University  of  Washington  

Sec.on  2:  Integer  &  Floa.ng  Point  Numbers  

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Representa.on  of  integers:  unsigned  and  signed  

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Unsigned  and  signed  integers  in  C  

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Arithme.c  and  shiBing  

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Sign  extension  

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Background:  frac.onal  binary  numbers  

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IEEE  floa.ng-­‐point  standard  

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Floa.ng-­‐point  opera.ons  and  rounding  

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Floa.ng-­‐point  in  C  

 

Frac.onal  Values  

University  of  Washington  

Frac.onal  Binary  Numbers  

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What  is  1011.101

2

?  

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How  do  we  interpret  frac.onal  decimal  numbers?  

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e.g.  107.95

10  

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Can  we  interpret  frac9onal  binary  numbers  in  an  analogous  way?  

Frac.onal  Values  

University  of  Washington  

• •

•

b

–1

.

Frac.onal  Binary  Numbers  

Frac.onal  Values  

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Representa.on  

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Bits  to  right  of  “binary  point”  represent  frac9onal  powers  of  2  

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Represents  ra9onal  number:  

b

i

b

i–1

b

2

b

1

b

0

b

–2

b

–3

b

–j

• • •

• • •

1

2

4

2

i–1

2

i

• • •

1/2

1/4
1/8

2

–j

b

k

⋅2

k

k =− j

i

∑

University  of  Washington  

Frac.onal  Binary  Numbers:  Examples  

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Value

 Representa.on  

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5  and  3/4    

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2  and  7/8    

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63/64

   

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Observa.ons  

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Divide  by  2  by  shiMing  right  

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Mul9ply  by  2  by  shiMing  leM  

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Numbers  of  the  form  0.111111…

2

 are  just  below  1.0  

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1/2  +  1/4  +  1/8  +  …  +  1/2

i

 +  …  →  1.0  

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Shorthand  nota9on  for  all  1  bits  to  the  right  of  binary  point:  

1.0  –  

ε

 

Frac.onal  Values  

101.11

2

10.111

2

0.111111

2

University  of  Washington  

Representable  Values  

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Limita.ons  of  frac.onal  binary  numbers:  

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Can  only  exactly  represent  numbers  that  can  be  wriWen  as  

x  *  2

y  

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Other  ra9onal  numbers  have  repea9ng  bit  representa9ons  

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Value

 Representa.on  

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1/3

 0.0101010101[01]…

2

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1/5

 0.001100110011[0011]…

2

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1/10

 0.0001100110011[0011]…

2

Frac.onal  Values  

University  of  Washington  

Fixed  Point  Representa.on  

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We  might  try  represen.ng  frac.onal  binary  numbers  by  

picking  a  fixed  place  for  an  implied  binary  point  

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“fixed  point  binary  numbers”  

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Let's  do  that,  using  8-­‐bit  fixed  point  numbers  as  an  example  

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#1:  the  binary  point  is  between  bits  2  and  3  

       b

7

 b

6

 b

5  

b

4

 b

3

   [.]  b

2

 b

1

 b

0  

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#2:  the  binary  point  is  between  bits  4  and  5  

       b

7

 b

6

 b

5

 [.]  b

4

 b

3

 b

2

 b

1

 b

0  

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The  posi.on  of  the  binary  point  affects  the  range  and  

precision  of  the  representa.on  

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range:  difference  between  largest  and  smallest  numbers  possible  

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precision:  smallest  possible  difference  between  any  two  numbers  

Frac.onal  Values  

University  of  Washington  

Fixed  Point  Pros  and  Cons  

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Pros  

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It's  simple.    The  same  hardware  that  does  integer  arithme9c  can  do  

fixed  point  arithme9c  

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In  fact,  the  programmer  can  use  ints  with  an  implicit  fixed  point  

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ints  are  just  fixed  point  numbers  with  the  binary  point    

to  the  right  of  b

0  

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Cons  

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There  is  no  good  way  to  pick  where  the  fixed  point  should  be  

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Some9mes  you  need  range,  some9mes  you  need  precision  –  the  

more  you  have  of  one,  the  less  of  the  other.    

Frac.onal  Values