1. Number Representation
ECE 645: Lecture 1
Number Representation
Part 1

2. Required Reading B. Parhami,
Computer Arithmetic: Algorithms and Hardware Design
Chapter 1, Numbers and Arithmetic, Sections
Chapter 2, Representing Signed Numbers

3. Recommended Reading J-P. Deschamps, G. Bioul, G. Sutter,
Synthesis of Arithmetic Circuits: FPGA, ASIC and
Embedded Systems,
Chapter 1, Introduction
Chapter Weighted systems
Chapter 3.2, Integers

4. Historical Representations: Ancient Egyptians Numerals (1)
~4000 BC
“Sum of Symbols”
= (34)10

5. Historical Representations: Ancient Egyptians Numerals (2)
What number does this stone carving
from Karnak represent?

6. Historical Representations: Ancient Egyptians Numerals (3)

7. Historical Representations: Ancient Egyptians Numerals (4)
Symbol for a fraction (part):
Special symbols for most commonly used fractions:

8. Historical Representations: Ancient Babylonian Numerals (1)
First known positional system
3100 BC
Radix 60 (sexagesimal)
Two symbols:
= = 10
− Integers and fractions were represented
identically - a radix point was not written but
rather made clear by context

9. Babylonian Numerals: Example
1 x x x 600
= (4,856)10

10. Historical Representations: Ancient Babylonian Numerals (2)
Digits from 1 to 59

11. Positional Code with Zero
Zero Represented by Space
Partial solution
What about trailing zeros?
Babylonians Introduced New Symbol or
4th to 1st Century BC
Zero Allows Representation of Fractions
Fractions started with zero

12. Mixed System Roman Numerals Sum of all symbols
I=1 V=5 X=10 L=50 C=100 D=500 M=1000
Difficult to do arithmetic
e.g.,
MCDXLVII IX ?

13. Hindu-Arabic Numeral System
Brahmi numerals, India, 400 BC-400 AD
Evolution of numerals in early Europe

14. Positional Code Decimal System
Documented in the 9th century
Position of coefficient determines its value
Coefficient in position is multiplied by radix (10) raised to the power determined by its position, e.g.,

15. Migration of Positional Notation
~750 AD
Zero spread from India to Arabic countries
~1250 AD
Zero spread to Europe
Importance of Zero
Ease of arithmetic which leads to improved commerce

16. Binary Number System Binary Positional number system
Two symbols, B = { 0, 1 }
Easily implemented using switches
Easy to implement in electronic circuitry
Algebra invented by George Boole ( )
allows easy manipulation of symbols

17. Modern Arithmetic and Number Systems
Modern number systems used in digital arithmetic can be broadly classified as:
Fixed-point number representation systems
Integers
Rational numbers of the form x = a/2f, a is an integer, f is a positive integer
Floating-point number representation systems
x * bE, where x is a rational number, b the integer base, and E the exponent
Note that all digital numbers are eventually coded in bits {0,1} on a computer

18. Encoding Numbers in 4-Bits
fixed point
floating point

19. Classification of number systems (1)
Positional
Non-positional
Fixed-radix
Mixed-radix
Conventional
Unconventional
Binary
Decimal
Hexadecimal
Signed-digit
Non-redundant
Redundant

20. Classification of number systems (2)
Positional
wi - weight of the digit xi
Fixed-radix
r - radix of the number system
Conventional fixed-radix
r integer, r > 0
xi  {0, 1, …, r-1}

21. Classification of number systems (3)
Unconventional fixed-radix
xi  {-, …,  }
Signed-digit >0  negative digits
Non-redundant number of digits =  +  + 1  r
Redundant number of digits =  +  + 1 > r

22. Fixed-point representation
Integral and fractional part
X = xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l
Integral part
Fractional part
Radix point
NOT stored in the register
understood to be in a fixed position

23. Fixed-Radix Conventional (Unsigned) Representations

24. Fixed-Radix Conventional Number Systems
Fixed Point Number system
Positional
Non-positional
Fixed-radix
Mixed-radix
Conventional (unsigned)
Unconventional (signed)
Binary
Decimal
Hexadecimal
Signed-digit
Non-redundant
Redundant

25. Range of numbers Xmin Xmax Number system Decimal
X = (xk-1 xk-2 … x1 x0.x-1 … x-l)10
10k - 10-l
Binary
X = (xk-1 xk-2 … x1 x0.x-1 … x-l)2
2k - 2-l
Conventional fixed-radix
X = (xk-1 xk-2 … x1 x0.x-1 … x-l)r
rk - r-l
Notation:
unit in the least significant position
unit in the last position
ulp = r-l

26. Number of digits in the integer part necessary to cover the range
0..Xmax
Number system
Decimal
Binary
Conventional
fixed-radix

27. Radix Conversion
Whole part
Fractional part
u = w . v
= ( xk–1xk– x1x0 . x–1x– x–l )r Old
= ( XK–1XK– X1X0 . X–1X– X–L )R New
Example: (31)eight = (25)ten
Two methods:
Option 1) Radix conversion, using arithmetic in the old radix r
Convenient when converting from r = 10 or familiar radix
Option 2) Radix conversion, using arithmetic in the new radix R
Convenient when converting to R = 10 or familiar radix
From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

28. Option 1: Arithmetic in old radix r
Converting whole part w: (105)ten = (?)five
Repeatedly divide by five Quotient Remainder
105 0
21 1
4 4
Therefore, (105)ten = (410)five
Converting fractional part v: ( )ten = (410.?)five
Repeatedly multiply by five Whole Part Fraction
Therefore, ( )ten  ( )five
From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

29. Radix Conversion of the Integral Part
R - destination radix
XI = (xk-1 xk-2 … x1 x0)R = =
= ((...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 ) R + x0
Quotient
Remainder
(...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 x0
...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 x1
……….
……….
xk-1
xk-2
xk-1

30. Radix Conversion of the Fractional Part
R - destination radix
XF = (. x-1 x-2 … x-l+1 x-l)R = =
= R-1 (x-1 + R-1 (x-2 + R-1 (…. + R-1 ( x-l+1 + R-1 x-l )….)))
Integer part
Fractional part
x-1
R-1 (x-2 + R-1 (….. + R-1 ( x-l+1 + R-1 x-l)….))
x-2
R-1 (….. + R-1 ( x-l+1 + R-1 x-l)….)
………………………………….
x-l+1
R-1 x-l
x-l
...

31. Option 2: Arithmetic in new radix R
Converting fractional part v: ( )five = (105.?)ten
( )five  55 = (22033)five = (1518)ten
1518 / 55 = 1518 / =
Therefore, ( )five = ( )ten
Converting (22033)five = (?)ten
((((2  5) + 2)  5 + 0)  5 + 3)  5 + 3
|-----| : : : :
: : : :
| | : : :
: : :
| | : :
: :
| | : :
| |
1518
Horner’s
rule or
formula
From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

32. Option 2 cont'd: Horner's rule for fractions
Horner’s rule is also applicable: Proceed from right to
left and use division instead of multiplication
Converting fractional part v: ( )five = (?)ten
(((((3 / 5) + 3) / 5 + 0) / 5 + 2) / 5 + 2) / 5
|-----| : : : :
: : : :
| | : : :
: : :
| | : :
: :
| | : :
| |
2.4288
| |
Horner’s
rule or
formula
From: Parhami, Computer Arithmetic: Algorithms and Hardware Design

33. Radix Conversion Shortcut for r=bg, R=bG
r=bg  b  R=bG
4=22   8=23
( )4 = ( )2 = (261.62)8
Trick here is to first convert to a number in radix b, then to R
Cluster in groups of 3 (because 23 = 8) moving away from binary point

34. Signed Number Representations

35. Representations of signed numbers
Signed-magnitude
Biased
Complement
Diminished-radix
complement
(Digit complement)
Radix-complement
r=2
r=2
Two’s complement
One’s complement

36. Signed- magnitude Two’s complement One’s complement
Biased

37. Signed-magnitude representation of signed numbers
k-1
k-2
magnitude
sign
Advantages:
conceptual simplicity
symmetric range: -(2k-1-1) .. -(2k-1-1)
simple negation
Disadvantages:
addition of numbers with the same sign and with
a different sign handled differently

38. Biased (excess-B) representation of signed integers
R(X) = X + B
B = 2k-1, k=4
-2k-1 ≤ X ≤ 2k-1-1 X R
R(X)

39. Biased representation
with radix 2
Signed number X
Representation
mapping
Unsigned Representation R(X)
Binary mapping
Bit vector (xk-1xk-2...x0.x-1...x-l)

40. Signed Number Representations
Complement
Signed Number Representations

41. Complement representations
with radix 2
Signed number X
Representation
mapping
Unsigned Representation R(X)
Binary mapping
Bit vector (xk-1xk-2...x0.x-1...x-l)

42. Useful dependencies 1 – xi = xi X when X  0 - X when X < 0 xi1 1 1

43. One’s complement transformation
 0
def
OC(A) = A = 2k – 2-l - A
k-1 k l
Properties:
0  OC(A)  2k – 2-l
OC(OC(A)) = A –
Ak-1 Ak-2 … A0 . A-1 A A-l
Ak-1 Ak-2 … A0 . A-1 A A-l

44. One’s Complement Representation of Signed Numbers
For –(2k-1 – 2-l)  X  2k-1 – 2-l
X for X > 0
0 or OC(0) for X = 0
OC(|X|) for X < 0
def
R(X) =
0  R(X)  2k – 2-l

45. One’s complement representation of signed integers
X>0
X<0
k=4 0,
2k-1 X
X+2k-1 = 2k-1 - |X|

46. One’s complement representation of signed numbers

47. Two’s complement transformation (1)
 0
A + 2-l = 2k – A for A > 0
def
TC(A) =
for A = 0
Properties:
2k – A = 2k – A – 2-l + 2-l =
= (2k – 2-l – A)+2-l = A + 2-l
0  TC(A)  2k – 2-l
TC(TC(A)) = A

48. Two’s complement transformation (2)
 0
def
TC(A) =
A + 2-l mod 2k = 2k – A mod 2k

49. Two’s Complement Representation of Signed Numbers
For –2k-1  X  2k-1 – 2-l
X for X  0
TC(|X|) for X < 0
def
R(X) =
0  R(X)  2k – 2-l

50. Two’s complement representation of signed integers
X>0
X<0
k=4 X
X+2k = 2k - |X|

51. Two’s complement representation of signed integers

52. Signed-magnitude representation of signed numbers
X>0
X<0
k=4 0,
2k-1 X
| X|+2k-1 = -X+2k-1

53. Signed-magnitude representation of signed numbers

54. Biased representation of signed numbers
X>0
X<0
B = 2k-1, k=4
X+B B
X+B

55. Biased representation of signed numbers

56. Arithmetic Operations in Signed Number Representations

57. Unsigned addition vs. signed addition
Programmer
Machine
Unsigned
mind
Signed
mind
weight
carry 1 1 1 X Y S + = x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 x0 y0 FA FA FA FA FA FA FA FA c8 c7 c6 c5 c4 c3 c2 c1 s7 s6 s5 s4 s3 s2 s1 s0

58. Out of range flags Carry flag - C C = 1 if result > MAX_UNSIGNED or
out-of-range for unsigned numbers
C = if result > MAX_UNSIGNED or
result < 0
otherwise
where MAX_UNSIGNED = for 8-bit operands
216-1 for 16-bit operands
Overflow flag - V
out-of-range for signed numbers
V = if result > MAX_SIGNED or
result < MIN_SIGNED
otherwise
where MAX_SIGNED = for 8-bit operands
for 16-bit operands
MIN_SIGNED = for 8-bit operands
for 16-bit operands

59. Overflow for signed numbers
Indication of overflow
Positive
+ Positive
= Negative
Negative
+ Negative
= Positive
Formulas
Overflow2’s complement = xk-1 yk-1 sk-1 + xk-1 yk-1 sk-1 =
= ck  ck-1

60. Two’s complement representation of signed integers

61. Addition and subtraction
Two’s complement
Numbers of the same sign
Numbers of the opposite sign
carry but not overflow
no carry but overflow
carry but not overflow

62. Two's Complement Adder/Subtractor Architecture
Can replace this with k xor gates

63. Arithmetic Shift Two’s complement Sh.L {001012 = +5} = 010102 = +10
Shift left may cause overflow
overflow
Sh.R { = +5} = = rem 1
Sh.R { = -5} = = rem 1
Shift right requires sign extension

64. Addition and subtraction
One’s complement
Numbers of the same sign
Numbers of the opposite sign 1 + 1 + 5
-15
Disadvantage: Need another adder after the addition is complete!
end-arround carry

65. Addition and subtraction
Signed-magnitude
Numbers of the same sign
Numbers of the opposite sign
sign bit
magnitude
sign bit
magnitude + + 17
11 > 6
carry = overflow – 1 5

67. Signed Number Representations
Summary

68. Representing k-bit signed binary numbers
Representation
for X>0
Representation
Representation
for 0
Representation
for X<0 0,
2k-1
Signed-
magnitude X
2k-1+|X|
Biased
X+B B
X+B
typical B=2k-1 or 2k-1-ulp
Complement X
0, M mod 2k
M-|X|=M+X
Two’s
complement X
2k-|X|=
One’s
complement
0, 2k-ulp
2k-ulp-|X|= X

69. Value of a number in the signed representations
(xk-1 xk-2 … x1 x0.x-1 … x-l)
Representation
Signed-
magnitude
Biased
Two’s
complement
One’s
complement

70. Extending the number of bits of a signed number
xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l X Y
yk’-1 yk’-2 … yk yk-1 yk-2 … y1 y0 . y-1 y-2 … y-l y-(l+1) … y-l’
signed-magnitude
xk xk-2 … x1 x0 . x-1 x-2 … x-l
two’s complement
xk-1 xk-1 xk xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l
one’s complement
xk-1 xk-1 xk xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l xk xk-1
biased
xk-1
. . .
xk-2 … x1 x0 . x-1 x-2 … x-l

71. Generalized Complement Representation

72. Generalized complement representation
of signed integers
M-N > P
M ≥ N+P+1

73. Generalized complement representation
of signed integers