ECE456: Number Systems (review)
Instructor: Dr. Honggang Wang
II-209B, hwang1@umassd.edu
ECE Dept., Fall 2013
Administrative Issues (9/16/13)
• Project team set-up due Wednesday, Sept. 25
• If you missed the first class, go to the course
website for syllabus and 1st lecture.
http://www.faculty.umassd.edu/honggang.wang/teaching.html
Dr. Wang
1 Mega bytes = 2? bytes
a)
b)
c)
d)
26 bytes (x)
217 bytes (x)
1024 bytes (x)
216 bytes (x)
e)
f)
g)
h)
From Background Survey
Dr. Wang
1E7 bytes (x)
210 bytes (x)
232 bytes (x)
No answer (x)
Number Systems
• Do you know the equivalent hexadecimal, octal, and
decimal values of the binary number 11001010?
• What is the equivalent binary number of the decimal
number 63?
From Background Survey
Dr. Wang
Conventions
Term
Normal Usage
Usage as a Power of 2
Kilo (K)
103
210 =1,024
Mega (M)
106
220 =1,048,576
Giga (G)
109
230 =1,073,741,824
Tera (T)
1012
240 =1,099,511,627,776
Mili (m)
10-3
Micro (m)
10-6
Nano (n)
10-9
Pico (p )
10-12
• Powers of 2 are most often used in describing memory capacity.
– Ex: 1Kilobyte (KB) =1024 bytes= 210 bytes
• Powers of 10 are used to describe the CPU clock frequencies:
cycles per second (Hz)
– Ex: Pentium 4 --1.8GHz = 1.8x109 Hz
Dr. Wang
Definitions
Term
Definition
bit
0 or 1
byte (B)
a group of 8 bits
nibble (nybble)
half a byte (4 bits)
word (w)
a group of bits that is processed simultaneously.
a word may consist of 8/16/32/other number of bits
machine dependent
(ex: 8086 – 16 bits; 80386/80486/Pentium – 32 bits)
double word
2 words
msb (most significant bit)
the leftmost bit in a word
lsb (least significant bit)
the rightmost bit in a word
Hz (hertz)
reciprocal of second
Dr. Wang
Review of Number Systems
• Overview
• Number systems conversions
Chapter 19 (online chapter)
Or Appendix A in 7th edition
Dr. Wang
Number Systems
• Two basic types of number systems:
– Non-positional
• Ex: Roman numerals: I, II, III, IV, V … X, XI … C
• Normally only useful for small numbers
– Positional
• Ex: the decimal systems
• Each position in which a digit/symbol is written has a different
positional value, which is a power of the base
Dr. Wang
Positional Number Systems
(Example)
Decimal number systems
1. a base of 10 (determines the magnitude of a place).
2. is restricted to 10 re-usable digits/symbols (0,1,2,3,4,5,6,7,8,9)
3. the value of a digit depends on its position
(digit x positional value = digit x baseposition)
4. sum of the value of all digits gives the value of the number.
58710 = 5 x 102 + 8 x 101 + 7 x 100
= 5 x 100 + 8 x 10 + 7 x 1
= 500 + 80 + 7
= 587
Dr. Wang
Positional Number Systems
Decimal (base is 10):
375.1710 = 3 x 102 + 7 x 101 + 5 x 100 + 1 x 10-1 + 7 x 10-2
= 3 x 100 + 7 x 10 + 5 x 1 + 1 x 0.1 + 7 x 0.01
= 300 + 70 + 5 + 0.1 + 0.07
= 375.17
In general (base is b),
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ...
Increase by 1
Dr. Wang
0
Decrease by 1
Exercise (1)
•
Specify the value of the digit 5 in the following decimal
numbers:
25
51
4538
Dr. Wang
• Binary
– Base 2
– 2 symbols:0,1
• Octal
– Base 8
– 8 symbols: 0,1,2,3,4,5,6,7
• Decimal
– Base 10
–
– ,3,4,5,6,7,8,9
10 symbols: 0,1,2,3,4,5,6,7,8,9
• Hexadecimal
– Base 16
– 16 symbols: 0,1,2,3,4,5,6,7,8,9,
A,B,C,D,E,F
– More compact representation
of the binary system
Dr. Wang
Decimal
(base 10)
Binary
Octal
(base 2) (base 8)
Hexadecimal
(base 16)
0
0
0
0
1
1
1
1
2
10
2
2
3
11
3
3
4
100
4
4
5
101
5
5
6
110
6
6
7
111
7
7
8
1000
10
8
9
1001
11
9
10
1010
12
A
11
1011
13
B
12
1100
14
C
13
1101
15
D
14
1110
16
E
15
1111
17
F
16
10000
20
10
17
10001
21
11
Example of Equivalent Numbers
Binary: 1 0 1 0 0 0 0 1 0 1 0 0 1 1 12
Octal: 502478
Decimal: 2064710
Hexadecimal: 50A716
Notice how the number of digits gets
smaller as the base increases.
Dr. Wang
Agenda
• Overview of number systems
– Positional and non-positional
– Base, positional value, symbol value
– Binary, decimal, octal, hexadecimal
• Number systems conversions
Dr. Wang
Number Systems Conversions
•
•
•
•
•
Binary, Octal, and Hex to Decimal
Decimal to Hex, Octal, and Binary
Binary
Hex
Binary
Octal
Hex
Octal
Dr. Wang
1. Binary, Octal, Hex  Decimal
Multiply the decimal equivalent of each digit by its
positional/place value (a power of the base) and sum
these products
In general (base is b: 2 for binary, 8 for Octal, 16 for Hex),
N = ...P3P2P1P0 . P-1P-2P-3...
= ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ...
Dr. Wang
Exercise (2)
• Convert the following numbers to their decimal
equivalents
10011012
1101.112
1AB.616
173.258
Dr. Wang
2. Decimal  Binary, Octal, or Hex
To convert decimal numbers to any base we
divide with the corresponding base until the
quotient is zero and write the remainders in
reverse order.
Dr. Wang
Decimal  Octal, Binary, Hex
• Divide the decimal number successively by
8 (for Octal), 2 (for Binary), 16 (for Hex)
• After each division record the remainder
– Octal: 0,1,…, or, 7
– Binary: either a 1 or 0
– Hex: 1, 2,…, or,9, or A, B, …, or F
• Continue until the result of the division
(quotient) is 0
• Write the remainders in reverse order
Dr. Wang
Exercise (3)
• Convert 123|10 to Base 8
• Convert 59|10 to Base 2
• Convert 42|10 to Base 16
Dr. Wang
Number Systems Conversions
(Agenda)
 Binary, Octal, and Hex to Decimal
 Decimal to Hex, Octal, and Binary
• Binary
Hex
• Binary
Octal
• Hex
Octal
Dr. Wang
Binary
Dr. Wang
Hex
Binary to Hexadecimal Conversion
10100010111001|2 = ?|16
Work from right to left
Divide into 4-bit groups
##10 1000 1011 1001
2
8
B
9
NOTE: # is a place holder for zero!
Dr. Wang
Decimal
Binary
Hexadecimal
0
0000
0
1
0001
1
2
0010
2
3
0011
3
4
0100
4
5
0101
5
6
0110
6
7
0111
7
8
1000
8
9
1001
9
10
1010
A
11
1011
B
12
1100
C
13
1101
D
14
1110
E
15
1111
F
Hexadecimal to Binary Conversion
FACE|16 = ?|2
F
A
C
E
1111 1010 1100 1110
FACE|16=1111101011001110|2
Dr. Wang
Decimal
Binary
Hexadecimal
0
0000
0
1
0001
1
2
0010
2
3
0011
3
4
0100
4
5
0101
5
6
0110
6
7
0111
7
8
1000
8
9
1001
9
10
1010
A
11
1011
B
12
1100
C
13
1101
D
14
1110
E
15
1111
F
Binary
Dr. Wang
Octal
Binary to Octal Conversion
10101110001101|2=?|8
Work from right to left
Divide into 3 bit groups
#10 101 110 001 101
2
5
6
1
5
 10101110001101|2=25615|8
Dr. Wang
Binary Octal
000
0
001
1
010
2
011
3
100
4
101
5
110
6
111
7
Octal to Binary Conversion
Binary Octal
1247|8=?|2
1
2
001 010
4
7
100
111
1247|8=001010100111 |2
=1010100111|2
Note: one need not write
the leading zeros
Dr. Wang
000
0
001
1
010
2
011
3
100
4
101
5
110
6
111
7
Hexadecimal
Dr. Wang
Octal
How do we convert from
hexadecimal to octal and
vice versa?
Convert to binary first!
Dr. Wang
Exercise (4)
•
Do you know the equivalent hexadecimal, octal, and
decimal values of the binary number 11001010?
______Yes _______No
If you answered Yes,
please indicate them below:
–
–
–
•
Equivalent hexadecimal number:________________
Equivalent octal number: __________________
Equivalent decimal number: __________________
What is the equivalent binary number of the decimal
number 63? _____________________________
From Background Survey
Dr. Wang
Exercise (5)
• Convert 18110 to binary and hex
• Convert 121F16 to decimal
• Convert 010101011002 to hex
• Convert A17F16 to octal
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Summary
1. Basic number systems concepts (base,
positional/place value, symbol value)
2. Convert back and forth between decimal
numbers and their binary, octal, and
hexadecimal equivalents
3. Abbreviate binary numbers as octal or
hexadecimal numbers
4. Convert octal and hexadecimal numbers to
binary numbers
Dr. Wang
Solution (1)
•
Specify the value of the digit 5 in the following decimal
numbers:
the 5 in 25 = 5 x 100 = 5
the 5 in 51 = 5 x 101 = 50
the 5 in 4538 = 5 x 102 = 500
Dr. Wang
Solution (2)
10011012 = 1 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x 21+1 x 20
= 64 + 0 + 0 + 8 + 4 + 0 + 1
= 7710
1101.112 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 1 x 2-2
= 8 + 4 + 0 + 1 + 1/2 + 1/4
= 13.7510
1AB.616 = 1 x 162 + A x 161 + B x 160 + 6 x 16-1
= 1 x 256 + 10 x 16 + 11 x 1 + 6 x 16
= 256 + 160 + 11 + 0.375
= 427.37510
173.258 = 1 x 82 + 7 x 81 + 3 x 80 + 2 x 8-1 + 5 x 8-2
= 1 x 64 + 7 x 8 + 3 x 1 + 2/8 + 5/64
= 64 + 56 + 3 + 0.25 + 0.078125
= 123.32812510
Dr. Wang
Solution (3-1)
Convert 123|10 to Base 8:
Base you are converting to
8 )123
8 )15 R 3
8 )1 R 7
0 R 1
Read Up!
Therefore, 123|10 = 173|8
Dr. Wang
Solution (3-2)
• Convert 59|10 to Base 2:
59|10 =1110112
• Convert 42|10 to Base 16:
16 )42
16 )2 R A
0 R 2
Read Up!
Therefore, 42|10 = 2A|16
Dr. Wang
Solution (4)
•
Do you know the equivalent hexadecimal, octal, and
decimal values of the binary number 11001010?
______Yes _______No
If you answered Yes,
please indicate them below:
–
–
–
•
Equivalent hexadecimal number:__CA___________
Equivalent octal number: _______312___________
Equivalent decimal number: _____202_____________
What is the equivalent binary number of the decimal
number 63? ________111111_____________
From Background Survey
Dr. Wang
Solution (5)
• Convert 18110 to binary (10110101) and hex (B5)
• Convert 121F16 to decimal (4639 10 )
• Convert 010101011002 to hex (2AC16)
• Convert A17F16 to octal (1205778)
Dr. Wang