1. Chapter 2 Data Types and Representations
Decimal, Binary, Octal, and Hexadecimal Number System
Complement Number System
Fixed Point and Floating Point Numbers
Character Codes, Cube Representation
Chapter 2
Decimal, Binary, Octal, and Hexadecimal Number System
Complement Number System
Fixed Point and Floating Point Numbers
Character Codes, Cube Representation

2. Positional Number System
Each number is represented by a string of digits, in which the position of each digit has an associated weight
= 1 ∙ ∙ ∙ ∙ ∙ ∙ 0.01
In general, any decimal number D of the form
has the value
dm – 1 dm – 2 …d1 d0 .d–1 d–2 …d–n
Least significant digit
Radix point
Most significant digit
Positional Numbers
D = dm – 1 ∙ 10m – 1 + …+ d0 ∙ d–1 ∙ 10–1 + …+ d–n ∙ 10–n =
Radix

3. Binary Number System General form of a binary number:
Its value is equivalent to
Examples
bm – 1 bm – 2 …b1 b0 b–1 b–2 …b–n
Least significant bit (LSB)
Binary point
Most significant bit (MSB)
B =
Radix 2
Binary Number System
= 1 ∙ ∙ ∙ ∙ ∙ 1 = 2110
= 1 ∙ ∙ ∙ ∙ ∙ ∙ 1 = 5310
= 1 ∙ ∙ ∙ ∙ ∙ =
= 1 ∙ ∙ ∙ ∙ =

4. Octal and Hexadecimal Numbers
Binary
Octal
Decimal
Hexadecimal 1 10 2 11 3
100 4
101 5
110 6
111 7
1000 8
1001 9
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F
10000 20
10001 21
Numbers
Radix
Binary 2
Octal 8
Decimal 10
Hexadecimal 16
Octal and Hexadecimal Numbers
Different representation of numbers 0 through 17
Back to Conversion of Numbers (Part II)

5. Conversion of Numbers (Binary, Octal, Hexadecimal)
Binary to Octal Conversion
Start at the binary point and work left. Separate the bits into groups of three, replace each group with corresponding octal digit.
Binary to Hexadecimal Conversion
Start at the binary point and work left. Separate the bits into groups of four, replace each group with corresponding hexadecimal digit.
= = 12348
Conversion of Numbers (Part I)
= = 29C16

6. Conversion of Numbers (Binary, Octal, Hexadecimal)
Conversion of Fractions
Start at the binary point. Group the binary digits on the right into groups of three (Octal) or four (Hexadecimal).
Conversion to Binary Numbers
Replace each octal or hexadecimal digit with corresponding 3-bit or 4-bit binary string from the Octal and Hexadecimal Table.
= = 0.568
= = 0.B816
Conversion of Numbers (Part II)
7658 = =
FED16 =
FED.CBA16 =

7. Conversion to Decimal Numbers
S = S ∙ r + di
i = m – 1
S = 0
Start
i = 0
i = i – 1
Done
D = = ((…((dm – 1)r + dm – 2)r + …)r + d1)r + d0
yes
Conversion to Decimal Numbers no

8. Conversion from Decimal Numbers
Divide S by r
S = quotient
di = remainder
i = 0
S = D
Start
i = m – 1
i = i + 1
Done no
yes
D = = ((…((dm – 1)r + dm – 2)r + …)r + d1)r + d0
Dividing the top equation by r, we obtain the quotient Q and remainder R
Q = (…((dm – 1)r + dm – 2)r + …)r + d1
R = d0
Conversion from Decimal Numbers

9. Conversion of Decimal Numbers Examples
Problem: Convert (a) 179 to binary, (b) 467 to octal, and (c) 3417 to hexadecimal.
Solution:
(b) 4 6 7 ÷ 8 = 5
remainder 3 (LSD)
remainder 2
remainder 7 (MSD)
Therefore,
46710
7238
(a) 1 7 9 ÷ 2 = 8
remainder 1 (LSD) 4
remainder 1
remainder 0 5
remainder 1 (MSD)
Therefore,
17910
(c) 3 4 1 7 ÷ 16 = 5 8
remainder 9 (LSD) 2
remainder 5
remainder 13 (MSD)
Therefore,
341710
D5916
Examples: Convert (a) 179 to binary, (b) 467 to octal, and (c) 3417 to hexadecimal.

10. Addition of Binary Numbersxi yi ci
ci+1 si 1
ci+1, si = xi + yi + ci
c0 = 0
i = 0
Start
i = m
i = i + 1
Done no
yes x 9 8 7 y 1 2 3
Carries
x + y
Decimal Addition
Addition of Binary Digits
512
256
128 64 32 16 8 4 2 1
x (987)
y (123)
Carries
x + y (1110)
s10 s9 s8 s7 s6 s5 s4 s3 s2 s1 s0
Addition of Binary Numbers
Binary Addition

11. Subtraction of Binary Numbersxi yi bi
bi+1 di 1
bi+1, di = xi – yi – bi
b0 = 0
i = 0
Start
i = m
i = i + 1
Done no
yes x 9 8 7 y 1 2 3
Borrows
x – y 6 4
Decimal Subtraction
Subtraction of Binary Digits
512
256
128 64 32 16 8 4 2 1
x (987)
y (123)
Borrows
x – y (864) d9 d8 d7 d6 d5 d4 d3 d2 d1 d0
Subtraction of Binary Numbers
Binary Subtraction
Back to Two’s-complement Subtraction

12. Sign Magnitude Representation
Start
addition
subtraction
s2 = s′2
s1 = s2
mr = m1 + m2
sr = s1
m1 > m2
m1 = m2
mr = m1 – m2
mr = 0
sr = 0
mr = m2 – m1
sr = s2
Done
yes no
A sign magnitude number <s, m>, consists of two parts:
Sign and Magnitude
The sign is either + or –
The magnitude is an integer between 0 and the largest representable value
Examples:
Sign Magnitude Representation =
= –12310
D1 = < s1, m1 > and D2 = < s2, m2 >
yields the result Dr = < sr, mr >

13. Complement Number System
Radix-complement of a number D = is equal to:
If digit complement d′ = (r – 1) – d then
Therefore,
is a negative number of D , since
(rm – 1) – D = ( (r – 1) (r – 1) … (r – 1) – (dm – 1 dm – 2 … d0 ) )
= ( (r – 1) – dm – 1) ( (r – 1) – dm – 2 ) … ( (r – 1) – d0 )
= d′m – 1 d′m – 2 … d′0 = = D′
Complement Number System (Part I)

14. Complement Number System
Decimal
Two’s Complement
Sign- Magnitude -8
1000 - -7
1001
1111 -6
1010
1110 -5
1011
1101 -4
1100 -3 -2 -1
0000
1000 or 0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111
Digit
Binary
Octal
Decimal
Hexa-decimal 1 7 9 F 6 8 E 2 - 5 D 3 4 C B A
Complement Number System (Part II)
Digit Complements
Two’s Complement and Sign-Magnitude Representations

15. Two’s-complement Addition
Adding two positive numbers generates a correct result
Adding two negative numbers generates a correct result if carry is ignored 1
(+2) +
(+4)
(+6)
Two’s-complement Addition (Part I) 1
(–2) +
(–4)
Ignored carry = 1
(–6)

16. Two’s-complement Addition
Adding a positive and a negative number generates a correct result
Adding large numbers may generate an incorrect result because of an overflow 1
(+2) +
(–4)
(–2) 1
(+4) +
(+5)
(–7) 1
(–4) +
(–5)
Ignored carry = 1
(+7)
Two’s-complement Addition (Part II)
Overflow rule: an overflow occurs when the sign of the sum is different than the signs of both operands

17. Two’s-complement Subtraction
Direct subtraction using subtraction standard procedure
Subtraction using two’s complement
Start
addition
subtraction
B2 = B2‘ + 1
Br = B1 + B2
Done
(+2) 1
Two’s complement of (+4) +
Carries
(–2)
(+2) 1
(+4) –
Borrows
(–2) ignore borrow =1
(–4) 1
Two’s complement of (–8) +
Carries
(+4) ignore carry =1
Two’s-complement Subtraction

18. Binary Multiplication
Decimal Multiplication
Binary Multiplication 1 4
multiplicand × 3
multiplier 2
3 × multiplicand
1 × multiplicand 8
product 1
multiplicand (14) ×
multiplier (13)
product (182)
Binary Multiplication

19. Shift-and-add Multiplication
MR = multiplier bm–1 …b1 b0 (m-bit)
MD = multiplicand (n-bit)
PP = partial product
Start
Example of shift-and-add multiplication
PP = 0
i = 0 1
multiplicand (14) ×
multiplier (13)
initial partial product
shifted multiplicand
second partial product
shifted zeros
third partial product
fourth partial product
product (182)
bi = 1 no
yes
PP = PP + (2i × MD)
i = i + 1
Shift-and-add Multiplication
i = m no
yes
Product = PP
Done

20. Two’s-complement Multiplication
Use multiplication procedure for unsigned numbers
Negate multiplicand if multiplier sign is negative 1
multiplicand (–14) ×
multiplier (–13)
extended partial product
extended multiplicand
extended shifted multiplicand
all zeros
extended, shifted, and negated multiplicand
ignore carry
product (182)
Note:
Carry out of MSB in the third PP is ignored before sign extension
Carry out of MSB is ignored in the final product
Sign truncation is needed in the final product
Two’s-complement Multiplication

21. Binary Division Decimal Division Binary Division 1 3 quotient 14 ) 8 6
dividend 4
shifted (divisor x 1)
reduced dividend 2
shifted (divisor x 3)
remainder 1
quotient (13)
1110 )
dividend (186)
shifted divisor
reduced dividend
remainder (4)
Binary Division

22. Floating-point Numbers
Floating-point numbers have the form
mantissa × (radix) exponent
Since radix is implicit, only mantissa and exponent must be represented explicitly
Floating-point numbers are fixed-point numbers given by the mantissa, whose radix point is specified by the exponent
Exponent is represented in the excess-code format called characteristic, obtained by adding a bias to the exponent:
bias = radixs – 1
where s is equal to the number of bits in the exponent field
Floating-point Numbers
Mantissa
sign
Signed
exponent
magnitude
Sign
Excess-127 characteristic
Normalized
Fraction
General format
32-bit standard
Implied binary point
Sign
Excess-1032 characteristic
Normalized
Fraction
64-bit standard
Implied binary point

23. Fixed-point vs. Floating-point
The range is the interval of numbers from the largest to the smallest representable number
The precision is the amount of numbers in a number interval
4-digit fixed number
4-digit floating-point number
Integer
Mantissa
Exponent
Representable numbers
0 – 9999
0 – 99 × 10
Range
~ 104
~ 10101
Precision
~ 100×
~ 1×
Example
1001 numbers between 1000 and 2000
11 numbers between 1000 and 2000
Fixed-point vs. Floating-point

24. Binary Codes for Decimal Numbers
Binary-coded Decimals (BCD)
2421 Code
Excess-3 Code
Biquinary Code
Decimal digit
BCD (8421)
2421
Excess-3
Biquinary
0000
0011 1
0001
0100 2
0010
0101 3
0110 4
0111 5
1011
1000 6
1100
1001 7
1101
1010 8
1110 9
1111
Binary Codes for Decimal Numbers

25. American Standard Code for Information Interchange (ASCII)
Character Codes
b3b2b1b0
b6b5b4
000
001
010
011
100
101
110
111
0000
NUL
DLE SP @ P ‘ p
0001
SOH
DC1 ! 1 A Q a q
0010
STX
DC2 “ 2 B R b r
0011
ETX
DC3 # 3 C S c s
0100
EOT
DC4 $ 4 D T d t
0101
ENQ
NAK % 5 E U e u
0110
ACK
SYN
& 6 F V f v
0111
BEL
ETB ’ 7 G W g w
1000 BS
CAN ( 8 H X h x
1001 HT EM ) 9 I Y i y
1010 LF
SUB * : J Z j z
1011 VT
ESC + ; K [ k {
1100 FF FS ,
< L \ l |
1101 CR GS - = M ] m }
1110 SO RS .
> N ^ n ~
1111 SI US / ? O _ o
DEL
NUL Null
SOH Start of heading
STX Start of text
ETX End of text
EOT End of transmission
ENQ Enquiry
ACK Acknowledge
BEL Bell
BS Backspace
HT Horizontal tab
LF Line feed
VT Vertical tab
FF Form feed
CR Carriage return
SO Shift out
SI Shift in
SP Space
DLE Data link escape
DC1 Device control 1
DC2 Device control 2
DC3 Device control 3
DC4 Device control 4
NAK Negative acknowledgement
SYN Synchronize
ETB End transmission block
CAN Cancel
EM End of medium
SUB Substitute
ESC Escape
FS File separator
GS Group separator
RS Record separator
US Unit separator
DEL Delete or rubout
Character Codes
American Standard Code for Information Interchange (ASCII)

26. Cube Representation
0 1
(a)
10 11
00 01
Any n-bit string can be represented by an n-cube with 2n vertices, each corresponding to a particular string of n bits
Any m-subcube has the same n – m bits and m bits that take all 2n possible combinations of 0s and 1s
Distance between two vertices is equal to 1 + the number of vertices on the shortest path between two vertices
(c)
(b)
(d)
Cube Representation
n-cubes for n = 1, 2, 3, and 4