Maths and the making of the modern world The maths behind Google and the Ipod Chris Budd Some common views on maths and mathematicians Mathematics is completely useless Mathematicians are evil souless geeks All Mathematicians are mad! The truth is rather different! The modern world would not exist without maths With maths you can tell the future and save lives Maths lies at the heart of art and music
Spot the mathematician, and why are they important? Maxwell and the discovery of electromagnetic waves B D M, H = + J, t t .D = , .B = 0. E =
Electromagnetism, radio, WiFi,TV, radar, mobile phones, microwaves all come from the work of Maxwell! A few other examples . Google: Linear algebra, graph theory, SVD Internet: Network theory Security: Fermat, RSA Error correcting codes: Galois theory Medical imaging: Radon Transform
Communications: FFT, Shannon Medical Statistics: Nightingale Mathematicians really have made the modern world possible The key to the modern world is information! It is important that we store, transmit and search this information carefully and without making mistakes Maths helps us to do this Storing information by telling the truth Pick a number
0,1,2,3,,7 Answer the following questions truthfully Q1. Is your number 4,5,6,7? Q2. Is your number 2,3,6,7? Q3. Is your number 1,3,5,7? Binary numbers 0 000
001 010 011 100
101 110 111 3 Bit Binary Number: x x
represented by three digits a b c a,b,c are 0 or 1 x = 4*a + 2*b + c eg. 101 = 4+0+1 = 5 011 = 0+2+1 = 3 eg. 101 1, 0 are called bits of information All information in a computer is made up of bits
Simplest information has ONE BIT Are you OK? Binary numbers 0..7 have 3 bits Usually binary numbers have more than 3 bits eg. 10011011 has 8 bits A symbol of 8 bits is called a byte. You can have 256 such symbols Letters A,B,C, are converted into 8 bit ASCII
Other languages eg. Japanese use 32 bit Unicode Camera takes picture made up of PIXELS 8 BITS per pixel .. 256 range of intensity = 1 byte 1 000 000 Pixels per Picture 3 colours One bite memory How does a monster count to 25?
On his fingers! Using binary you can count from 0 to 31 on one hand with 5 bit binary numbers eg. 10110 = 16 + 4 + 2 = 22 11001 = 16 + 8 + 1 = 25 How to avoid errors. Sometimes we make mistakes Mean to send 11100011 Make a mistake on one bit and send 11101011
Can we tell if we have made a mistake? Answer the following questions. Either tell the truth or lie at most once Pick a number between 0 and 7 Q1 Is it 4,5,6,7? Q2
Is it 2,3,6,7? Q3 Is it 1,3,5,7? Q4 Is it 1,2,4,7?
Can we find the liar? 0 0 0 0 0 1 0 0 1 1 2 0 1 0 1
3 0 1 1 0 question 4 1 0 0 1 5 1 0 1 0 6
1 1 0 0 answer to last 7 1 1 1 1 If all true there are an: even number of 1s If one lie there is an: odd number of 1s Last digit/question is called a parity bit and tells us if we have made a mistake
Technique was used by ancient scribes when copying biblical texts Once we spot an error we can either Discard the whole message . OR . Ask for the information to be sent again . OR . We can try to correct it Error correcting codes.
Used to store the numbers 0,1,2,3,4,5,6,7 and other data in such a way that any errors can not only be detected but corrected. Answer the following questions .. You can either tell the truth or lie at most once Choose a number 0,1,2,3,4,5,6,7 Q1 Is the number 4,5,6,7? Q2
Is the number 2,3,6,7? Q3 Is the number 1,3,5,7? Q4 Is the number
1,3,4,6? Q5 Is the number 1,2,5,6? Q6 Is the number 2,3,4,5? 0
000 000 1 001 110 2 010 011 3 011 101
4 100 101 5 101 011 Binary number 6 Correcting number 110 110
Start with a binary number 110110 Telling the truth doesnt change the number Lying once changes the number by one digit Hamming Distance: Take two binary numbers. How many digits do we have to change to turn one into the other? 110110
100110 0 000 000 1 001 110 All are a Hamming distance of 3 2 010 011
3 011 101 4 5 100 101 101 011 Binary number 6
Correcting number 110 110 apart Choose the closest number to the one you are sent. This must be correct. Error correcting codes. Extend these ideas using Finite Field Theory (Galois) These are widely used in
CDs Digital TV and Radio Mobile phones Invented in the 1940s by Satellites Hamming in the Bell Labs
Different types of code are used depending upon The Level of noise Whether the noise is random or comes in bursts These are used in IPODs IPOD also compress the information. For example Instead of sending this message which has lots of vowels in it which we dont
really need W cn snd ths mssg nstd whch ds nt hv ny vwls t ll Nw try ths fr yrslf MPEG file also compresses sound waveforms Decompose into a sum of harmonics. Only store the first few of these The Maths Behind Google Google searches for information stored on many web-sites. Web-sites are linked together by a network showing which web-site points
to which other web-site It RANKS web-sites in order of the importance of the information that they contain. IDEA. A website is important if lots of other websites link to it. PAGE RANK Each Web-site has a rank R Divide R by the number N of web-sites that this web-site links, to get S=R/N For each web-site, calculate R by adding up the 4 values of S for every web-site that connects to 12
9 it. 9 4 3 4 3 2 6
4 2 Now do the same for 1 000 000 0000 more computers Need to calculate the unit eigenvector of an extremely large matrix! And make many $$$$$$$$$$$$$$$$$
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