MATHEMATICS IN EVERYDAY LIFE

Group Members: Ang Wei Chun Chan Cheng Yow Koid Cin Yik

ROLE OF MATHEMATICS IN MODERN TECHNOLOGIESMathematics and sciences both play

major roles in modern technologies. Without them, many things are impossible to be done. There are few roles of Mathematics that we would like to focus here.

Mathematics as languageMathematics as securityMathematics as database

MATHEMATICS AS LANGUAGE IN MODERN TEHCNOLOGIES

Mathematics is the languages used in digital devices, in fact computer are born in Mathematics languages.

When we are transferring real world information like picture and audio through digital devices, they are converted to binary numeric.

Binary numeric is the raw languages that can be understand by digital devices.

WHY BINARY?

This is because digital devices can only process the data in binary form. The digital devices are build with electrical circuits, which are either on or off. Just two states to work with. So the natural number system for use in an digital electronic device is base 2 (called the binary number system).

Each binary digit is called bit which can hold either 1 or 0. A string of 8 bits long data is called byte.

TEXT TO DIGITAL

When we want to transfer text information through computer, we will use keyboard as input device.

Each time when we enter a text character, it will trigger a string of 8 binary digits. As a result, computer will search for the corresponding text character and shows the text character on the screen.

The system used is called the ASCII code (American Standard Code for Information Interchange).

EXAMPLE When we press the button a on the

keyboard.

The computer will detect the input as binary numeric which is 01100001.

After that, processor will compute it to base 10 which is 97.

By referring to ASCII chart from the system memory, the computer search for the corresponding character which is “a”.

The text character of “a” will be displayed on the screen and can be transfer to the world with internet.

ASCII Representation of Characters (just a sample)

Character Base 10 Base 2

(return) 13 00001101

(space) 32 00100000

! 33 00100001

1 49 00110001

2 50 00110010

@ 64 01000000

A 65 01000001

B 66 01000010

C 67 01000011

a 97 01100001

b 98 01100010

c 99 01100011

(delete) 127 01111111

PICTURE TO DIGITAL

If you look closely at your display screen, you can see that the image on it is made up of lots of little spots, called picture elements (which is more commonly shortened to pixel).

Each pixel in a screen image might be represented by three bytes in the computer

The numbers in the bytes tell the display how much red, blue, and green light should be mixed together to make the color of the pixel (three bytes can represent millions of possible colors for each pixel).

MATHEMATICS AS SECURITY IN MODERN TECHNOLOGIES As new technologies are emerging every day, the threat of

security had become higher. Personal private information can be easily fall into unwanted hand if it was not handle properly.

Mathematics play important roles in modern security system.

Cryptography is used to make this possible. It is the practice and study of hiding information.

Modern cryptography intersects the disciplines of mathematics, computer science, and engineering.

Applications of cryptography include ATM cards, computer passwords, and electronic commerce.

BASIC TERMINOLOGY OF CRYPTOGRAPHY

BASIC TERMINOLOGY OF CRYPTOGRAPHY

A cryptographic algorithm, also called a cipher, is the mathematical function used for encryption and decryption.

The security of a modern cryptographic algorithm is based on a secret key. This key might be any one of a large number of values. The range of possible key values is called the keyspace.

Both encryption and decryption operations are dependent on the key K and this is denoted by the K subscript in the functions EK(P) = C and DK(C) = P.

EXAMPLE OF TRADITIONAL CYPTOGRAPHY

Scenario: Alice wants to send a message (plaintext p) to Bob. The communication channel is insecure and can be eavesdropped by Trudy. If Alice and Bob have previously agreed on an encryption scheme (cipher), the message can be sent encrypted (ciphertext c)

Alice Bob

cencrypt decryptp c p

Trudy

12

EXAMPLE (S. SINGH, THE CODE BOOK, 1999) Ciphertext PCQ VMJYPD LBYK LYSO KBXBJXWXV BXV ZCJPO EYPD KBXBJYUXJ

LBJOO KCPK. CP LBO LBCMKXPV XPV IYJKL PYDBL, QBOP KBO BXV OPVOV LBO LXRO CI SX'XJMI, KBO JCKO XPV EYKKOV LBO DJCMPV ZOICJO BYS, KXUYPD: 'DJOXL EYPD, ICJ X LBCMKXPV XPV CPO PYDBLK Y BXNO ZOOP JOACMPLYPD LC UCM LBO IXZROK CI FXKL XDOK XPV LBO RODOPVK CI XPAYOPL EYPDK. SXU Y SXEO KC ZCRV XK LC AJXNO X IXNCMJ CI UCMJ SXGOKLU?'

OFYRCDMO, LXROK IJCS LBO LBCMKXPV XPV CPO PYDBLK

Any Guesses???

Cryptography

13

CodeX Z A V O I D B Y G E R S P C F H J K L M N Q T U WA B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Plaintext Now during this time Shahrazad had borne King Shahriyar three sons. On the thousand and first night, when she had ended the tale of Ma'aruf, she rose and kissed the ground before him, saying: 'Great King, for a thousand and one nights I have been recounting to you the fables of past ages and the legends of ancient kings. May I make so bold as to crave a favour of your majesty?’ Epilogue, Tales from the Thousand and One Nights

Traditional cryptography are simple and can be easily cracked by third party using frequency analysis.

Here how it is done..

15

FREQUENCY ANALYSIS

Identyfying comon letters, digrams and trigrams... PCQ VMJYPD LBYK LYSO KBXBJXWXV BXV ZCJPO EYPD KBXBJYUXJ LBJOO KCPK. CP

LBO LBCMKXPV XPV IYJKL PYDBL, QBOP KBO BXV OPVOV LBO LXRO CI SX'XJMI, KBO JCKO XPV EYKKOV LBO DJCMPV ZOICJO BYS, KXUYPD: 'DJOXL EYPD, X LBCMKXPV XPV CPO PYDBLK Y BXNO ZOOP JOACMPLYPD LC UCM LBO IXZROK CI FXKL XDOK XPV LBO RODOPVK CI XPAYOPL EYPDK. SXU Y SXEO KC ZCRV XK LC AJXNO X IXNCMJ CI UCMJ SXGOKLU?'

OFYRCDMO, LXROK IJCS LBO LBCMKXPV XPV CPO PYDBLK First guess: LBO is THE

16

FREQUENCY ANALYSIS

Assuming LBO represents THE we replace L with T, B with H, and O with E and get

PCQ VMJYPD THYK TYSE KHXHJXWXV HXV ZCJPE EYPD KHXHJYUXJ THJEE KCPK. CP THE THCMKXPV XPV IYJKT PYDHT, QHEP KHO HXV EPVEV THE LXRE CI SX'XJMI, KHE JCKE XPV EYKKOV THE DJCMPV ZEICJE HYS, KXUYPD: 'DJEXT EYPD, ICJ X LHCMKXPV XPV CPE PYDHLK Y HXNE ZEEP JEACMPTYPD TC UCM THE

IXZREK CI FXKL XDEK XPV THE REDEPVK CI XPAYEPT EYPDK. SXU Y SXEE KC ZCRV XK TC AJXNE X IXNCMJ CI UCMJ SXGEKTU?'

EFYRCDME, TXREK IJCS THE LHCMKXPV XPV CPE PYDBTK More guesses…?

In modern world, the best known cryptography in use are Public Key Cryptography which used very complex Mathematics algorithm to avoid hacker to crack the protected information.

MATHEMATICS AS DATABASE IN MODERN TECHNOLOGIES

One of the uses of Mathematics in database is the invention of barcode system.

A barcode is an optical machine-readable representation of data, which shows certain data on certain products.

There are many type of Symbology for linear barcode, the most common in use is the EAN-13 digit barcode.

HOW TO ENCODE THE BARCODE?

The picture shows a sample of EAN-13 barcode.

EAN-13 barcode system contains 13 digits of number. It divided into four areas: number system, manufacturer code, product code, and check digit.

The number system is the first 3 digits of barcode, it shows the country of the product company according to the GS1 Member Organisation code.

000 - 019 GS1 US

030 - 039 GS1 US

060 - 139 GS1 US

300 - 379 GS1 France

400 - 440 GS1 Germany

450 - 459 & 490 - 499 GS1 Japan

460 - 469 GS1 Russia

471 GS1 Taiwan

480 GS1 Philippines

489 GS1 Hong Kong

690 - 695 GS1 China

750 GS1 Mexico

800 - 839 GS1 Italy

880 GS1 South Korea

885 GS1 Thailand

888 GS1 Singapore

890 GS1 India

893 GS1 Vietnam

896 GS1 Pakistan

899 GS1 Indonesia

955 GS1 Malaysia

958 GS1 Macau

977 Serial publications (ISSN)

978 - 979 Bookland (ISBN)

The manufacturer and product code are unique for it's manufacturer and product.

The check digit is an additional digit used to verify that a bar code has been scanned correctly.

To check whether the barcode are scan correctly, 1) we need to sum up all the digits in the odd position which are

(7+0+0+4+3+1) =15

2) Then sum up all the digits in even position and times 3 which are (5+1+5+5+0+0)x3 = 48

3) Sum the product of both which are 15+48=63. Find 63+x =modulo 10. Therefore x = 7

As you can see, there is no price information encoded in a bar code.

When the scanner at the checkout line scans a product, the cash register sends the barcode number to the store's central POS (point of sale) computer to look up the barcode number. The central computer sends back the actual price of the item at that moment.

This approach allows the store to change the price whenever it wants, for example to reflect sale prices.

If the price were encoded in the bar code, prices could never change. On the other hand, not encoding a fixed price gives the store an easy way to rip off customers. When you hear about "scanner fraud" in the news, that is what the newsperson is talking about. It is incredibly easy for a store to mistakenly or purposefully overprice an item.

MATHEMATICS AS AN ONGOING CULTURAL ACTIVITY

Golden ratio

The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras

Geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle

ARCHITECTURE Pyramids

Golden ratio in the design of the ancient monuments. base edges range from 755–756 height of the structure is 481.4 feet. bisector of the side of the pyramid comes out to 612 feet. If

we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619.

ART

Mona lisa her face actually appears in a golden rectangle, which also

makes her face appear more beautiful to human eyes.

Also another masterpiece, the Last Supper, contains Golden Ratios

TESSELLATIONS

Tessellations are a combination of math, art and fun.

Tessellations are observed in some works of great artists like M.C. Escher.

Tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps.

ARCHITECTURE In architecture, we can always see tessellations can be

found in quilts, floor tiling, wallpaper, brick and many things.

ART

Paper cutting

GEOMETRY Art

The Platonic solids and other polyhedral are a recurring theme in Western art.

Salvador Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

ARCHITECTURE

CONTEMPORARY MATHEMATICS

Studying contemporary mathematics can help us to understand the world around us. Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions. The study of mathematics also leads to the ability to think logically and solve problems

Mathematics is used throughout the world as an essential tool in many fields, such as:

Natural science Engineering Medicine Social sciences Business

BUSINESS MATHEMATICS

Mathematics used in: Accounting Inventory management Marketing Sales forecasting Financial analysis

BUSINESS MATHEMATICS

Mathematics typically used in commerce includes elementary arithmetic, elementary algebra, statistics and probability.

Business management can be made more effective in some cases by use of more advanced mathematics such as calculus, matrix algebra and linear programming.

MATERIALS SCIENCE

Materials science is concerned with the synthesis and manufacture of new materials, the understanding and prediction of material properties and evolution and control of these properties over a time period.

Fluid with magnetic particles or electrically charged particles enhance the effect of brake fluid and shock absorbers in the car. Mathematicians have developed new tools in functional analysis have been able to estimate or compute the effective properties of composites.

ARMY

Mathematics in army include mathematics for materials, security issues and future system. These are used in energy efficiency like optimal control, information or data mining on move and secured wireless communication systems in the weapons.