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Enters! conversi a base b

em divisions sccessives i ens /edem amb

els restes.

Eemple! $ en base

$=11+111="+1

"=+1

=1+0

$=10111%

Fixeu-vos que s'escriu en el sentitinvers: el primer reste s la xifra de les

"unitats", el segon "desenes", ... i en la

divisi ltima el quocient s la primera

xifra.

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Representaci de reals en pnt

=dp

dp-1

...d1

d0

.d-1

d-

d-$

...(b

,amb 0di

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Reals en pnt ! conversi a base b

3a part entera %es/erra pnt decimal4 la

tractem com n enter5 la part *raccion6rias'obt7 *ent sccessis prodctes i /edant-seamb la part entera.

Eemple! $.1" en base $=10111%

0.1"=0.$ 0.=0.#

0.$=0.8 0.#=0.&

0.8=1. 0.&=1.8 0.8=1. 9.

0.1"=0.0010011001%

$.1"=10111.0010011001...%

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Representaci de reals en pnt :otant

=%-14s;%b

bE on s 0,1> 7s el si?ne,

E %eponent4 7s n enter,

; %mantissa4=d0.d

-1d

-d

-$... ,0d

i

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odicaci d'enters

(mb nbits es representen nvalors di*erents

Enters positis! entre 0i n-1! en base

Enters positis i ne?atis! complement a 2, entre -n-1i n-1-1

Clcul del complement a 2:

Ci el nombre 7s positi %0AAn-1-14 es representa en base amb nbits

Ci 7s ne?ati %-n-1A

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Eemple! complement a

n=1

000=1111101000%

-000

anvia 0 D 1

+1

011111010000

011111010000 100000101111

100000110000

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Eemple! -n-1

n=1

11=100000000000%

-11

anvia 0 D 1

+1

100000000000

100000000000 011111111111

100000000000

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Eemple n="

15 01111

14 0111013 01101

12 01100

11 01011

10 01010

9 010018 01000

7 00111

6 00110

5 00101

4 001003 00011

2 00010

1 00001

0 00000

15 01111

14 0111013 01101

12 01100

11 01011

10 01010

9 010018 01000

7 00111

6 00110

5 00101

4 00100

3 00011

2 00010

1 00001

0 00000

-1 11111

-2 11110-3 11101

-4 11100

-5 11011

-6 11010

-7 11001-8 11000

-9 10111

-10 10110

-11 10101

-12 10100-13 10011

-14 10010

-15 10001

-16 10000

-1 11111

-2 11110-3 11101

-4 11100

-5 11011

-6 11010

-7 11001-8 11000

-9 10111

-10 10110

-11 10101

-12 10100

-13 10011

-14 10010

-15 10001

-16 10000

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Cmes en complement a

5+1 = ?5 00101

1 00001

6 00110

5+1 = ?5 00101

1 00001

6 00110

14-7 = ?

14 01110

-7 11001

7 100111

14-7 = ?

14 01110

-7 11001

7 100111

Aritmtica a/(2n)Aritmtica a/(2n)

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odicaci de reals en pnt :otant

=%-14s;%b

bE on s 0,1> 7s el si?ne,

E 7s n enter, ;=F0.F

1F

F

$... ,0F

i

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3'est6ndard JEEE 2"#

Es treballa en base , per tant d0@0 implica d

0=1

%no es ?arda4, m= 1.d1d

...d

p=1.*

Krecisi simple (float4 es ?arda en $ bits

1 bit! si?ne %s4 L0 per positiM

& bits! E+12 %e, ep modicat4 L12=8-1-1, biaixM

$ bits! $ bits de *racci %f4

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3'est6ndard JEEE 2"#

Krecisi doble %double4 es ?arda en 8# bits

1 bit! si?ne %s4 L0 per positiM

11 bits! E+10$ %e4 L10$=11-1

-1, biaixM" bits! " bits de *racci %f4

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alors JEEE 2"# %precisi simple4

e="" * @ 0, NaN%Not a Nmber4

e="" * = 0, (-1)s

0

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alors JEEE 2"# %precisi doble4

e=0#2 * @ 0, NaN%Not a Nmber4

e=0#2 * = 0, (-1)s

0

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=10. en precisi simple

O=10.=".1 1=."" =1.2" &

%posem 1!% '2, o P les ve?ades necess6ries 4

1.2"=1.010 0011 0011 0011 00119.

E=&, e=&+12=1$0 = 10000010%

s=0 %positi4

01000001001000110011001100110011

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Qin valor 7s exactament

01000001001000110011001100110011

10000010%=1&0, E=1&0-12=$

s=0 %positi4

m=1.01000110011001100110011%

1+-2+-$+-#+-10+-11+-1+-1*+-18+-1++-22+-2&=

1+%21+1#+1$+1&+12+++8+*++1+14P2& =

1+$08&82P&$&&80& =

1!2#++++#$1*81208+8*00!!!!

,("2&) ,10!1+++++80+2$*1&$#18#*0000000!!!

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Qin valor el se?ei

(1!2#++++#$1*81208+8*00!!!! .2-2&) "2&

01000001001000110011001100110100

10000010%=1&0, E=1&0-12=$

s=0 %positi4

m=1.01000110011001100110100%

1+-2+-$+-#+-10+-11+-1+-1*+-18+-1++-21=

1+%21+1#+1$+1&+12+++8+*++4P2& =

1+$08&8&P&$&&80& ,1!2#*0000+*&$#&1$0$2*0000000!!!! ("2&) ,

10!200000#$2+&+*&12*000000!!!

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=-0.1 en precisi simple

=-0.1=-0. -1=-0.# -=-0.& -$=-1.8 -

1.8=1.1001 1001 1001 1001 1001 10019.

E=-, e=-+12=1$ = 01111011%

s=1 %ne?ati4

10111101110011001100110011001101

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Qin valor 7s exactament

10111101110011001100110011001101

011110110%=12&, E=12&-12=-#

s=1 %ne?ati4

m=1.10011001100110011001101%

1+-1+-+-*+-8+-++-12+-1&+-1$+-1#+-20+-21 +-2&

, 1+"0$$18"P&$&&80&

,1!$0000002&818*#+101*$2*000000!!!!

-("2-) ,-0!100000001+011$11+&8#$*$2*000!!!

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Qin valor el se?ei

-(1!$0000002&818*#+101*$2*000!! -2-2&) /2-

10111101110011001100110011001100

011110110%=12&, E=12&-12=-#

s=1 %ne?ati4

m=1.10011001100110011001100%

1+-1+-+-*+-8+-++-12+-1&+-1$+-1#+-20+-21 ,

1+"0$$18#P&$&&80& =

1! *++++++0$&2*$8&*+*0000!!!!

-("2-) ,-0!0+++++++0&+*&**22$0+*000!!!

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Els nombres :otantsS en la recta real

Copyright Journal of Artificial Societies and Social Simulation, [2006]

http://jasss.soc.surrey.ac.uk/admin/copyright.htmlhttp://jasss.soc.surrey.ac.uk/admin/copyright.html