CHAPTER 1
DIGITAL SYSTEMS AND BINARY
NUMBERS
1
LECTURE 1
OUTLINE OF CHAPTER 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Digital Systems
Binary Numbers
Number-base Conversions
Octal and Hexadecimal Numbers
Complements
Signed Binary Numbers
Binary Codes
Binary Storage and Registers
Binary Logic
2
2
DIGITAL SYSTEMS AND BINARY
NUMBERS
Digital computers
General purposes
Many scientific, industrial and commercial
applications
Digital systems
Digital Telephone
Digital camera
Electronic calculators
Digital TV
3
3
ANALOG AND DIGITAL SIGNAL
Analog system
The physical quantities or signals may vary continuously over
a specified range.
Digital system
The physical quantities or signals can assume only discrete
values.
X(t)
X(t)
t
Analog signal
t
Digital signal
4
4
BINARY DIGITAL SIGNAL
For digital systems, the variable takes on
discrete values.
Two level, or binary values.
Binary values are represented abstractly by:
Digits 0 and 1
Words (symbols) False (F) and True (T)
Words (symbols) Low (L) and High (H)
And words Off and On
5
DECIMAL NUMBER SYSTEM
Base (also called radix) = 10
Digit Position
Weight = (Base)
1
0
5 1 2
-1
-2
7 4
Position
Magnitude
2
Integer & fraction
Digit Weight
10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
100
10
1
0.1 0.01
10
2
0.7 0.04
Sum of “Digit x Weight”
Formal Notation
500
d2*B2+d1*B1+d0*B0+d-1*B-1+d-2*B-2
(512.74)10
6
6
OCTAL NUMBER SYSTEM
Base = 8
Weights
Weight = (Base)
Position
Magnitude
8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }
Sum of “Digit x Weight”
Formal Notation
64
8
1
1/8 1/64
5 1 2
7 4
2
-1
1
0
-2
5 *82+1 *81+2 *80+7 *8-1+4 *8-2
=(330.9375)10
(512.74)8
7
7
BINARY NUMBER SYSTEM
Base = 2
Weights
2 digits { 0, 1 }, called binary digits or “bits”
Weight = (Base)
Position
Magnitude
Sum of “Bit x Weight”
Formal Notation
Groups of bits
8 bits = Byte
11000101
4
2
1
1/2 1/4
1 0 1
0 1
2
-1
1
0
-2
1 *22+0 *21+1 *20+0 *2-1+1 *2-2
=(5.25)10
(101.01)2
8
8
HEXADECIMAL NUMBER SYSTEM
Base = 16
Weights
Weight = (Base)
Position
Magnitude
16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }
Sum of “Digit x Weight”
Formal Notation
256
16
1
1/16 1/256
1 E 5
7 A
2
-1
1
0
-2
1 *162+14 *161+5 *160+7 *16-1+10 *16-2
=(485.4765625)10
(1E5.7A)16
9
9
THE POWER OF 2
n
2n
n
2n
0
20=1
8
28=256
1
21=2
9
29=512
2
22=4
10
210=1024
3
23=8
11
211=2048
4
24=16
12
212=4096
5
25=32
20
220=1M
Mega
6
26=64
30
230=1G
Giga
7
27=128
40
240=1T
Tera
Kilo
10
10
ADDITION
Decimal Addition
1
+
1
1
Carry
5
5
5
5
1
0
= Ten ≥ Base
➔ Subtract a
Base
11
11
BINARY ADDITION
Column Addition
1 1 1 1 1 1
1 1 1 1 0 1
= 61
1 0 1 1 1
= 23
1 0 1 0 1 0 0
= 84
+
≥ (2)10
12
12
BINARY SUBTRACTION
Borrow a “Base” when needed
1
2
0 2 2 0 0 2
−
= (10)2
1 0 0 1 1 0 1
= 77
1 0 1 1 1
= 23
0 1 1 0 1 1 0
= 54
13
13
BINARY MULTIPLICATION
Bit by bit
1 0 1 1 1
x
1 0 1 0
0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
1 1 1 0 0 1 1 0
14
14
DECIMAL (INTEGER) TO BINARY CONVERSION
Divide the number by the ‘Base’ (=2)
Take the remainder (either 0 or 1) as a coefficient
Take the quotient and repeat the division
Example: (13)10
Quotient Remainder
13 / 2 =
6/2=
3/2=
1/2=
Answer:
6
3
1
0
1
0
1
1
Coefficient
a0 = 1
a1 = 0
a2 = 1
a3 = 1
(13)10 = (a3 a2 a1 a0)2 = (1101)2
15
MSB
LSB
15
DECIMAL (FRACTION) TO BINARY CONVERSION
Multiply the number by the ‘Base’ (=2)
Take the integer (either 0 or 1) as a coefficient
Take the resultant fraction and repeat the multiplication
Example: (0.625)10
Integer Fraction Coefficient
0.625 * 2 =
0.25 * 2 =
0.5
*2=
Answer:
1
0
1
.
.
.
25
5
0
a-1 = 1
a-2 = 0
a-3 = 1
(0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2
MSB
LSB
16
16
DECIMAL TO OCTAL CONVERSION
Example: (175)10
Quotient Remainder
175 / 8 =
21 / 8 =
2 /8=
21
2
0
Answer:
Coefficient
a0 = 7
a1 = 5
a2 = 2
7
5
2
(175)10 = (a2 a1 a0)8 = (257)8
Example: (0.3125)10
Integer Fraction Coefficient
0.3125 * 8 =
0.5
*8=
Answer:
2
4
.
.
5
0
a-1 = 2
a-2 = 4
(0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8
17
17
BINARY − OCTAL CONVERSION
8 = 23
Each group of 3 bits represents an
octal digit
Assume Zeros
Example:
( 1 0 1 1 0 . 0 1 )2
( 2
6
. 2 )8
Octal
Binary
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
Works both ways (Binary to Octal & Octal to
Binary)
18
18
BINARY − HEXADECIMAL CONVERSION
16 = 24
Each group of 4 bits represents a
hexadecimal digit
Assume Zeros
Example:
( 1 0 1 1 0 . 0 1 )2
(1
6
. 4 )16
Hex
Binary
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Works both ways (Binary to Hex & Hex to
Binary)
19
19
OCTAL − HEXADECIMAL CONVERSION
Convert to Binary as an intermediate step
Example:
( 2
6
.
2 )8
Assume Zeros
Assume Zeros
( 0 1 0 1 1 0 . 0 1 0 )2
(1
6
.
4 )16
Works both ways (Octal to Hex & Hex to
Octal)
20
20
DECIMAL, BINARY, OCTAL AND HEXADECIMAL
Decimal
Binary
Octal
Hex
00
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
00
01
02
03
04
05
06
07
10
11
12
13
14
15
16
17
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
21
21
0