Number System: Representation and code

1
Number System Representation and code
2
Data Representation Unit and Processing Unit

• In order to make a computer work, it is necessary
  to convert the information we use in daily life
  into a format that can be understood by the
  computer.
• Binary (base 2)
• Binary digit Bits 0, 1Bytes 00000000
  11111111Words 0000000000000000
  1111111111111111
• Octal (base 8)
• Octal digital 0,1,2,3,4,5,6,7
• Hexadecimal (base 16)
• Hexadecimal digit 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
  
• base N

3
base N Example
Weight summation method
4
Decimal to Binary Conversion

• Weight summation method
• Repeating Division method
• Repeating Multiplication method

5
Repeating Division Method
4310
43 2 21 remainder 1
21
21 2 10 remainder 1
10 2 5 remainder 0
1010112
5 2 2 remainder 1
2 2 2 remainder 0
1 2 0 remainder 1
6
Repeating Multiplication method
0.312510
0.3125 x 2 0.625 carry 0
0.625
0.01012
0.625 x 2 1.25 carry ?
0.25
1
0.25 x 2 0.5 carry 0
0.5 x 2 1.0 carry 1
7
Conversion Between Bases

• Base 2 Base 2
• Base 3 Base 3
• Base 4 Decimal Base 4
• Base R Base R
• There are shortcuts for Base 2,8,16

Generally, conversion between bases can be done
using decimal numbers
8
Conversion Binary-Octal/Hexadecimal

• BinarygtOctal Grouped 3-bit from radix point
• OctalgtBinary Invert
• BinarygtHexadecimal Grouped 4-bit from radix
  point
• HexadecimalgtBinary

9
Binary Arithmetic Operation

• Addition
• Subtraction
• Multiplication

10
Binary Arithmetic Operation

• ADDITION
• Similar to decimal
• Example
• 110112 64710
• 100112 53710
• ----------------------
  ----------------------
• 1011102 118410
• ----------------------
  ----------------------

11
Binary Arithmetic Operation

• SUBTRACTION
• Example
• 110112 64710
• 100112 - 53710 -
• ----------------------
  ----------------------
• 001102 9010
• ----------------------
  ----------------------

12
Binary Arithmetic Operation

• MULTIPLICATION
• Example
• 110112 64710
• 100112 x 53710 x
• -------------------------------------------
  --------------------------------------------
  11011 4529
• 110110 19410
• 0000000 323500
• 00000000
• 110110000 -----------------------------
  ---------------
• -------------------------------------------
  34743910
• 10000000012

13
Signed Number Representations

• In mathematics, negative numbers in any base are
  represented in the usual way, by prefixing them
  with a "-" sign.
• However, on a computer, there are various ways of
  representing a number's sign.
• Four methods of extending the binary numeral
  system to represent signed numbers
• sign-and-magnitude
• ones' complement
• two's complement
• excess-N

14
Sign-and-magnitude

• Negative number always written with sign at the
  front
• Example
• -2010
• -10010
• In computer memory, sign is represent by number
• 0 for
• 1 for -

Magnitude
Sign
15
Sign-and-magnitude

• Some early binary computers (e.g. IBM 7090) used
  this representation.

Value 0 0 0000000 (0)10 1 0000000 - (0)10
16
Ones' complement

• The ones' complement form of a negative binary
  number is the bitwise NOT applied to it the
  complement of its positive counterpart.
• Like sign-and-magnitude representation, ones'
  complement has two representations of 0 00000000
  (0) and
• 11111111 (-0).

17
Ones' complement

• This numeric representation system was common in
  older computers the PDP-1 and UNIVAC 1100/2200
  series, among many others, used ones'-complement
  arithmetic.

18
First Complement

• Number x,
• n-bit can represent first complement
• -x 2n - x - 1
• Example
• -(1)10 (28 - 1 - 1)10
• 25410
• 111111101s

19
Ones' complement

• ISSUE
• Addition of -1 to 2

20
Two's complement

• The problems of multiple representations of 0 and
  the need for the end-around carry are
  circumvented by a system called two's complement.
  
• In two's complement, negative numbers are
  represented by the bit pattern which is one
  greater (in an unsigned sense) than the ones'
  complement of the positive value.
• In two's-complement, there is only one zero
  (00000000).
• Negating a number (whether negative or positive)
  is done by inverting all the bits and then adding
  1 to that result.
• 11111101s
• 11111112s

11111112s 00000102s ------------------ 10000
0012s
Value for -1
Ones' complement
Twos complement
Addition of -1 to 2
Ignore
21
Second Complement

• Number x,
• n-bit can represent second complement
• -x 2n - x
• Example
• -(1)10 (28 - 1)10
• 25510
• 111111112s

22
Excess-N

• Excess-N, also called biased representation, uses
  a pre-specified number N as a biasing value.
• A value is represented by the unsigned number
  which is N greater than the intended value.
• Thus 0 is represented by N, and -N is represented
  by the all-zeros bit pattern.

23
Base -2

• In conventional binary number systems, the base,
  or radix, is 2 thus the rightmost bit represents
  20, the next bit represents 21, the next bit 22,
  and so on.
• However, a binary number system with base -2 is
  also possible.
• The rightmost bit represents (-2)0 1, the next
  bit represents (-2)1 -2, the next bit (-2)2
  4 and so on, with alternating sign.

24
Comparison table
25
N Complement
-x 2n - x Bn - X B Base n no. of bit X
value

• -x 2n - x - 1
• (Bn - 1) - X
• B Base
• n no. of bit
• X value
• one complement for (0101)2
• (24-1)-01011s
• 1111 - 01011s 10101s
• ninth complement for (22)10
• (102-1)-229s
• 99 - 229s779s

two complement for (0101)2 (24-0101)2s (10000 -
00101)2s 10112s tenth complement for (22)10
(102-22)10s (100 22)10s7810s
26
Subtraction using B Complement

• Given two n-digit base-B unsigned numbers,
• I J, Subtraction for (I-J) is as
• Add I to B-complement for J
• ? (5 N - 3 )10
• (2 N)10 N 2n 28
• (2 256)10
• (100000010)2s ? (00000010)2s
• ? (5 - 3 10n)10s N 10n 101
• (2 10)10s
• (12)10s ? (12 - 101)10
• ? (2)10

510 (-3)10
-x 2n - x (two complement)
E.g. 510 - 310
I (Bn - J) (I - J) Bn
n 1 bit
If I?J, there is one final carry Bn, ignore
final carry to obtain answer as I-J
27
Subtraction using B Complement

• If I?J, no final carry Bn,
• (I - J) Bn
• (10 N - 22 )10
• (-12 N)10 N 2n 28
• (-12 256)10
• (11110100)2s
• (10 - 22 10n)10s N 10n 102
• (-12 100)10s
• (88)10s? (88 -102)10
• ? - (12)10

E.g. 1010 - 2210
I (Bn - J) (I - J) Bn
n 2 bits
28
Exercise

• Identify the following result using i) Two and
  ii) Tenth Complement
• 33 10 - 2210
• 01102 - 00102
• 00112 01112
• (33 N - 22)10 ? (11 256)10 ? 1000010112s?
  000010112s
• (33 N - 22)10 ? (11 102)10s ? (111)10s ?
  (11)10
• ? (111-102)10 ? (11)10
• (0110 N - 0010)2 ? (0100 28)2s ? (100000100)2s
  ? (00000100) 2s
• (0100 24)2s ?
  (10100) 2s ? (0100) 2s
• (0110 N - 0010)10 ? (0100101)10s? (410)10s ?
  (14)10s ? (4)10
• ?
  (14-101)10 ? (4)10
• (0011 N - 0111)2 ? (-0100 28)2s ? (11111100)
  2s

29
Subtraction using B-1 Complement

• Add I to B-1 complement for J
• I (Bn - 1 - J) (I - J - 1) Bn
• If I?J, there is one final carry Bn,
• ignore final carry to obtain answer as I-J
• If I?J, no final carry Bn

30
Subtraction using B-1 Complement
-x 2n - x - 1

• I (Bn - 1 - J) (I - J -1) Bn
• (5 N - 3 - 1)10
• (1 N)10 N 2n 28
• (1 256)10
• (100000001)1s
• ? (00000001 1)2
• ? (00000010)2
• (5 10n - 3 - 1)10 N 10n 101
• (1 10)9s
• (11)9s
• ? (1 1)10
• ? (2)10

E.g. 510 - 310
n 1 bit

• If I?J, there is one final carry Bn,
• ignore and add the final carry to obtain answer
  as I-J

31
Subtraction using B-1 Complement

• If I?J, no final carry Bn,
• (I - J - 1) Bn
• E.g. 1010 - 2210 (10 N - 22 - 1)10
• (-13 N)10 N 2n 28
• (-13 256)10
• (243)10
• ? (11110011)1s
• 1210 (00001100)2
• -1210 (11110011)1s
• (10 - 22 - 1 10n)9s N 10n 102
• (-13 100)9s
• (87)9s
• ? (871)10s
• ? 88 - 10210
• ? - (12)10

-x 2n - x 1
-x 10n -1 - x (ninth complement)
n 2 bits
-22 (102 -1- 22)9s (87)9s