sounds

Binary Numbers

+ynapsesNeurons

Te%t, numbers,images, sounds

'%ternal (epresentation o! )n!ormation


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)n!ormation "e need to represent

Numbers )ntegers 2&4, 43

ositi$enegati$e $alue 5166, 52&

7loating point numbers 12#&4,

Te%t Characters letters, digits, symbols

9ther Graphics, +ound, ideo, ;


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Numbering +ystems

*e use the decimal numbering system 16 digits: 6, 1, 2, &, 4, , 3, 1 > ? >16 L >6,

C>166

?)) > 12, entium )

*hat i! "e only had one symbol/ ))))) ))))) )) > 12


7/64 1%2> 1%211 1%2 1%21414 ### 1%2 ### 1%211 1%2 1%266> 3,&> 3,&


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(epresenting +igned )ntegers

0o" does the computer represent signed$alues/

Three schemes ha$e been used:

+igned magnitude

1Es complement

2Es complement

*hile unsigned binary is in some sense -natural.signed binary arithmetic is a human construct

There isa best "ay 5 "hich is "hat "e all use today


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(epresenting +igned )ntegers

+igned Fagnitude

7irst bit is the signbit: 6> $e 1> 5$e

(emainder is the magnitudei#e# numeric $alue

14 in signed magnitude representation

514 in signed magnitude representation

roblems @rithmetic logic is comple%

There are t"o Heros:

6666666666666666

1666666666666666

6111666666666666

+ign bit#

6111666666666661


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(epresenting +igned )ntegers 1Es Complement

There is no signbit but an algorithm:

To con$ert !rom $e to 5$e and $ice $ersa:

in$ert E!lipE all the bits

14 in 1Es complement representation


roblems

@lthough arithmetic is much simpler, there aret"o Heros:

6666666666666666

1111111111111111

6111666666666666

1666111111111111


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(epresenting +igned )ntegers 2Es Complement

There is no signbit but an algorithm:

To con$ert !rom $e to 5$e and $ice $ersa:

in$ert E!lipE all the bits

then add 1



+imple arithmetic like 1Es complement 9nly one Hero:

6666666666666666 -6., con$ert:

1111111111111111

1

6666666666666666

6111666666666666

6166111111111111


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2# Dsing normaliHed scienti!ic notation

+cienti!ic notation : K F % B K' B is the base, F is the mantissa , ' is the e%ponent#

'%ample: decimal, base>16 & > & % 166 e#g# & 1

266 > 2#6 % 16& e#g# 2#6 1666

'asy to con$ert to scienti!ic notation: 161#11 % 26

NormaliHe to get the -#. in !ront o! !irst

le!tmost 1 digit )ncrease e%ponent by one !or each location -#.

mo$es le!t decreases i! "e ha$e to mo$e right

161#11 % 26> #16111 % 2&


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+tore the normaliHed number

1166666666111616

#16111 % 2 #16111 % 2&&

'%ponent 3 bitsFantissa 16 bits

@ssumed binary point

Base 2 implied, not stored


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(epresenting Te%t

0o" can "e represent te%t in a binary!orm/ @ssign to each character a positi$e integer $alue

!or e%ample, @ is 3, B is 33, ;

Then "e can store the numbers in their binary

!ormM

The mapping o! te%t to numbers Code mapping

Need standard code mappings "hy/: @+C)) @merican +tandard Code !or )n!ormation

)nterchange > each letter =5bits

only 23 di!!erent characters can be represented 2=

Dnicode > each letter 135bits sometimes


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@+C)) Code mapping Table

;

H

y

%

;C

B

@

Char

;

122

121

126

;316 @NA ?R

?6 @NA ?O26 True

7alse


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? is 16 and R is 1'%amples

?>16 9( ?R

?6 9( ?O26

Truth Table !or 9(

Let a and b be any Boolean e%pressions, then

TrueTrueTrue

True7alseTrue

TrueTrue7alse

7alse7alse7alse

a 9( bba

True

True

h bl !


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& a 9( b

d> N9T a 9( b @NA N9T b

Circuits Aiagram and Boolean


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g'%pr

Aeri$ing Boolean e%pressions !or theoutput#

NOT ((a OR b) AND (NOT b))

(a OR b) (a OR b)a

b

(a OR b) AND (NOT b)

(NOT b)

Circuits Aiagram and Boolean '%pr


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Circuits Aiagram and Boolean '%pr

(emember, "hen "riting Booleane%pressions !or circuit diagrams, "e canuse a di!!erent notationM

(a + b) (a + b)a

b

b

(a + b) b

(a + b) b

'%ample


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Transistor

*e sa" ho" to build logic gates @NA, 9(,

N9T: Transistors

Gates

Boolean logic

the building blocks of number system

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