Post on 03-Jan-2016
MATH 1020: MATH 1020: Mathematics For Non-science
Chapter 4: Chapter 4: Information in a networked age
1
Instructor: Prof. Ken Tsang
Room E409-R9
Email: kentsang@uic.edu.hk
Transmitting Information
– Binary codes– Data compression– Encoding with parity-check sums– Cryptography– Model the genetic code
2
Information , data & numbersInformation , data & numbers
Today information are transmitted all over the world through the internet
Information is just collection of data– Pictures – jpg, tif …– Sound – mp3, mp4– Video – wmv, mvb
Data consisted of numbers
3
4
Decimal Number SystemDecimal Number System As human normally counts with hands and there are
totally 10 fingers on both hands, this probably explains the origin of the decimal number system.
10 digits:– 0,1,2,3,4,5,6,7,8,9
Also called base-10 number system, – Or Hindu-Arabic, or Arabic system
Counting in base-10– 1,2,…,9,10,11,…,19,20,21,…,99,100,…
Decimal number in expanded notation– 234 = 2 * 100 + 3 * 10 + 4 * 1
Hindu–Arabic numeral systemHindu–Arabic numeral system The Brahmi (ancient Indian) numerals at the basis of the system
predate the Common Era. The development of the positional decimal system occurred during the
Gupta period (笈多王朝 , 320 to 540 CE). Aryabhata, a Gupta period scholar, is believed to be the first to come up
with the concept of zero.
These Indian developments were taken up in Islamic mathematics in the 8th century.
A young Italian in the 12th century, Fibonacci, traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time, recognizing that arithmetic with Hindu-Arabic numerals is simpler and more efficient than with Roman numerals.
5
Fibonacci (1170-1250 CE)Fibonacci (1170-1250 CE) Italian mathematician, Leonardo Fibonacci
(through the publication in 1202 of his Book of Calculation, the Liber Abaci) introduced the Arabic numerals, the use of zero, and the positional decimal system to the Latin world.
Liber Abaci showed the practical importance of the new numeral system, by applying it to commercial bookkeeping.
6
The numeral system came to be called "Arabic" by the Europeans. It was used in European mathematics from the 12th century, and entered common use from the 15th century.Fibonacci significantly influenced the evolution of capitalist enterprise and public finance in Europe in the centuries that followed.
7
Positional Numbering SystemPositional Numbering SystemThe value of a digit in a number
depends on:– The digit itself– The position of the digit within the number
So 123 is different from 321– 123: 1 hundred, 2 tens, and 3 units– 321: 3 hundred, 2 tens, and 1 units
Roman numeralsRoman numeralsRoman numerals are numeral system of ancient
Rome based on the letters of the alphabetThe first ten Roman numerals are I, II, III, IV, V, VI,
VII, VIII, IX, and X. (no zero)Tens: X; hundreds: C; thousands: MNon-positional: e.g.
– 321 CCCXXI– 982 CMLXXXII– 2010 MMX
8
Non-decimal Number SystemsNon-decimal Number Systems
The Maya civilization and other civilizations of pre-Columbian Mesoamerica used base-20 (vigesimal), as did several North American tribes (two being in southern California).
Evidence of base-20 counting systems is also found in the languages of central and western Africa.
The Irish language also used base-20 in the past. Danish numerals display a similar base-20 structure.
9
10
Base Base rr Number System Number SystemFor any value For any value rrValue is based on the sum of a power
series in powers of rr
rr is called the base, or radix
11
Binary Number SystemBinary Number SystemBinary number system has only two digits
– 0, 1– Also called base-2 system
Counting in binary system– 0, 1, 10, 11, 100, 101, 110, 111, 1000,….
Binary number in expanded notation– (1011)2 = 1*23 + 0*22 + 1*21 + 1*20
– (1011)2 = 1*8 + 0*4 + 1*2 + 1*1 = (11)10
12
Why Binary? Why Binary? Computer is a Binary machine It knows only ones and zeroes Easy to implement in electronic
circuitsReliableCheap
13
Gottfried Leibniz (1646-1716)Gottfried Leibniz (1646-1716)Leibniz, German mathematician
and philosopher, invented at least two things that are essential for the modern world: calculus, and the binary system.
He invented the binary system around 1679, and published in 1701. This became the basis of virtually all modern computers.
Leibniz & Leibniz & I-Ching I-Ching (( 易经易经 ))
As a Sinophile, Leibniz was aware of the I-Ching and noted with fascination how its hexagrams correspond to the binary numbers, and concluded that this mapping was evidence of major Chinese accomplishments in the sort of philosophical mathematics he admired.
14
15
An ancient Chinese binary An ancient Chinese binary number system in Yi-Jing (number system in Yi-Jing ( 易经易经 ))
Two symbols to represent 2 digits Zero: represented by a broken line One: represented by an unbroken line “—” yan 阳爻,“ --” yin 阴爻。
16
HexadecimalHexadecimalHexadecimal number system has 16 digits
• 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F• Also called base-16 system
Counting in Hexadecimal– 0,1,…,F,10,11,…,1F,20,…FF,100,…
Hexadecimal number in expanded notation
– (FF)16 = 15*161 + 15*160 = (255)10
17
Conversion between binary & Conversion between binary & hexadecimal hexadecimal for the first 16 numbersfor the first 16 numbers
18
Why Hexadecimal?Why Hexadecimal?Hexadecimal is meaningful to humans, and
easy to work with for a computerCompact
– A BYTE is composed of 8 bits– One byte can thus be expressed by 2 digits in
hexadecimal– 11101111 EF– 11101111b EFh
Simple to convert them to binary
20
Conversions Conversions Between Number SystemsBetween Number Systems
Octal to Decimal
– (32)8 = (?)10
What’s wrong?
– (187)8 = 1*64 + 8*8 + 7*1
21
Conversions Conversions Between Number SystemsBetween Number Systems
Decimal to Binary
Reading the remainders from bottom to top, we have 32110 = 1010000012
remainderquotient
321 / 2 = 160 1 160 / 2 = 80 0 80 / 2 = 40 0 40 / 2 = 20 0 20 / 2 = 10 0 10 / 2 = 5 0 5 / 2 = 2 1 2 / 2 = 1 0 1 / 2 = 0 1
32110 = ?2
22
Analogue DataAnalogue Data Analogue: something that is analogous or
similar to something else (Webster) Analogue Data: The use of continuously
changing quantities to represent data. A mercury thermometer is an analogue device.
The mercury rises and falls in a continuous flow in the tube in direct proportion to the temperature.
The mathematical idealization of this smooth change as a continuous function leads to “Analogue Data”, an infinite amount of data
23
From Analogue to Digital dataFrom Analogue to Digital dataData can be represented in one of two ways:
analogue or digital:
Analogue data: A continuous representation (using mathematical function or smooth curve) , analogous to the actual information it represents
Digital data: A discrete representation, breaking the information up into separate elements (data)
24
Digitized InformationDigitized InformationComputers, cannot work with analogue
information So we digitize information by breaking it into
pieces and representing those pieces separatelyWhy do we use binary?
– Modern computers are designed to use and manage binary values because the devices that store and manage the data are far less expensive and far more reliable if they only have to represent one of two possible values.
25
Bit and Byte Bit and Byte BIT = Binary digIT, “0” or “1”State of on or off ( high or low) of a
computer circuitKilo 1K = 210 = 1024 ≈ 103 Mega 1M = 220 = 1,048,576 ≈ 106 Giga 1G = 230 = 1,073,741,824 ≈ 109
26
Bit and Byte Bit and Byte Byte is the basic unit of addressable memory 1 Byte = 8 BitsThe right-most bit is called the LSB
Least Significant BitThe Left-most bit is called the MSB
Most Significant Bit
27
Binary RepresentationBinary RepresentationOne bit can be either 0 or 1 (“on” & “off”
electronic signals)Therefore, one bit can represent only two things To represent more than two things, we need
multiple bitsTwo bits can represent four things because
there are four combinations of 0 and 1 that can be made from two bits: 00, 01, 10, 11
28
Binary RepresentationBinary RepresentationIn general, n bits can represent 2n things because there
are 2n combinations of 0 and 1 that can be made from n bits
Note that every time we increase the number of bits by 1, we double the number of things we can represent
Questions:– How many bits are needed to represent 128 things?– How many bits are needed to represent 67 things?
29
ASCIIASCIIASCII stands for American Standard
Code for Information InterchangeThe ASCII character set originally used
seven bits to represent each character, allowing for 128 unique characters
Later ASCII evolved so that all eight bits were used which allows for 256 characters
31
Unicode charactersUnicode charactersExtended version of the ASCII character set is
not enough for international use The Unicode character set uses 16 bits per
character– Therefore, the Unicode character set can represent 216, or over
65 thousand, characters
Unicode was designed to be a superset of ASCII– The first 256 characters in the Unicode character set
correspond exactly to the extended ASCII character set
With the Unicode, all text (in most languages) information can be represented.
33
To digitize the signal we periodically measure the voltage of the signal and record the appropriate numeric value– this process is called sampling
In general, a sampling rate of around 40,000 times per second is enough to create a reasonable sound reproduction
Representing Audio Representing Audio InformationInformation
35
• A compact disk (CD) stores audio information digitally • On the surface of the CD are microscopic pits that represent Binary digits•A low intensity laser is pointed as the disc•The laser light reflects strongly if the surface is smooth and reflects poorly if the surface is pitted
Representing Audio Representing Audio InformationInformation
36
Audio Formats– WAV, AU, AIFF, VQF, and MP3
MP3 is dominant – MP3 is short for MPEG (Moving Picture Experts Group)
audio layer 3 file– MP3 employs both lossy and lossless compression– First it analyzes the frequency spread and compares it to
mathematical models of human psychoacoustics (the study of the interrelation between the ear and the brain), then it discards information that can’t be heard by humans
– Then the bit stream is compressed to achieve additional compression
Representing Audio Representing Audio InformationInformation
Image BasicsImage Basics
00000000000000000011110000000000000000 00000000000000001100001100000000000000 00000000000000010000000010000000000000 00000000000000100000000001000000000000 00000000000000100010001001000000000000 00000000000001000111011100100000000000 00000000000001000010001000100000000000 00000000000001000000000000100000000000 00000000000001000000000000100000000000 00000000000001001000000100100000000000 00000000000000100100001001000000000000 00000000000000100011110001000000000000 00000000000000010000000010000000000000 00000000000000001100001100000000000000 00000000000000000011110000000000000000 00011110010000000000000000000000000000 01100010010000000000000000000000000000 11000100100000000000000000000000000000 00000100100001110001011000101100100100 00111111110010010001101000110101100100 00001001000100100111001011100101001000 00010010000101101010010101001011011010 00010010000110110111111011111101101100 00000000000000000100000010000000011000 00000000000000001100000110000000110000 00000000000000001000000100000000100000
38
41
Representing ColorRepresenting ColorColor is often expressed in a computer as an
RGB (red-green-blue) value, which is actually three numbers that indicate the relative contribution of each of these three primary colours
For example, an RGB value of (255, 255, 0) maximizes the contribution of red and green, and minimizes the contribution of blue, which results in a bright yellow
RGB ModelRGB ModelRGB Color Model
– Red – Green – Blue– Additive model combines varying amounts of
these 3 colors
42
Image BasicsImage Basics
(0, 0, 0) is black
(255, 255, 255) is white
(255, 0, 0) is red
(0, 255, 0) is green
(0, 0, 255) is blue
(0, 255, 255) is cyan
(255, 0, 255) is magenta
(255, 255, 0) is yellow
45
46
DAC
Composing color imageComposing color imageStore the actual intensities of R, G, and B
individually in the framebuffer24 bits per pixel = 8 bits red, 8 bits green, 8
bits blue
49
Digitized Images and GraphicsDigitized Images and Graphics
Digitizing a picture is the act of representing it as a collection of individual dots called pixels
The number of pixels used to represent a picture is called the resolution
Several popular raster file formats including bitmap (BMP), GIF, and JPEG