uni/gds/zahlendarstellung-3.md

GDS Eingangstest Zahlensysteme

Versuch 3

1

$$ (27)_b + (28)_b = (54)_b $$

$$ (2b + 7 + 2b + 8){10} = (5b + 4){10} $$ $$ 4b + 15 = 5b + 4 $$ $$ 11 = b $$

2

$$ A=(1101)_2 $$ $$ B=(10100)_2 $$

 (10100)_2 * (1101)_2 = (100000100)_2
  10100000
 + 1010000
 +   10100
 100000100

$$ (10100)_2 * (1101)_2 = (100000100)_2 $$

3

$$ zk ... zweierkomplement $$

$$ v_{zk} = (001000100)2 $$ $$ (v{zk})_{10} = 2^2 + 2^6 = 4 + 64 = 68 $$

4

$$ ez ... exzess $$ $$ z_{ez} = (1100111)_2 $$

$$ z = -(8){10} $$ $$ (z{ez})_{10} = 2^0 + 2^1 + 2^2 + 2^5 + 2^6 $$

      $$ = 1 + 2 + 4 + 32 + 64 $$
      $$ = 103 $$

$$ e = (103 - (-8)){10} = (111){10} $$

$$ 111 mod 2 = 1 ; 110 / 2 = 55 $$ $$ 55 mod 2 = 1 ; 54 / 2 = 27 $$ $$ 27 mod 2 = 1 ; 26 / 2 = 13 $$ $$ 13 mod 2 = 1 ; 12 / 2 = 6 $$ $$ 6 mod 2 = 0 ; 6 / 2 = 3 $$ $$ 3 mod 2 = 1 ; 2 / 2 = 1 $$ $$ 1 mod 2 = 1 ; 0 / 2 = 0 $$

$$ (e)_2 = (1101111)_2 $$

5

$$ (10000.11001)2 = (x){10} $$

$$ (x)_{10} = 2^4 + 2^{-1} + 2^{-2} + 2^{-5} $$ $$ = 16 + 1/2 + 1/4 + 1/32 $$ $$ = 16 + 0.5 + 0.25 + 0.031 $$ $$ = 16.781 $$

1/32=0_031
100
  40
   8R

 0_5
+0_25
+0_031
=0_781

6

$$ (455.592){10} = (x){16} $$

455/16 = 28
135
  7R

$$ (455-7)/16 = 448/16 = 28 $$

28/16 = 1
12R

$$ (28-12)/16 = 16/16 = 1 $$

1/16 = 0
1R

$$ (x){16} = (1C7){16} + (0.592)_{10} $$

0.592 * 16 = 9.472
5.920
   12
  540
3.000

0.472 * 16 = 7.552
4.720
   12
  420
2.400

0.552 * 16 = 8.832
5.520
   12
  300
3.000

$$ (x){16} = (1C7.978){16} $$

7

$$ (DB4.E07)_{16} = (x)_2 $$

4 = 0100
7 = 0111
B = 1011
C = 1100
D = 1101
E = 1110

$$ (x)_2 = (110110110100.111...)_2 $$

8

$$ (356.077)8 = (x){10} $$

$$ (x)_{10} = 3*8^2 + 5*8^1 + 6*8^0 + 0*8^-1 + 7*8^-2 + 7*8^-3 $$ $$ = 3*64 + 5*8 + 6 + 0 + 7/64 + 7/512 $$ $$ = 192 + 40 + 6 + 56/512 + 7/512 $$

$$ 3*512 = 1024 + 512 = 1536 < 1560 $$

63/512 = 0.123...
630
1180
 1560
  ...R

 192
+ 40
+  6
=238

$$ (x){10} = (238.123...){10} $$

9

$$ (0.8125)_{10} = (0.11...)_2 $$

0.8125 * 2 = 1.625
    10
    40
   200
1.6000

0.625 * 2 = 1.25
   10
   40
1.200

$$ (0.11)2 = 1/2 + 1/4 = (0.75){10} $$

$$ (0.8125){10} - (0.75){10} = 0.0625 $$