Mathematical Ability

8/10/2019 Mathematical Ability

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FOUNDATION COURSE: BUILDING MATHEMATICAL ABILITY

Unit I

Numbers: The Magic of Secret Codes

1.1 Sherlock Holmes and the Story of the Dancing Figures

How many of you are aware of the famous nineteenth century private detective-Sherlock

Holmes-immortalized by Arthur Conan Doyle through his fictional stories? It would be a good

idea if you could visit the following website and read or acquaint yourself with some of the

thrilling adventures of Sherlock Holmes.

http://arthursbookshelf.com/adventure/doyle/complete-holmes.pdf

In fact, several movies, TV serials and radio programs have also been made based on these

stories. In particular and very relevant to our interaction here is the story of the dancing figures

http://www.youtube.com/watch?v=ISpuTNxZVck

After you have read the story Adventure of the dancing men or watched the TV

adaptation of the same discuss in the class as to how Sherlock Holmes cracked the code.

Develop a secret code of your own and see if your friends can crack it.

Coded message have played an important role in World War I and World War II. United State

kept out of World War I till 1917. It was during this time that British cryptographers deciphered

a telegram sent by the then State secretary of foreign affairs Arthur Zimmermann. You may wish

to read more about it at http://www.archives.gov/education/lessons/zimmermann/#documents.

The Zimmermann telegram as it is popularly known now had a series of numerals that

represented words in German. It was only after this telegram was intercepted and deciphered that

the United States was drawn into the war and the rest as they say was history.

Similarly in World War II Germany sent secret radio messages that they believed could not be

deciphered by their allies. It was due to the cryptographers at Britains Bletchley Park that the

German code was cracked. This led to the final defeat of Germany. You can read more about it at

http://www.history.co.uk/explore-history/ww2/code-breaking.html.

Many of us would like to send messages to our friends. Would it not be interesting to send secret

messages on Facebook to a particular friend of yours that the others cannot understand?


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Create a Facebook account of your class. Each one of the student in the class can put in

a secret message on a regular basis. The others can try to decipher it.

There are times when we do not want others to be privy to our mails or messages. There are datathat need to be kept as secret. For example your bank account number, your ATM pin, your

credit card number and so on. Just think as to how much data a bank needs to store for its

customer! And if each customer data has to be coded secretly what a huge work it would be. Do

you think it could be a good idea to have a secret symbol for every alphabet as in the story

Adventure of the dancing men? Why do you think Zimmermann used numbers in his telegram?

Before we discuss about numbers it would be a good idea to try out this fun activity.

Create a gibberish language that you can speak with your friends. You can get some

ideas from the site http://www.wikihow.com/Speak-Gibberish to make your own gibberish

language.

1.2Playing with numbers

Look around you. List out some instances where you encounter numbers daily.

What type of numbers do you encounter? Are they all made up of digits from 0 to

9? Do you think that there are any other ways of representing numbers? What do numbers

depict?

There is a lot you can do with numbers. How many meaningful words can be made with the

alphabets E and B? What about the numbers 2 and 4? How many meaningful arrangements can

be done using numbers 2 and 4?

Make your own number system

Use the following symbols , , , , , , to represent units, tens, hundreds and so on. Have

fun in trying out simple mathematical operations by representing numbers in this notation. For


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example you could express the sum of the two numbers 3456 + 1987 or their difference or their

products using your notations.

By now, you would have realized that a lot can be done with numbers. Cryptographers,

Computer analysts find it easier to deal with numbers rather than figures or alphabets. In fact all

your data in computers, ATM card, Identity card are stored as numbers.

Let us look at the statements:

Rita scored 345 marks out of 500.

The price of mango is Rs.60 per kg.

Samir stands sixth in a class of 30 students.

I take bus number 861 to reach to the college.

In each of the above statement, we have used numbers with different purpose. Numbers have an

inherent simplicity that makes it very fascinating. Although a complete book can be written on

their interesting facts and properties, we shall limit ourselves to special types of numbers.

Lets us begin by looking at the first 20 natural numbers. We start with the number 1.What all

numbers can divide 1 completely? When we talk about dividing a number completely we mean

that it should not leave any remainder. For example 2 divides 3 and we get an answer 1.5. But it

does not divide completely as it will leave a remainder 1. This is similar to you having three

apples that you wish to divide amongst yourself and your friend without cutting or breaking it.

Each one of you shall have one apple and one will be left. Therefore, 1 can only be divided by 1

itself. In a similar way 2 can be divided by 1 and 2. The number 3 can be divided by 1 and 3.

Let us now look at number 4. It can be divided by 1, 2 and 4.

We make a table for the first few numbers.

Number Divisible by

1 1

2 1, 2

3 1, 3

4 1, 2, 4

5 1, 5

6 1, 2, 3, 6

7 1, 78 1, 2, 4, 8

9 1, 3, 9

10 1, 2, 5, 10

11 1, 11

12 1, 2, 3, 4, 6, 12

13 1, 13


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14 1, 2, 7, 14

15 1, 3, 5, 15

16 1, 2, 4, 8, 16

17 1, 17

18 1, 2, 3, 6, 9, 18

19 1, 1920 1, 2, 4, 5, 10, 20

(a) How many numbers can be divided by exactly one number?

(b) How many numbers can be divided by exactly two numbers?

(c) How many numbers can be divided by more than two numbers?

Numbers that can be divided by exactly two numbers are called primes.

Numbers that can be divided by more than two numbers are called composite numbers.

The number 1 that has only one factor is neither prime nor composite.

Activities

Activity 1.2.1: Note down the registration number of any ten vehicles. You can also click these

registration numbers by your mobile. The registration number should preferably be of four digits.

Try to find which of these numbers are prime numbers. If they are not prime, list out the numbers

that divide them?

Activity 1.2.2:Check whether your college / XII Board examination roll number is a prime. If

not try to search for the prime number that is nearest to your roll number. You may use the prime

number calculator at http://math.about.com/library/blprimenumber.htm or

http://www.math.com/students/calculators/source/prime-number.htm

Activity 1.2.3: Pick out any two prime numbers and place them side by side to form another

number. For example we can form two numbers from 2 and 3, namely, 23 and 32. See whetheryou can get another prime. In the above example 23 is prime whereas 32 is not prime. Repeat the

activity with three primes. You can also repeat the activity with two digits and three digits prime

numbers.


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Activity 1.2.4:Every number has a story to tell Pick up any prime number. Write down some

facts about it. You may want to visit the site http://primes.utm.edu/curios/page.php/101.htmlto

know about some facts for 101. You could do a similar exercise for your prime.

Activity 1.2.5:Have a discussion in your class as to why these numbers are called prime. In

what other respect is the word primeused? Would you like to name these numbers something

else? Defend your choice of the terminology for these numbers.

Activity 1.2.6:If you did not have the prime calculator in Activity 1.2.2, how would you find out

whether a number is prime?

1.3Into the captivating world of the fascinating numbers

Prime numbers have fascinated mathematicians for a number of years. Prof D Zagier in his

inaugural lecture The first 50 million prime numbers at the Bonn University had quoted "...there

is no apparent reason why one number is prime and another not. To the contrary, upon

looking at these numbers one has the feeling of being in the presence of one of the

inexplicable secrets of creation." You could go through the complete inaugural address at

http://www.wstein.org/simuw/misc/zagier-the_first_50_million_prime_numbers.pdf

Make a list of some interesting facts that he mentions in his lecture. Have a

discussion on the observations he has made on the prime numbers.

The popular recreational mathematics writer Martin Gardener also remarked "No branch of

number theory is more saturated with mystery than the study of prime numbers: those

exasperating, unruly integers that refuse to be divided evenly by any integers except

themselves and 1. Some problems concerning primes are so simple that a child can understand

them and yet so deep and far from solved that many mathematicians now suspect theyhave no

solution. Perhaps they are undecideable. "

If the prime numbers are so fascinating and interesting dont you think that we should venture

out in search of them? So let us begin our adventure with prime numbers by finding them outfirst.

Have look at the short videohttp://www.youtube.com/watch?v=9m2cdWorIq8.


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Find out about Eratosthene and his contributions.

There is a lot that we know about the prime numbers now. We know that there will always be a

prime number between a natural number nand 2n ( Bertrands postulate) . We also know that

except 2 there is no other prime that is an even number. Also, any prime number will always be

of the form 6n 1 or 6n+ 1. We give you a hint to prove this.

Any number will be of the form 6n, 6n+ 1, 6n+ 2, 6n+ 3, 6n+ 4, 6n + 5 (6n+ 6 will again be of

the form 6n). Can 6n, 6n+ 2, 6n+ 4 be prime? Can 6n+ 3 be prime?

Although Euclid was the first one to prove that there are infinite numbers of prime numbers1,

there are several proofs available. Some of these proofs are available athttp://primes.utm.edu/notes/proofs/infinite/index.html. In fact, a mathematician who was being

interviewed remarked thatEuclid proved that there are infinitely many primes over 2,000 years

ago. The host immediately asked Are there still infinitely many primes?

Try to list out the prime numbers from 1 to 10. How many prime numbers are there?

How many prime numbers are there from 1 to 50? How many prime numbers are there from 1 to

100?

Though we know a great deal about prime numbers, a lot still needs to be discovered. There is

the twin prime conjecture2that states that there are infinitely many twin primes. Twin primes

are prime numbers that differ by a difference of 2. For example (3, 5), (5, 7), (17, 19) are all

examples of twin primes. This interesting fact about the prime numbers was used by Paolo

Gierdano in his novel The solitude of prime numbers.

Mathematicians call them twin primes: pairs of prime numbers that are close to each other,

almost neighbors, but between them there is always an even number that prevents them from

truly touching. Numbers like 11 and 13, like 17 and 19, 41 and 43. If you have the patience to goon counting, you discover that these pairs gradually become rarer. You encounter increasingly

isolated primes, lost in that silent, measured space made only of ciphers, and you develop a

distressing presentiment that the pairs encountered up until that point were accidental, that

solitude is the true destiny. Then, just when youre about to surrender, when you no longer have

1An outline of the proof is given at the end of the chapter

2In mathematics we use conjecture for a statement that is believed to be true but is not yet proven to be true


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the desire to go on counting, you come across another pair of twins, clutching each other tightly.

There is a common conviction among mathematicians that however far you go, there will always

be another two, even if no one can say where exactly, until they are discovered.

Mattia thought that he and Alice were like that, twin primes, alone and lost, close but not close

enough to really touch each other. He had never told her that. When he imagined confessing

these things to her, the thin layer of sweat on his hands evaporated completely and for a good

ten minutes he was no longer capable of touching anything.

Excerpt from The solitude of prime numbersby Paolo Giardano

Then there is the Goldbach conjecturewhich states that any even prime number greater than 2

can be expressed as a sum of two prime numbers . Mathematicians have also found that there is

always a prime between n2and (n+ 1)2but have not yet been able to prove it yet.

These are all open problems in mathematics. Any person who can either give an example to

show that these statements are false or give a proof that supports these statements would become

very famous.

Let us also try to explore some interesting properties of prime numbers. A grid of first 300

natural numbers is given below. The grid has 10 columns and 30 rows and some of the boxes are

colored.

2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110


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111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150

151 152 153 154 155 156 157 158 159 160

161 162 163 164 165 166 167 168 169 170

171 172 173 174 175 176 177 178 179 180

181 182 183 184 185 186 187 188 189 190

191 192 193 194 195 196 197 198 199 200

201 202 203 204 205 206 207 208 209 210

211 212 213 214 215 216 217 218 219 220

221 222 223 224 225 226 227 228 229 230

231 232 233 234 235 236 237 238 239 240

241 242 243 244 245 246 247 248 249 250

251 252 253 254 255 256 257 258 259 260

261 262 263 264 265 266 267 268 269 270

271 272 273 274 275 276 277 278 279 280

281 282 283 284 285 286 287 288 289 290

291 292 293 294 295 296 297 298 299 300

a.

What can you say about the numbers in the red colored boxes?b. What about the numbers in the blue coloured boxes?

c. Try to find patterns in the boxes colored with the same colour.

d. Is there anything that strikes you about the numbers in the white boxes?

e. Take the numbers in the last row and list out the numbers that divide them

f. Twin primes are two prime numbers whose difference is two. The first twin prime

numbers are 3 and 5. List out all the twin prime numbers between 1 300.


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g. 3, 5 and 7 go up in step of two and all three are primes. Is there any such triplet in

your grid? If you extend the grid, will there be a possibility of finding such triplet?

h. Find prime numbers that are one less than perfect square. How many such prime

numbers exist in the grid? How many such prime numbers will you be able to get if

you extend the grid?

i.

Find prime numbers that are one more than a perfect square. How many such prime

numbers exist in the grid?

j. Repeat part e and f for cubes and power four.

k. Find prime numbers that can be expressed as 2n- 1. What can you say about n?

l. Find prime numbers that can be expressed as 2n+ 1. What can you say about n?

m. Repeat part h and i for 3n- 1, 4n- 1, 3n+ 1 and 4n+ 1

n. Generalise and analyse as to what could mand nbe if mn- 1 and m

n+ 1 are prime.

o. Check if there is a prime between nand 2nfor n> 1 in the grid.

p. Check if there is a prime between n2and (n+1)2for every natural number n

q.

Show that every even integer greater than 2 in the grid can be written as a sum of twoprime numbers. Is it true for every odd integer greater than 2? Can you express every

integer greater than 5 as a sum of three prime numbers? Verify that all odd numbers

greater than 7 can be expressed as sum of three odd primes.

r. Take any even number. Can you show that it can be written as a difference of two

prime numbers? Is this a unique representation?

Activities

Activity 1.3.1:Make a grid of 20 natural numbers, Find out all the prime numbers. Extend your

grid to first 40, 100 and 200 natural numbers to find the prime numbers. Discuss that if you wish

to find out all prime numbers less than or equal to n. Up to what number should the above

process be repeated so that you are sure that the numbers left behind are prime?

Activity 1.3.2:Make a grid consisting of the years in this century, that is, from 2000 to 2099.

Use the method of Sieve of Eratosthenes to find all the primes from 2000 to 2099.

(a)What numbers can you immediately strike out?

(b)What difficulties do you find in finding the prime numbers?

(c) Is there any way that you can simplify the process?

(d)Do you think the method of the Eratosthenes sieve is efficient for finding large primes?

Activity 1.3.3:Some of the questions have been posed to you regarding the grid above. Try to

frame some more questions on the patterns that you observe.


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Activity 1.3.4:In the above grid there are 10 columns. Make a grid with 9 columns and list the

numbers. Do you have the same pattern?

Activity 1.3.5:Some patterns for primes emerge out of Ulams spiral. Discuss the Ulams spiral

in the class. Think about different shapes you can use to form spirals and see if some patterns

emerge out.

Activity 1.3.6:Watch the video http://www.youtube.com/watch?v=UYkLz8BIS8k

1.4Big money for big prime

January 25th2013 largest prime number having 17,425,170 digits was discovered by Dr Curtis

Cooper. It earned him a whopping cash amount of $3000. Did you know that you could also win

this amount for finding the next really big prime number?And if you find a prime with

100,000,000 digits or more then you are eligible for hold your breath an award of $50,000.

Wont the work of finding prime be simpler if we could have a mathematical formula that would

generate prime numbers a sort of a prime number generating machine.

Generate the first few terms of the quadratic polynomial n2 + n + 17. This formula

generates some prime numbers. What is the least value of nfor which this is not prime? Why?

Also try the cubic polynomial n3+ n

2+ 17

Generate the first few terms of the quadratic polynomial n2+ n+ 41. Does this

expression also generate prime numbers? If yes, then for what values of n?

Visit the site http://mathworld.wolfram.com/Prime-GeneratingPolynomial.htmlto look

up more number of prime generating polynomials.

Do you think a polynomial of the form n2+ n+ p, where p is aprime number will

generate primes?


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Pierre De Fermat in 1660 stated that all numbers that can be written in the form = 2+1(called the Fermats number) will be a prime. Let us check out the first few numbers of theseform

= 2

+ 1 =_____________

= 2+ 1 =_____________= 2+ 1 =_____________= 2+ 1 =65537= 2+ 1 =4294967297Use the prime number calculator at http://math.about.com/library/blprimenumber.htm or

http://www.math.com/students/calculators/source/prime-number.htm to check that while F4 is

prime, whereas F5is not. One of the divisors of F5is 641. It was Leonhard Euler who found that

that one of the divisors of F5 is 641. In fact it has been proved that Fermats numbers from F5to

F21 are composite. It is being conjectured that all Fermats numbers greater than F4 are

composite.

Another mathematician, Marin Mersenne, a contemporary of Fermat was investigating number

of the form Mn= 2n 1. These numbers are called Mersenne numbers. Let us check out the first

few Mersenne numbers. In the table below some entries are filled where as some are left blank.

Try to fill them up.

n Mn= 2n 1 Prime/ Composite/ Neither If composite then the prime factors1 1 Neither

2 3

3

4

5

6 63 Composite 3, 7

7

8

9

10

11 2047 23, 89

12

13 8191

14

15

16

17


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18

19

20

For what values of ndo you get prime? Is it always true?

The prime numbers of the form Mn= 2n 1 are called Mersenne primes. The largest Mersenne

prime Discovered by Dr Curtis Cooper isM57885161. As already mentioned before, it has

17,425,170 numbers of digits.

So far we have seen that there are no general formulas that can generate prime numbers. But is

there a way we can see how many prime numbers will be there?

Here is a table that gives you the prime numbers less than a number that is a power of 10

Number (n) Number of prime numbers(p) less than thegiven number(n) Ratio of primenumbers 10 10

14 0.4

100 102 25 0.25

1,000 103

168 0.168

10,000 104 1,229 0.1229

1,00,000 105 9,592 0.09592

10,00,000 106

78,498 0.078498

1,00,00,000 107 6,64,579 0.0664579

10,00,00,000 108 57,61,455 0.05761455

1,00,00,00,000 109 5,08,47,534 0.050847534

10,00,00,00,000 1010 45,50,52,511 0.0455052511

Each number (n) is raised to some power (y) of 10 in the table above. For example, if n = 1000

then y = 3 and if n = 10,00,000 then y = 6. Let us now try to find the value of x1and x2that are

given by the expressions

x= . and x= (. )

n y x1 x210 1 4.344 7.68

100 2 21.720 27.746

1,000 3 144.801 169.319

10,000 4 1086.021 1218.323

1,00,000 5 8688.907 9514.747

10,00,000 6 72400.81 78051.826


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1,00,00,000 7 620578.379 661638.216

10,00,00,000 8 5430060.81 5741846.577

1,00,00,00,000 9 48267207.25 50715082.665

10,00,00,00,000 10 434404865.334 454132606.721

If you look at the number of primes greater than n and the value of x2you will find that these two

values are approximately same for large values of n. In fact the theorem of prime numbers states

that the number of primes (p) less than a given number (n) is approximately equal to .

Activities

Activity 1.4.1:Using a calculator complete the following table

Number (n) Power to

which 2 israised (y)

Number of prime

numbers(p) less thanthe given number(n) x=

n0.6931 y

x= n(0.6931 y 1)2

4

8

16

32

64

128

256

512

1024

Activity 1.4.2:You can repeat Activity 1.4.1 for numbers that are obtained from powers of 3 and

5. The numerical factor in the denominator will be replaced by 1.0986 and 1.609 respectively.

Activity 1.4.3:Search the internet for the history of prime number theorem. Also investigate the

improvements that have been made for approximating the number of primes. Why do you think

it is important to have an estimation of primes?

Activity 1.4.4:Starting with 1 (red colored cell) write down numbers in a spiral way as shown

below.

37 36 35 34 33 32

17 16 15 14 13 31

18 5 4 3 12 30

19 6 1 2 11 29

20 7 8 9 10 27


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21 22 23 24 25 26

Some of the numbers along the diagonals (green colored) are prime. Get another grid by

changing the number 1 in the shaded cell. For example you may wish to start from number 3.

Find the largest square grid so that all the numbers on the diagonal (bottom left to top right) are

prime.

.*Activity 1.4.5:Look at the paper Eric S Rowland A natural prime generating recurrence,

Journal of integer sequences, Vol. 11, 2008, Article 08.2.8. Write a two page summary of the

paper.

(https://cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.pdf).

Activity 1.4.6:Watch the video http://www.youtube.com/watch?v=3RfYfMjZ5w0

1.5Euclids Division Algorithm

Recall long division that you have done in earlier classes

In this case the remainder 3 is less than the divisor 6. The quotient 2 is less than the divisor 6 and

the remainder 3.

Think and discuss the following

Will the remainder be always less than the divisor?

Will the remainder be always less than the dividend?

Should the quotient be less than the remainder?

Should the quotient be less than the divisor?

Should the quotient be less than the dividend?

What is the relation between the divisor, quotient, dividend and the remainder?

156

2

12

3

Divisor Dividend

Quotient

Remainder


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Euclids algorithm is an efficient method of finding the greatest common divisor of two numbers

by repeated division.

Example 1.5.1 Let us do a repeated division for the numbers 6 and 15

Step I:

Step II:

Let us look at the numbers that divide 15 completely: 1, 3, 5, 15

Numbers that divide 6 completely are: 1, 2, 3, 6

What is the greatest number that divides 6 and 15 completely?

Do a similar exercise for (16,25), (24,12), (13,7), (45, 25).

Example 1.5.2

Step I: Take the two numbers as 112 and 147

Step II: dividend = 147, divisor = 112

Step III: remainder = 35, quotient = 1

156

2

12

3

Divisor Dividend

Quotient

Remainder

63

2

6

0

Divisor Dividend

Quotient

Remainder


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Step IV: dividend = 112, divisor = 35

Step V: remainder = 7, quotient = 3

Step VI: dividend = 35, divisor = 7

Step V: remainder = 0, quotient = 5

The numbers that divide 112 completely are: 1, 2 4, 7, 8, 14, 16, 28, 56, 112

The numbers that divide 147 completely are: 1, 3, 7, 21, 49, 147.

The greatest number that divides 112 and 147 completely is 7.

This greatest number that divides two numbers completely is called the greatest common

divisor (gcd) or the Highest Common Factor (HCF).

Two numbers whose HCF is 1 are called relatively prime or relative prime numbers.

Find the HCF of (16,25). Are 16 and 25 prime? Are they relatively prime?

Discuss whether two prime numbers will always be relatively prime or not.

Watch the video http://www.youtube.com/watch?v=4xANqGj7nnI

Note that

HCF(15, 6) = 3 15 2 x 6 = 3

HCF(18, 10) = 2 2 x 10 18 = 2

HCF(49, 50) = 1 50 49 = 1

We see that if HCF (a, b) = f, then f = ma+ nbfor any integers mand n.

Try to express HCF (28,20) = 4 as 28 a+ 20 bwhere aand bare integers. Do a similarexercise for (16,25), (24,12), (13,7), (45, 25).


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The Euclids algorithm can be visualized by using the tiling analogy. Suppose we wish to cover a

floor area of 6 x 15 exactly by square tiles and we wish to know the size of the largest square tile

that can be used. So we start by a rectangle of size 6 x 15 (Figure I)

FIGURE I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

2

3

4

5

6

We first cover the rectangle by squares of side 6 (the smaller of the two numbers 6 and 15). Weare now left with a smaller rectangle of size 3 x 6 (Figure II)

FIGURE II

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

23

4

5

6


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We first cover the smaller rectangle of side 3 x 6 by squares of side 3 (the smaller of the twonumbers 3 and 16). We see that we have covered the whole of the bigger rectangle. (Figure III)

FIGURE III

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

2

3

4

5

6

Therefore, the gcd (6, 15) = 3.

Do a similar exercise for the numbers 10 and 18.

Activities

Activity 1.5.1:Visualise the Euclids algorithm for the pair of numbers (16,25), (24,12), (13,7),

(45, 25).

Activity 1.5.2:Take different dimensions of a floor plan and find the maximum dimension of a

square tile that can cover the floor. See if you can use tiles of different shapes to tile the floor.

Activity 1.5.3:Have fun with tessellations athttp://www.mathcats.com/explore/tessellationtown.html

*Activity 1.5.4:Reverse Euclids algorithm to find aand bfor the equation HCF (28,20) = 4 =28 a+ 20 b. Use this to find aand bsuch that HCF(735, 3024) = 735a+ 3024b.


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*Activity 1.5.5:Write a code for the Euclids algorithm in any programming language that youare familiar with.

*Activity 1.5.8:Read about continued fractions. Can you relate a continued fraction for FigureIII ?

1.6Importance of prime numbers in encryption

Did you know that you use prime numbers to keep the personal information secure every timeyou or your parents shop on the internet using your credit card? How? Using RSA cryptography.Cryptography comes from the Greek words kryptos that means hidden and graphein whichmeans to write. So cryptography is a language that converts messages into gibberish, so that theycannot be read by persons other than the intended recipient. One of the simplest method is thesubstitution cipher.

Watch the video http://www.youtube.com/watch?v=fbGlKAA95Jw

If you mix two colors red and blue what color do you get? Does the shade of the new color

depend upon the amount of each color you are mixing? Is it possible to separate out the red and

blue color from the new shade that you have obtained?

Similarly can you multiply 5 and 7 and tell the answer? What about the product of 23 and 47?

How about multiplying 193 and 197?

Let us ask the question in another way. Can you tell the factors of 15? Can you factorise 1081?

What about the factors of 38021?

Just like mixing of the two colors is an easier process, multiplication of two numbers is easy. The

reverse process of factoring a number becomes difficult. More so, if the factors happen to be

large primes. The fact that it is difficult to factorize a number that have large primes as factors is

used in encrypting messages. One such method of encryption is the RSA algorithm named after

Ron Rivest, Adi Shamir and Leonarf Addleman who first publicly described the algorithm.

Watch the video http://www.youtube.com/watch?v=kYasb426Yjk

Find out about Ron Rivest, Adi Shamir and Leonarf Addleman


20/31

Revisit the Sherlock Holmes story The Adventure of the Dancing Men by Sir

Arthur Conan Doyle.

The RSA method involves coding and decoding of a message. Let us assume that our message is

a single word CAGE from the set of alphabets A B C D E F G H I. Each alphabet is associated

with a number starting from 1.

A B C D E F G H I1 2 3 4 5 6 7 8 9

Therefore the numerical representation of CAGE is 3 1 7 5. The steps involved in coding thismessage are

Step I: Choose any two prime numbers. Let us choose 2 and 5

Step II: Multiply the two numbers to get the module. In this case it is 10

Step III: Calculate the product (2 1) x (5 1) = 4

Step IV: Choose a number greater than 1 and less than 4 (the number obtained in Step III) which

is relatively prime to 4. The only choice for us in this case is 3. This is our public key.

Step V: Raise all the powers of the associated number to the power of the public key obtained in

Step IV.

33= 27, 13= 1, 73 = 343, 53= 125

Step VI: Divide each new number obtained in Step V by the module obtained in Step II and note

the

remainder. In this case the remainders are 7, 1, 3, 5

Step VII: The coded message is 7135.

Let us try to repeat the algorithm for larger values of prime numbers.

Step I: Let the two prime numbers be 7 and 11.

Step II: The module is 7 x 11 = 77

Step III: (7 1) x (11 1) = 60


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Step IV: The public key has to be a number greater than 1 and less than 60 that is relativelyprime to 60. We can choose any one of 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53 or 59.Let us choose it as 13.

Step V: 313

= 1594323, 113

= 1, 713

= 96889010407, 513

= 1220703125

Step VI: Divide each number in step V by 77 and note the remainder. The coded message is38 1 35 26

Find the coded message if you use the key as 7, 11, 17 or 19.

Let us now see how we can decode the message. To decode the message we have to generate a

private key or decoder. When we subtract 1 from the product of the private key and the public

key it should divide the number obtained in Step III completely.

In the first case the public key is 3 and the number is 4. So the private key can be 7 or 11 or 15,

etc. Apply step V and VI to the coded message replacing the public key by the private key. For

example if the private key is 7 then we have

77= 823543, 17= 1, 37= 2187, 57= 78125

The decoded message is 3175.

Activities

Activity 1.6.1:What are the possibilities for the private key in the second example when theprime numbers used are 7 and 11?

Activity 1.6.2:Implement the RSA algorithm using the primes 2 and 5 to encode DEAF. Whatdo you observe? Why? What will happen if we use the prime numbers 7 and 11?

Activity 1.6.3:Can the public key be used to decode the message the first example where theprime numbers used are 2 and 5? Can it be used for prime numbers 7 and 11? Try to find out ifthe public key can be used to decrypt the message. Then discuss what makes this method ofdecoding a good method.

Activity 1.6.4:Associate a different number to each alphabet arbitrarily. Disclose the module m

and the key kto your friend. (Assume your friend is aware of this algorithm). See if your friendcan decipher the message.

Activity 1.6.5:The number of alphabets must be taken less than the module. Discuss whathappens if it is more than the module

Activity 1.6.6:Think about the various steps you can take so that it becomes difficult to decodeyour message.


22/31

Activity 1.6.7:Watch the movie http://www.youtube.com/watch?v=wXB-V_Keiu8

1.7Matrices

1.7.1Introduction to Matrices

Each one of us have had a look at calendars. The days of a particular month are arranged in a

rectangular block that looks somewhat like this

Sun Mon Tue Wed Thu Fri Sat

1 2 3 4 5

6 7 8 9 10 11 12

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30 31

The dates could also be arranged as

Sun 6 13 20 27

Mon 7 14 21 28

Tue 1 8 15 22 29

Wed 2 9 16 23 30

Thu 3 10 17 24 31

Fri 4 11 18 25

Sat 5 12 19 26


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Writing of numbers in such rectangular forms is known as matrices. In the boxes above there are

cells that are colored in dark grey. These cells do not contain any number. When we write down

matrices all cells would have numbers (positive, zero or negative). A matrices having six

numbers could be written in any one of the forms given below

1 2 3 4 5 6 or

1

2

34

5

6 or 1 2 3

4 5 6 or 1 23 4

5 6

The first form has only one row, the second form has only one column, the third form has two

rows and three columns and the last form has three rows and two columns. We will restrict

ourselves only to matrices that have two rows.

1.7.2Addition of Matrices

Suppose that Mr Ahmed has two shops at locations A and B. Both the shops keep clothes for

boys and girls in two price ranges

Price range I: Rs 200 399 and

Price range II: Rs 400 599.

The number of clothes in each shop are represented as

80 3646 90 , 75 3066 85

He wishes to find the total number of clothes for boys and girls in the two price ranges.

Therefore, the total clothes in

Price range I for boys = 80 + 75 = 155

Price range II for boys = 46 + 66 = 112

Price range I for girls = 36 + 30 = 66Price range II for girls = 90 + 85 = 175

In matrix form this can be represented as

80+ 75 36 + 3046+ 66 90 + 85


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So we see that the addition of two matrices is obtained by adding the corresponding elements.

1.7.3Multiplication of Matrices

In the previous example let us assume that Mr Ahmed offers a scheme of selling all the clothesfor boys at a flat rate of Rs 300 and all the clothes girls at a flat rate of Rs 450 in his first shop.

We wish to find the total revenue he would generate in each range, assuming he sells out all the

clothes.

Revenue generated by clothe in price range of Rs 200 Rs 399: 80 x 300 + 36 x 450 = Rs 40200

Revenue generated by clothe in price range of Rs 400 Rs 399: 46 x 300 + 90 x 450 = Rs 53700

The matrix representation would be

80 36

46 90 300

450 = 80 300 + 36 45046 300 + 90 450 =

40200

53700

Suppose he comes up with another scheme: All the clothes for boys at a flat rate of Rs 350 and

all the clothes girls at a flat rate of Rs 400 in his first shop. Then the total revenue generated in

each range would be given as

80 36 46 90

350400

= 80 350 + 36 40046 350 + 90 400 = 4240052100

Both the calculation in a single step can be represented as

80 36

46 90 300 350

450 400 = 80

300

+36

450 80

350

+36

400

46 300+ 90 450 46 350 + 90 400

= 40200 4240053700 52100

Suppose we interchange the order of multiplication and write

300 350450 400

80 36 46 90

= 300 80+ 350 46 300 36 + 350 90450 80+ 400 90 450 36 + 400 90

= 40100 42300

53700 52200The resultant matrices in the two cases are not same. Why did this happen?

Let us just try to analyse the first elements.

80 300 + 36 450: (No of clothes for boys in range I) x (flat rate for boys clothesin 1st


25/31

scheme) + (No of clothes for girls in range I) x (flat rate for girls

clothesin 1stscheme)

= Revenue generated by clothes in price range I in the first scheme

300

80

+350

46: (flat rate for boys clothesin 1

stscheme) x (No of clothes for boys in

range I) + (flat rate for boys clothesin 2nd scheme) x (No of clothes forboys in range II)

= Total revenue generated in 1stscheme by boys clothes in price range I

and in 2nd

scheme by boys clothes in price range II

We see that the interpretation of the two terms have changed. Therefore the order in which we

multiply the two matrices is very important.

Let us look at step wise multiplication of two matrices

Step I:

Step II:

Step III:

Step IV:

Let us see what we get if we interchange the order of the matrices

Step I:

1

3

2 5

7 3

4

2

X = 18

1

3

2 5

7 3

4

2

X = 18 17

1

3

2 5

7 3

4

2

X = 18 17

34

1

3

2 5

7 3

4

2

X = 18 17

34 16

4 1

2 3

5

3

2

7

X = 15


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Step II:

Step III:

Step IV:

1.7.4Some special Matrices

In the last subsection we saw that the order in which we multiply two matrices is important. Is it

always true that if we interchange the order of the matrices we will get a different matrix each

time? We give here two special matrices. If we interchange the order we will get the same result.

Case I:

Case II: Let the matrix be

5

3

4 1

2 3

2

7

X = 15 23

5

3

4 1

2 3

2

7

X = 15 23

25

5

3

4 12 3

2

7

X = 15 2325 19

5

3

1 0

0 1

2

7

X = 2 5

7 3

0

1

2 5

7 3

1

0

X = 2 5

7 3

2 5

7 3


27/31

We form another matrix as follows

Step I:

Step II: The new matrix obtained from step I is

Step III: Multiply the original matrix and the matrix in step II

Let us interchange the order

2 5

7 3

Interchange these two

entries

Replace these two entries by

their negative signs

3 -5

-7 2

3 -5

-7 2

-29 0

0 -29

2 5

7 3

X =

3 -5

-7 2

X2 5

7 3

=-29 0

0 -29


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In each of the cases I and II we get the same resultant matrix irrespective of the order in which

we multiply.

In the first case we have a special matrix that has only 1 and 0 as its entries. This matrix is called

a unit matrixor identity matrix.

If we change the order of these entries does the result still remains the same?

In the second case given a matrix we follow a certain procedure to obtain a second matrix. If

these two matrices are multiplied, irrespective of the order, we get the same result. The resultant

matrix is again a special case that has a certain constant (- 29 in this case) and 0 as entries.

Will we always get a similar pattern for any matrix that we pick up? Take up some

matrices and verify it.

1.7.5Inverse of a matrix

In the previous subsection we saw that if we multiply and

we get a matrix with 0 and a constant 29 as its entries. Let us divide each element in the

second matrix by 29 and now multiply the first matrix by this new matrix.

The matrix is called the inverse of the matrix

2 5

7 3

X-3/29 5/29

7/29 2/29

=1 0

0 1

2 5

7 3

X-3/29 5/29

7/29 2/29= 1 0

0 1

-3/29 5/29

7/29 2/29

2 5

7 3

2 5

7 3

3 -5

-7 2


29/31

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30/31

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31/31

*Activity 1.8.5: The above method requires that the decoder knows the inverse of the 2 x 2

matrix which is used for the conversion of the 2 x n matrix. Taking the hint from the RSA

method check if it is possible to device a method so that the inverse of the 2 x 2 matrix is not

known to the decoder.

*Euclids proof of the infinitude of primes

Let us try to replicate the proof of Euclid. We know that the first few primes are 2, 3, 5, 7, 11,

13, . For the sake of arguing let us assume that there are only two primes: 2 and 3. Consider

the number 2 x 3 + 1 = 7. We get a prime number greater than 2 and 3. Therefore, our

assumption that there are only two primes is not correct. Let us now assume that there are only

three primes: 2, 3, and 5. We now consider the number 2 x 3 x 5 + 1 = 31 which is a prime

greater than 2, 3, and 5. Again our assumption that there are only three primes is wrong.Do you think that if we multiply some prime numbers and add 1 to the result we will always get

a prime? Try out multiplying the first four prime numbers and add 1.

2 x 3 x 5 x 7 + 1 = ? Is this number prime?

Consider now that there are only six primes: 2, 3, 5, 7, 11 and 13. We again consider the number

2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 (You may use a calculator to do so). Is the new number

prime? Is it composite? The number is too big to answer these questions straight away. But let us

do some deductive analysis. There can only be two cases:

(a)

30031 is prime. In this case we have got a prime number greater than 2, 3, 5, 7, 11 and13. So our assumption that there are only six primes is wrong.

(b)30031 is composite. In this case 30031 must have prime factors. We have assumed that

there are only six primes 2, 3, 5, 7, 11 and 13. By using a calculator you can verify that

30031 is not divisible by 2, 3, 5, 7, 11 or 13. Therefore there must be another prime

number different from these six numbers that will divide 30031 if it is a composite

number. So our assumption that there are only six primes is wrong. Incidentally, the

prime factors of 30031 are 59 and 509.

So if we assume that there are finitely many primes we can always show that there will be a

prime number greater than these finitely many numbers which will be prime. This will always bein contradiction to our assumption of finitely many primes. Hence, the number of primes are

infinite.

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Mathematical Ability