MATHEMATICS IN DAILY LIFE

LEGENDRE Jérémy

PHANTHACHITH Jimmy

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TABLE OF CONTENTS

INTRODUCTION

I. HOW TO WIN USING MATHEMATICS ? ( Jérémy )

- Basic elements about probability

- Application in logical games

II. MATHEMATICS IN SECURITY SYSTEMS . ( Jimmy )

- The purpose of cryptography

- Terminology

- Caesar cipher

- Hill cipher

III. DESIGNING AND MODELLING WITH MATHEMATICS . ( Jérémy )

From the tile of your living-room to the Parisian subway…

CONCLUSION

ABSTRACTS VOCABULARY L ISTS

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INTRODUCTION :

A mathematics teacher, in front of his pupils, is making his lesson, when one of them puts up one hand… Teacher: Yes? Have you got any question? Pupil: Yes… I wondered… Are mathematics helpful?

It is probably a very delicate situation, which most of teachers may have faced. Some would answer that mathematics are very useful, without ever quoting any concrete example, whereas some others would find several daily applications of maths.

Being students in this abstract science makes us used to this question. Indeed, it is well-known, that many people wonder what mathematics are used for. Nevertheless, concrete uses are very common. In truth, from the tile of your bathroom, to the construction of bridges, passing by forecasts of the course of the stock exchange, mathematics are present in multiple fields. The purpose of this presentation is to give some examples which use this fabulous science…

Then, we will introduce several basic elements so as to explicate different applications in everyday life.

First, we will start off by a little of probability. Among others, the famous Monty Hall Problem will be detailed. Next, we will move on the utilisation of cryptography, so as to explain how to create a secret message, using two different ways. Finally, some various applications in geometry and arithmetic will be treated in the case of the Parisian subway, or to show how to tile a rectangular room.

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1 HOW TO WIN USING MATHEMATICS ? s we will discover throughout this presentation, there are many applications of mathematics in every day life. In this section, we will show how to understand a situation which uses probabilities…

Don’t you think it will be interesting to be able to win one bet thanks to mathematical elementary knowledge?

1.1 Basic elements about probability

Definition : • A mathematical assertion is a proposition, which can be true or false.

• If A is a mathematical assertion, we will note A the contrary assertion. • An issue of an experience is named an event.

Examples: A : “I love chocolate.”

A : “I hate chocolate.” B : “All the students were born at the different dates.”

B : “There are two students, born at the same date.” Definition :

• Given A an event of an experience. We will note ( )P A the probability that A is realized.

• Two events are independent if the probability of one does not depend on the probability of the other.

Theorem: For all independent events A and B,

0 ( ) 1P A≤ ≤ .

( ) 1 ( )P A P A= −

( ) ( ) ( ) ( )P A B P A P B P A B∪ = + − ∩

( ) ( ). ( )P A B P A P B∩ =

Example: Let us roll the dice… A : “we obtain 6”

A : “we don’t obtain 6” ( we obtain 1,2,3,4 or 5 ).

So, 1

( )6

P A = and 1 5

( ) 16 6

P A = − =

A

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1.2 The Monty Hall problem It is a probability puzzle based on the American television game show “Let’s make a deal”. Context :

Considering three doors. One of those hides a car… The two others a goat. The candidate is asking to choose one of the doors…

Assuming the car is hidden behind the second door. Let us note A : “the candidate win the car”.

So, at the beginning of the game, 1

( )3

P A =

• First, the candidate chooses one of the doors. • Second, the anchor, who know where is the car, opens one of the two others doors,

behind which is hidden a goat. Then he asks the candidate to change his choice, there are also two possibilities :

The candidate does not change. Then, 1

( )3

P A = .

The candidate changes his choice. Then, 2

( )3

P A = .

In order to prove this result, we will distinguish three cases.

1. The candidate chooses the door N°1. Then, the anchor opens the door N°3. o If the candidate does not change, he looses. o If he changes, he wins.

2. The candidate chooses the door N°2. Then, the anchor opens the door N°1 or N°3. In

the both cases, o If the candidate does not change, he wins. o If he changes, he looses.

3. The candidate chooses the door N°3. Then, the anchor opens the door N°1.

o If the candidate does not change, he looses. o If he changes, he wins.

Finally, we remark that if he decides to change, the probability he wins is 2

3, otherwise

the probability is 1

3.

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1.3 A small bet... Context:

Given a man, a woman and a classroom, counting 30 students. Man : I bet 50€ that there is at least two students in this class, born at the same date. Woman : Oh ! I don’t think! But I make the bet… Who has most chance to win 50€ ?

Let us note A : “There are two students, born at the same date”.

So, A: “All the students were born at different dates.”

Then, calculate ( )P A … Settle the date of birth of the first student. So, if we want that the second is born at a different date, it rests 364 possibilities on 365. And it is important to note that the probability that the second student is born on one of the 364 other days is independent of the date of the first student ! For the third, it rests 363 possibilities… And so, we easily obtain that the thirtieth has 336 possibilities on 365. We also have :

365 364 363 336( ) ... 0.3

365 365 365 365P A = × × × × ≃

To conclude, ( ) 1 ( ) 0.7P A P A= − ≃ . Then, the man has more chance to win…

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AliceC→M

BobM→C

Eve

C

C ~ C'

BobM→C

AliceC'→M

BobM

AliceM'

EveM'

hen we used to be children, have we never written some secret messages in order to be read by only people you want so as to, for instance, talk with your lover, win games, hide something to someone, etc. In mathematics, there is a field called

Cryptography coming from the Greek “kruptos” which means to hide and “graphein” to write. Indeed, cryptography is the science of writing in secret code.

2.1 The purpose of cryptography

By the past, cryptography was only concerned with message confidentiality i.e. conversion of message from an understandable form into an incomprehensible one which is unreadable by interceptors whom do not own the secret knowledge (we will see some examples in the next section). However, in our present day, many information travel through, for instance, the Internet whether or not personal data.

In the aim to protect everyone from new generation offences, mathematicians target four points in which cryptography must be based on:

• Confidentiality/ Privacy: we want to transmit data on a non secured channel such as only the receiver is allowed to read it

• Integrity: when receiving the data, we want to be sure nothing change its content during the transmission

• Authentication: we want to be sure on the identity of the sender

• Non-repudiation: we want to be able to prove to a third party that a commitment made by someone has really been taken (such as he can't deny it)

2.2 Terminology

From now on, I will introduce some specific vocabulary as I may be understood (I hope :-) and we will focus on confidentiality).

2 MATHS IN SECURITY SYSTEMS

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Definition :

• Alphabet: Finite set of symbols used to write message (e.g. A→Z, 0→9, {0,1}) • Plain text: the original message written with letters in the alphabet • Cipher: method of encryption • Cipher text: the encrypted message • Encryption: a process which give you a cipher text from the plain text • Decryption: method in which you can get the plain text with the cipher text • Key: Piece of information in the cryptosystem which enable the encryption and the

decryption. Without the key, and even knowing the cipher, the decryption should be impossible.

In cryptography, there is this famous principle given by Auguste Kerckhoffs in the 19th century in which cryptography is based on.

Kerckoffs' principle: A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.

In other words, if someone wants to attack the cryptosystem, we assume that he or she knows the method of encryption and decryption, but the key is secret, then, the security of the system rely on the key.

2.3 Historical example : Caesar cipher

a) Methods of encryption

In 58 BC, Julius Caesar was planning on attacking the Gaul. In order to communicate with his generals, he imagined two method of encryption. The first one consists in changing Latin letters into Greek letters. The second process is a simple encryption: it consists in a shift of 3 positions to the right in the alphabet so that A may be D, B would be replaced by E, C would become F... In other words, it is just a substitution cipher. By his time, this cipher was probably reliable in spite of his simplicity, because numbers of people were illiterate or if his enemies intercept the message they would have assumed that the message was written in an unknown foreign language.

The Caesar cipher can be explained with mathematical formula by numbering each letter of the Latin alphabet with numbers from 0 to 25 in the lexicographic order.

Definition :

Let ∑ an alphabet and n an element of ∑. The Caesar cipher is given with the formula: f (n) = n+3 (mod 26)

Figure 1: Caesar cipher

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It is worth noticing that 3 is the key of this cryptosystem. Indeed, we can “generalize” the Caesar cipher by changing the number 3 (which corresponds to the letter C) by any number from 0 to 25 (or any letter from A to Z) so that, we can count the key number of the cryptosystem that is to say 26 keys. Obviously with such a poor number of key the Caesar cipher is easily breakable: a computer can test all of the 26 possibilities nowadays without any difficulties.

Remark: We can apply an affine transformation to the Caesar cipher so that the formula given above would be f(n) = an+b (mod 26), a, b elements of ∑. As a consequence, the number of useful keys (useful there are some affine transformation which give Id∑ -the identity-) increase to 312 (still computable with a computer).

b) The frequency of Al-Kindi The “generalised” Caesar cipher is a mono-alphabetical substitution i.e. it fits with the

two following conditions: 1. Two different letters are not encrypted the same way 2. A letter as a unique cipher letter.

In the 9th century, an Arab erudite named Abu Yusuf Al-Kindi suggests a method in order to decode mono-alphabetical substitution. Being used to analyse Islamic holy texts so as to verify its authenticity, Al-Kindi observes that the letters and syllables, in Arabic, have different frequencies. Thus, the letters a and l are the most used, whereas j appears 10 times less often. In consequence, Al-Kindi suggests calculating the frequency of each letter in the cipher text in the aim to compare it with the frequency observed in the language used to write the message. Below, you will see an example of Al-Kindi frequencies for English.

Figure 2 : Al Kindi frequencies for English language

c) Vigenère cipher

In the Caesar cipher, the key is just a single letter, so that nowadays only children play with it! With the Caesar cipher, we get the ciphertext just with a letter from the plaintext e.g.:

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I D O N T L I K E F R O G S

+ C C C C C C C C C C C C C C

F A L K Q I F H B C O L D P

Thanks to the method of frequencies by Al-Kindi, it enables us to break the Caesar cipher rather easily. In the 16th century, Blaise de Vigenère imagined a variant of the Caesar cipher in function of a key which is not only a letter of the Latin alphabet. The key could be a word, so that it can break the natural order of the letter in the ciphertext, in this case, if the key is long enough the Al-Kindi method should be useless. Definition :

Let ∑ an alphabet. A word is a concatenation of elements in ∑. ∑* is the set of all words generated by ∑.

• Example: I D O N T L I K E F R O G S

+ S W E E T S W E E T S W E E

A Z S R M D E O I Y J K K W

Vigenère used this table (Fig.3) for the encryption and decryption of his cipher, however, if looking at this table tire your eyes, you can also make the bijection between {A→Z} and {0→25} and just add the letters of the plaintext with the letters of the key modulo 26.

Figure 3: Vigenère table

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2.4 Hill cipher

a) Method of encryption

Another example is the Hill cipher which is completely different from the Caesar cipher, a way to show you that there is not just substitution cipher. The key is a squared matrix of n dimensions invertible modulo 26. The strategy is to cut the plain-text in blocks of vectors of n dimensions, and to multiply all of the vectors generated by the key (obviously, I forgot to say that the alphabet is created the same way as the Caesar cipher, except when the dimension of the key is greater than 26, the alphabet would be generated in another way but it is not the goal to explain it here).

b) Example

• Key:

• Plaintext: RENDEZ VOUS

c) Remarks

• If you want decrypt, we use the same process with the inverse matrix M-1 • The key depends only on its value modulo 26, in other words, two matrices M and M'

such as M mod 26 = M' mod 26 give the same key. Then, the number of keys is the numbers of 4-uplets (a,b,c,d) of integers with 0 ≤ a,b,c,d ≤ 25 such as a, b, c, d is primed with 26. We prove that this number is (2²-1)(2-1)(13²-1)(13-1) = 157 248. With matrices 5×5, the number of keys is about 1034, so it is big enough to avoid an attack by research of the key.

• However, the Hill cipher is weak to modern attack that I will not explain unfortunately here.

→blocks RE ND EZ VO US

→vectors (17,4) (13,3) (4,25) (21,14) (20,18)

×M (mod 26) (19,25) (2,11) (7,24) (3,7) (20,18)

Ciphertext TZ CL HY DH US

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3 DESIGNING AND MODELLING WITH MATHEMATICS

1 How to tile a room ? Context :

Considering a living-room, which is 6m60 long and 4m50 wide. If you want to tile this room, using only entire square tiles, having the biggest size, how do you proceed?

The answer is very simple, you have to use the GCD (Greatest Common Number) of the two numbers : 660 and 450. Indeed, the GCD can only apply to integers! So, you need to use the size in centimetre. Let us factor 660 and 450 in products of prime numbers…

2

2 2 2

660 3 220 3 4 55 2 3 5 11

450 9 50 3 2 25 2 3 5

= × = × × = × × ×

= × = × × = × ×

Then, ( )660,450 2 3 5 30GCD = × × =

Consequently, to succeed in your labour, you have to use square tiles of 30cm long. Your

living-room will also be made up of 22 tiles in the length and 15 in the width.

3.2 The secret of the Parisian subway...

The Parisian subway is very particular… Indeed, if two people, who waiting for the subway on the both platform, can talk each other just in murmur, as we wan see on the figure n°3.

Fig 1

Fig 2

The station « Arts et Métiers » of the very

famous Parisian subway

Fig 3

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Before explaining this strange phenomenon, have a look at the figure n°2… We can notice that the ceiling of the station is vaulted. In fact, it represents a semi-ellipsis! Definition :

F and F’ are named the focus of the ellipsis. Theorem:

Given M a point of the ellipsis (E). Let us note (N) the perpendicular straight line to the ellipsis in the point M. Then, (N) is the bisector of the angle FMF’.

This theorem also proves the result! Indeed, as the figures n°4 and 5 show us, if one person stands at the focus F of the ellipsis, all acoustic waves emitted will pass by the second focus F’, after being reflected on the edge of the ellipsis.

That explains why the two people can talk to each other in a whisper… Remark :

This phenomenon is also used in the Louvre Museum, in the Agrigentes’ Cathedral, or in the rotunda of the Capital Building in Washington.

Fig 4

Fig 5

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CONCLUSION :

From a very long time – the first elements of mathematics dated from more than

20.000 years -, humankind has developed many theories so as to explain how things work. Those theories help us to understand a lot of things in the nature, thanks to physicists who, describe more easily our environment, using mathematics.

Then, the importance of this science is not anymore to prove… We detailed some

applications of mathematics in this presentation, but there are many others in various fields! Every physicist could give us a lot more applications. Here, we just wanted to focus on a few examples in daily life. Of course, we could have tackled other problems… For instance, why the buttered-toast always falls upside-down! Or explain the importance of mathematics in architecture. Some other people could teach us how to win in Black Jack or in Poker.

In fact, there are so many answers to the question: “Are mathematics useful?”, from measuring an angle, to explaining how works a nuclear explosion. Moreover, the applications of mathematics are plentiful, at every level of knowledge, from the elementary school, to the highest ranked mathematicians. To conclude, let us add that our goal was to show different applications of basic elements of mathematics in everyday life. Hoping our objectives have been met…