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Compsci 001 3.1
Today’s topics
Binary Numbers Brookshear 1.1-1.6
Slides from Prof. Marti Hearst of UC Berkeley SIMS
Upcoming Networks
• Interactive Introduction to Graph Theoryhttp://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro
• Kearns, Michael. "Economics, Computer Science, and Policy." Issues in Science and Technology, Winter 2005.
Problem Solving
Compsci 001 3.2
Digital Computers
What are computers made up of? Lowest level of abstraction: atoms Higher level: transistors
Transistors Invented in 1951 at Bell Labs An electronic switch Building block for all modern electronics Transistors are packaged as Integrated Circuits (ICs)
40 million transistors in 1 IC
Compsci 001 3.3
Binary Digits (Bits)
Yes or No On or Off One or Zero 10010010
Compsci 001 3.4
More on binary
Byte A sequence of bits 8 bits = 1 byte 2 bytes = 1 word (sometimes 4 or 8 bytes)
Powers of two How do binary numbers work?
Compsci 001 3.5
Data Encoding Text: Each character (letter, punctuation, etc.) is
assigned a unique bit pattern. ASCII: Uses patterns of 7-bits to represent most symbols used in written English text
Unicode: Uses patterns of 16-bits to represent the major symbols used in languages world side
ISO standard: Uses patterns of 32-bits to represent most symbols used in languages world wide
Numbers: Uses bits to represent a number in base two Limitations of computer representations of numeric
values Overflow – happens when a value is too big to be represented
Truncation – happens when a value is between two representable values
Compsci 001 3.6
Images, Sound, & Compression
Images Store as bit map: define each pixel
• RGB• Luminance and chrominance
Vector techniques• Scalable• TrueType and PostScript
Audio Sampling
Compression Lossless: Huffman, LZW, GIF Lossy: JPEG, MPEG, MP3
Compsci 001 3.7
Decimal (Base 10) Numbers
Each digit in a decimal number is chosen from ten symbols:
{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
The position (right to left) of each digit represents a power of ten.
Example: Consider the decimal number 2307
2 3 0 7
position: 3 2 1 0
2307 = 2103 + 3102 + 0101 + 7100
Compsci 001 3.8
Binary (Base 2) Numbers
Each digit in a binary number is chosen from two symbols:
{ 0, 1 } The position (right to left) of each digit represents a power of
two. Example: Convert binary number 1101 to decimal
1 1 0 1
position: 3 2 1 0
1101 = 123 + 122 + 021 + 120
= 18 + 14 + 02 + 11 = 8 + 4 + 1 = 13
Compsci 001 3.9
Powers of Two
Decimal Binary Power of 2
1 1
2 10
4 100
8 1000
16 10000
32 100000
64 1000000
128 10000000
021222
42
32
526272
Compsci 001 3.10
Famous Powers of Two
Images from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
Compsci 001 3.11
Other Number Systems
Images from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
Compsci 001 3.12
Binary Addition
Images from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
Also: 1 + 1 + 1 = 1 with a carry of 1
Compsci 001 3.13
Adding Binary Numbers
101
+ 10
--------
111
101 + 10 = ( 122 + 021 + 120 ) + ( 121 + 020 )
= ( 14 + 02 + 11 ) + ( 12 + 01 )
Add like terms: There is one 4, one 2, one 1
= 14 + 12 + 11 = 111
Compsci 001 3.14
Adding Binary Numbers
1 1 carry
111+ 110---------
1101
111 + 110 = ( 122 + 121 + 120 ) + (122 + 121 + 020 )
= ( 14 + 12 + 11 ) + (14 + 12 + 01 )
Add like terms: There are two 4s, two 2s, one 1
= 24 + 22 + 11
= 18 + 14 + 02 + 11 = 1101
BinaryNumber Applet
Compsci 001 3.15
Converting Decimal to Binary
Decimal
012345678
Binary
0
1
10
11
100
101
110
111
1000
conversion
0 = 020
1 = 120
2 = 121 + 020
3 = 2+1 = 121 + 020
4 = 122 + 021 + 020
5 = 4+1 = 122 + 021 + 120
6 = 4+2 = 122 + 121 + 020
7 = 4+2+1 = 122 + 121 + 120
8 = 122 + 022 + 021 + 020
Compsci 001 3.16
Converting Decimal to Binary
Repeated division by two until the quotient is zero Example: Convert decimal number 54 to binary
012
32
62
132
272
542
1
1
0
1
1
0
Binary representation of54 is 110110
remainder
Compsci 001 3.17
Converting Decimal to Binary
1 32 = 0 plus 1 thirty-two
6 8s = 1 32 plus 1 sixteen
3 16s = 3 16 plus 0 eights
13 4s = 6 8s plus 1 four
27 2s = 13 4s plus 1 two
54 = 27 2s plus 0 ones
1
1
0
1
1
0
012
32
62
132
272
542
Subtracting highest power of two
1s in positions 5,4,2,1
54 - 25 = 22
22 - 24 = 6
6 - 22 = 2
2 - 21 = 0
110110
Compsci 001 3.18
Problems
Convert 1011000 to decimal representation
Add the binary numbers 1011001 and 10101 and express their sum in binary representation
Convert 77 to binary representation
Compsci 001 3.19
Solutions Convert 1011000 to decimal representation
1011000 = 126 + 025 + 124 + 123 + 022 + 021 + 020
= 164 + 032 + 116 + 18 + 04 + 02 + 01
= 64 + 16 + 8 = 88
Add the binary numbers 1011001 and 10101 and express their sum in binary representation
1011001 + 10101 ------------- 1101110
Compsci 001 3.20
Solutions
Convert 77 to binary representation
012
22
42
92
192
382
772
1
0
0
1
1
0
1
Binary representation of77 is 1001101
Compsci 001 3.21
Boolean Logic AND, OR, NOT, NOR, NAND, XOR Each operator has a set of rules for combining two
binary inputs These rules are defined in a Truth Table (This term is from the field of Logic)
Each implemented in an electronic device called a gate Gates operate on inputs of 0’s and 1’s These are more basic than operations like addition
Gates are used to build up circuits that • Compute addition, subtraction, etc• Store values to be used later• Translate values from one format to another
Compsci 001 3.22
Truth Tables
Images from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
Compsci 001 3.23
In-Class Questions
1. How many different bit patterns can be formed if each must consist of exactly 6 bits?
2. Represent the bit pattern 1011010010011111 in hexadecimal notation.
3. Translate each of the following binary representations into its equivalent base ten representation.
1. 1100
2. 10.011
4. Translate 231 in decimal to binary
