Number Systems: Negative Integers and Floating Point

1
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2
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• ?? ??? ?? ???? ????? ???????? ?? 10 ????? ???? ??
  ???
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• ??? ????? ?? ????? ?? ??? ????? ???? ????? ????
  ???
• ?? ??? ?? ???? ?? ??? ??? ?? ?? ??? ?? ?? ???
  ??? ????
• bit binary digit
• Byte 8 bits
• ??????? ????? ???? ?? ????? ?? ??? ????

3
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• Unsigned Integers
• Binary form,
• Useful for non-negative data
• Characters, Ages, Date,
• Examples 5 in 8 bits is 0000 0101
• N bits hold range 0.. 2N -1
• Signed Integers
• Sign Bit
• Twos Complement

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Negation By Sign bit
sign 1 bit
Number, 7bits

• Examples 52 0000 0101 , -52 1000 0101
• N bits Range -2N-1 -1... 2N-1 -1
• Problems
• 1000 0000 0 0000 0000
• Special algorithm for subtraction

5
???? 2

• Twos complement of a kbits binary number, N, is
  2k - N .
• Examples 2s comp of 5 in 8 bits is 28 - 5 251
• 5 0000 0101
• 251 1111 1011
• Simple conversion Logically NOT the number, then
  add a 1 .

6
???? 2

• ???? 2 ??? k ???? N ????? ??? ??2k N
• ????
• comp(5) 28 - 5 251
• 5 0000 0101? comp(5) 251 1111 1011
• comp(128) 1000 0000 256 128 128
• ???? ?????? ???? 2
• ?????? ??? ?? ?? ?? ? ?????? ?? ?? ?? ??? ?????
  ?? ????
• ??? ???? ?? ?? (1) ??? ?? ????

7
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• ?? ???? ??? ?????? ??? ??? ?? ?????? ??? ??? ???
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8
????? ?????? Fractional Numbers

• Examples 456.7810 4 x 102 5 x 101 6 x 100
  7 x 10-18 x 10-2
• 1011.112 1 x 23 0 x 22 1 x 21 1 x 20
  1 x 2-1 1 x 2-2
• 8 0 2 1
  1/2 ¼
• 11 0.5 0.25 11.7510
• ????? ?? ????? 2 ?? ????? 10
• Examples 111.112 1 x 22 1 x 21 1 x 20
  1 x 2-1 1 x 2-2
• 4 2 1 1/2
  ¼ 7.7510
• Examples 11.0112

9
????? ????? 10 ?? ????? 2

• Examples 7.7510 (?)2
• Conversion of the integer part repeated division
  by 2
• 7 / 2 3 (Q), 1 (R) ? 3 / 2 1 (Q), 1 (R) ?
  1 / 2 0 (Q), 1 (R) 710 1112
• Conversion of the fractional part perform a
  repeated multiplication by 2 and extract the
  integer part of the result
• 0.75 x 2 1.50 ? extract 1
• 0.5 x 2 1.0 ? extract 1
  0.7510 0.112
• 0.0 ? stop
• ? Combine the results from integer and
  fractional part, 7.7510 111.112
• How about choose some of
• Examples try 5.625B

4
2
1
1/2
1/4
1/8
0.25
0.125
0.5
10
????? ??????

• Exercise 1 Convert (0.625)10 to its binary
  form
• Exercise 2 Convert (0.6)10 to its binary form
• Solution

Solution 0.625 x 2 1.25 ? extract 1 0.25 x
2 0.5 ? extract 0 0.5 x 2 1.0 ? extract 1
0.0 ? stop ? (0.625)10 (0.101)2
11
????? ??????

• Exercise 3 Convert (0.8125)10 to its binary
  form

Solution 0.8125 x 2 1.625 ? extract 1
0.625 x 2 1.25 ?
extract 1 0.25 x 2 0.5 ? extract 0 0.5 x 2
1.0 ? extract 1 0.0 ? stop ? (0.8125)10
(0.1101)2
12
????? ????? ??????

• ??? x ???? ?????? ????? ????? ????? ?? ???? ???
• x f Base E
• f mantissa(??????)
• E exponent(???)
• Example 125.3210 0.12532 103
• mantissa
• - 125.3210 - 0.12532 103
• 0.054610 0.546 10 1
• ?????? ?? ???? ????? ???? ???? ?? ???. ??? ?? ???
  ?????? ??? ??? ?? ??? ????? 1 ???? ??? ??? ???
  ??? ???

13
????? ???? ?????

• Example
• 134.1510 0.13415 x 103
• 0.002110 0.21 x 10-2101.11B .1011 x 23 or
  1.011 x 22 (hidden1)0.011B .11 x 2-1 or 1.1 x
  2-2 (hidden1)AB.CDH .ABCD x 162
• 0.00ACH .AC x 16-2

14
????? ????? ??????

• ??? ?? ????? 16 ???? ???? ????? ????? ??? ?????
  ?????? ??????? ??????? ?? ?? ????
• ????? ??? ???? ????? ????? ??? ??? ( 0 ???? ? 1
  ????)
• 11 ??? ???? ???? ????? ??? ????? ?????? ?????.(
  (f
• ???? ??? ????? ???? ????? ???? ??? ?? ????? 7 ???
  (E). ???? ???? ??? ????? ???? ?? ??? ???207
  (0111). 218 (1000), 2-16 (0110)

x f Base E
f 1.?1?2?3?4?11 ?12?15
E converted to binary, b1b2b3b4
LSB
MSB
0 - 1
Excess-7
15
????

• ????? ??? 1110.111010111111 ?? ????????
• ??? ??? ????? ????
• 1110.111010111111
• 1.110111010111111 2 3
• ??? ??? ?????? ??? 1073
• ????? 16 ???? 0 11011101011 1010

16
???? ????? ?????

• ??? 0110 0000 0000 0100 B ?? ??? ????? ???. ???
  ???? ?? ?????
• 0 11000000000 0100 B
• 1.11B 2(4-7)-3 0.00111B
• 7/32 0.21875D

17
IEEE 754 Standard

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• 8 ??? ?????? ???? ??? ?? ???? ?? ??? ?? ??? 127
  ?? ?? ????? ??? ???.
• ????? ?????? double ?? ???? ?????? ??????? ??
  ????

18
IEEE 754 Standard

• ????? ???
• ??? ??? ? ?????? ??? ?????? ??? ??? ???
• ??? ??? ??? ????? 1 ???? ??????? ??? ???
• ??? ??? ?????? ????? ??? ?? ????? ???? ? ??????
  ??? ????? ??? ?? ????? ???
• ??? ??? ?????? ????? ??? ?? ????? ???? ? ??????
  ??? ??? ????? ??? ????? ???? ??? NaN Not a
  Number

19
IEEE 754 Standard

• IEEE 754 Examples Normalized Numbers
• 0 1000 0011 0000 0000 0000 0000 0000 000
• 1 x 24 16
• 0 0011 0001 0000 0000 0000 0000 0000 000
• 1 x 2-78 3.3087e-24
• 0 1000 0001 0100 0000 0000 0000 0000 000
• 1.25 x 22 5

20
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