Compsci Today’s topics l Binary Numbers Brookshear l Slides from Prof. Marti Hearst of UC Berkeley SIMS l Upcoming Networks Interactive Introduction to Graph Theory Kearns, Michael. "Economics, Computer Science, and Policy." Issues in Science and Technology, Winter Problem Solving
Compsci Digital Computers l What are computers made up of? Lowest level of abstraction: atoms Higher level: transistors l 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 Binary Digits (Bits) l Yes or No l On or Off l One or Zero l
Compsci More on binary l Byte A sequence of bits 8 bits = 1 byte 2 bytes = 1 word (sometimes 4 or 8 bytes) l Powers of two l How do binary numbers work?
Compsci Data Encoding l 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 l Numbers: Uses bits to represent a number in base two l 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 Images, Sound, & Compression l Images Store as bit map: define each pixel RGB Luminance and chrominance Vector techniques Scalable TrueType and PostScript l Audio Sampling l Compression Lossless: Huffman, LZW, GIF Lossy: JPEG, MPEG, MP3
Compsci Decimal (Base 10) Numbers l Each digit in a decimal number is chosen from ten symbols: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } l The position (right to left) of each digit represents a power of ten. l Example: Consider the decimal number position: = 2 10 0
Compsci Binary (Base 2) Numbers l Each digit in a binary number is chosen from two symbols: { 0, 1 } l The position (right to left) of each digit represents a power of two. l Example: Convert binary number 1101 to decimal position: = 1 2 0 =1 1= = 13
Compsci Powers of Two DecimalBinaryPower of
Compsci Famous Powers of Two Images from
Compsci Other Number Systems Images from
Compsci Binary Addition Images from Also: = 1 with a carry of 1
Compsci Adding Binary Numbers l = ( 1 2 0 ) + ( 1 2 0 ) = ( 1 1 ) + ( 1 1 ) l Add like terms: There is one 4, one 2, one 1 = 1 1 = 111
Compsci Adding Binary Numbers 1 1 carry l = ( 1 2 0 ) + (1 2 0 ) = ( 1 1 ) + (1 1 ) l Add like terms: There are two 4s, two 2s, one 1 = 2 1 = 1 1 = 1101 l BinaryNumber Applet
Compsci Converting Decimal to Binary Decimal Binary conversion 0 = 0 = 1 = 1 = 2+1 = 1 = 1 = 4+1 = 1 = 4+2 = 1 = = 1 = 1 2 0
Compsci Converting Decimal to Binary l Repeated division by two until the quotient is zero l Example: Convert decimal number 54 to binary 1 0 1 0 Binary representation of 54 is remainder
Compsci Converting Decimal to Binary l 1 32 = 0 plus 1 thirty-two l 6 8s = 1 32 plus 1 sixteen l 3 16s = 3 16 plus 0 eights l 13 4s = 6 8s plus 1 four l 27 2s = 13 4s plus 1 two l 54 = 27 2s plus 0 ones 1 0 1 0 l Subtracting highest power of two l 1s in positions 5,4,2, = = = = 0
Compsci Problems l Convert to decimal representation l Add the binary numbers and and express their sum in binary representation l Convert 77 to binary representation
Compsci Solutions l Convert to decimal representation = 1 2 0 = 1 1 = = 88 l Add the binary numbers and and express their sum in binary representation
Compsci Solutions l Convert 77 to binary representation 1 0 1 0 1 Binary representation of 77 is
Compsci Boolean Logic l AND, OR, NOT, NOR, NAND, XOR l 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) l 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 Truth Tables Images from
Compsci 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 in hexadecimal notation. 3. Translate each of the following binary representations into its equivalent base ten representation Translate 231 in decimal to binary