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ECE2030
Introduction to Computer Engineering
Lecture 2: Number System
Prof. Hsien-Hsin Sean Lee
School of Electrical and Computer Engineering
Georgia Tech
Decimal Number Representation
• Example: 90134 (base-10, used by Homo Sapien)
= 90000 + 0 + 100 + 30 + 4
= 9*104 + 0*103 + 1*102 + 3*101 + 4*100
• How did we get it?
10 90134
10 9013
10 901
10 90
9
2
4
3
1
0
2
Generic Number Representation
• 90134
= 9*104 + 0*103 + 1*102 + 3*101 + 4*100
• A4 A3 A2 A1 A0 for base-10 (or radix-10)
= A4*104 + A3*103 + A2*102 + A1*101 + A0*100
(A is coefficient; b is base)
• Generalize for a given number N w/ base-b
N = An-1 An-2 … A1 A0
N = An-1*bn-1 + An-2*bn-2 + … + A2*b2 + A0*b0
**Note that A < b
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Counting numbers with base-b
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
90
21
91
22
92
23
93
24 ….. 94
25
95
26
96
27
97
28
98
29
99
Base-10
4
100
101
102
103
104
105
106
107
108
109
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20
70
21
71
22
72
23
73
24 ….. 74
25
75
26
76
27
77
100
101
102
103
104
105
106
107
How about Base-8
4
How about base-2
0
1
5
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
5
How about base-2
0
1
6
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
6
How about base-2
0=0
1=1
10 = 2
11 = 3
100 = 4
101 = 5
110 = 6
111 = 7
1000 = 8
1001 = 9
1010 = 10
1011 = 11
1100 = 12
1101 = 13
1110 = 14
1111 = 15
Binary = Decimal
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Derive Numbers in Base-2
• Decimal (base-10)
– (25)10
• Binary (base-2)
– (11001)2
• Exercise
2 25
1
8
1
2 12
2 6
2 3
0
0
1
8
Base-2
• Decimal (base-10)
• Exercise
– (982)10
• Binary (base-2)
– (1111010110)2
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9
Base 8
• Decimal (base-10)
• Exercise
– (982)10
• Octal (base-8)
– (1726)8
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10
Base 16
• Decimal (base-10)
– (982)10
• Hexadecimal (base-16)
• Hey, what do we do when we
count to 10??
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•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
a
b
c
d
e
f
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Base 16
• (982)10 = (3d6)16
• (3d6)16 can be written as (0011 1101 0110)2
• We use Base-16 (or Hex) a lot in computer
world
– Ex: A 32-bit address can be written as
0xfe8a7d20 (0x is an abbreviation of Hex)
– Or in binary form
1111_1110_1000_1010_0111_1101_0010_0000
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Number Examples with Different Bases
• Decimal (base-10)
– (982)10
• Binary (base-2)
– (01111010110)2
• Others examples:
– base-9 = (1321)9
– base-11 = (813)11
– base-17 = (36d)17
• Octal (base-8)
– (1726)8
• Hexadecimal (base-16)
– (3d6)16
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Convert between different bases
• Convert a number base-x to base-y, e.g. (0100111)2 to (?)6
– First, convert from base-x to base-10 if x  10
– Then convert from base-10 to base-y
0100111 = 026 + 125 + 024 + 023 + 122 + 121 + 120 = 39
6 39
3
6 6
1
0
 (0100111)2 = (103)6
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Base-b Addition
Negative Number Representation
• Options
– Sign-magnitude
– One’s Complement
– Two’s Complement (we use this in this course)
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Sign-magnitude
• Use the most significant bit (MSB)
to indicate the sign
– 0: positive, 1: negative
• Problem
– Representing zeros?
– Do not work in computation
• We will NOT use it in this course !
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+0
000
+1
001
+2
010
+3
011
-3
111
-2
110
-1
101
0
100
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One’s Complement
• Complement (flip) each bit in a
binary number
• Problem
– Representing zeros?
– Do not always work in computation
• Ex: 111 + 001 = 000  Incorrect !
• We will NOT use it in this course !
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+0
000
+1
001
+2
010
+3
011
-3
100
-2
101
-1
110
0
111
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Two’s Complement
• Complement (flip) each bit in a
binary number and adding 1, with
overflow ignored
• Work in computation perfectly
• We will use it in this course !
3
011
-3
101
One’s complement
One’s complement
100
010
Add 1
-3
19
101
Add 1
3
011
19
Two’s Complement
• Complement (flip) each bit in a
binary number and adding 1, with
overflow ignored
• Work in computation perfectly
• We will use it in this course !
100
One’s complement
011
Add 1
100
20
The same 100 represents
both 4 and -4
which is no good
0
000
+1
001
-1
111
+2
010
-2
110
+3
011
-3
101
??
100
20
Two’s Complement
• Complement (flip) each bit in a
binary number and adding 1, with
overflow ignored
• Work in computation perfectly
• We will use it in this course !
100
One’s complement
011
Add 1
100
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MSB = 1 for negative
Number, thus 100
represents -4
0
000
+1
001
-1
111
+2
010
-2
110
+3
011
-3
101
-4
100
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Range of Numbers
• An N-bit number
N
– Unsigned: 0 .. (2 -1)
N-1
N-1
– Signed: -2 .. (2 -1)
• Example: 4-bit
0000 (0)
1110 (-8)
Unsigned numbers
1111 (15)
0111 (7)
Signed numbers
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Binary Computation
010001 (17=16+1)
001011 (11=8+2+1)
--------------011100 (28=16+8+4)
Unsigned arithmetic
010001 (17=16+1)
101011 (43=32+8+2+1)
--------------111100 (60=32+16+8+4)
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Signed arithmetic (w/ 2’s complement)
010001 (17=16+1)
101011 (-21: 2’s complement=010101=21)
--------------111100 (2’s complement=000100=4, i.e. -4)
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Binary Computation
The carry is
discarded
The carry is
discarded
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Unsigned arithmetic
101111 (47)
011111 (31)
--------------001110 (78?? Due to overflow, note that
62 cannot be represented
by a 6-bit unsigned number)
Signed arithmetic (w/ 2’s complement)
101111 (-17 since 2’s complement=010001)
011111 (31)
--------------001110 (14)
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BACKUP
Application of Two’s Complement
• The first Pocket Calculator “Curta”
used Two’s complement method for
subtraction
• First complement the subtrahend
– Fill the left digits to be the same length
of the minuend
– Complemented number = (9 – digit)
• 4’s complement = 5
• 7’s complement = 2
• 0’s complement = 9
• Add 1 to the complemented number
• Perform an addition with the
minuend
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Examples
• 13 – 7
– Two’s complement of 07 = 92 + 1 = 93
– 13 + 93 = 06 (ignore the leftmost carry digit)
• 817 – 123
– Two’s complement of 123 = 876 + 1 = 877
– 817 + 877 = 694 (ignore the leftmost carry digit)
• 78291 – 4982
– Two’s complement of 04982 = 95017 + 1 = 95018
– 78291 + 95018 = 73309 (ignore the leftmost carry digit)
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