Методические указания по английскому языку для студентов 1 курса математического факультета. 2004 (стр. 2 из 5)
5. If the centre of gravity lies on each of these lines …
6. If we were able to find area of the triangle in three different ways by starting with each side as base in turn …
7. If we were confident that the three medians did indeed concur …
6. Speak about medians of triangles and their properties, using expressions:
It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably.
Assignments for unit I:
1. Collect all the information about triangles, based on text “What is a triangle?”, according to your own plan.
2. Arrange the sentences, you have written out logically.
3. Unite all the sentences, using the following words and expressions: It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably, it is clear that…, considering, using hypotenuse as a side, to receive the square, is equal to, on the one hand, on the other hand, hence, it is evident that… It is known that…, we are quite familiar with…, every mathematician is sure of…, it should be pointed out…, in fact, thus, It is well known, consequently, therefore, so, it is obvious, it is evident, apparently, manifestly.
4. Do the same work (points 1, 2, 3) for each text: “Types of Triangles”, “Similarity criteria of triangles”, “Pythagorean Theorem”, “Medians of a triangle”.
5. Read the text, you have composed and make sure all the points of it are arranged logically.
6. Make changes if necessary.
7. Read the text one more time and think of the title.
Try to solve
Problem 1
Across the river
Jake Hardy was standing on the river bank, looking across to the far side. ‘How wide do you recon it is?’ asked Harold. Jake adjusted the rim of his hat, and turned to look downstream. He paused and then walked with deliberate paces along the river bank, then turned and called out, ‘About thirty meters, give or take a few.’
How did he estimate the width of the river?
Fig. 1.6
Smile -
About application of mathematics in linguistics
A teacher of English was ill and a teacher of mathematics replaced him.
He began to compose a table of irregular verbs:
Then he said:
- Okay, I mark this form as x . Then it’s possible to compose the proportion:
Unit II.
Prime numbers
Text 1
Prime numbers
Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other.
In Book of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.
There is then a long gap in the history of prime numbers during what is usually called the Dark Ages. The next important developments were made by Fermat at the beginning of the 17th Century. He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares. He devised a new method of factorizing large numbers which he demonstrated by factorizing the number 2027651281 = 44021
46061. He proved what has come to be known as Fermat's Little Theorem (to distinguish it from his so-called Last Theorem). This states that if p is a prime then for any integer a we have ap = a modulo p. This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n - 2 is divisible by n. The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31
11 is composite. Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers. Number of the form 2n - 1 attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers Mn because Mersenne studied them. Not all numbers of the form 2n - 1 with n prime are prime. For example 211 - 1 = 2047 = 23
89 is composite, though this was first noted as late as 1536. For many years numbers of this form provided the largest known primes. In 1952 the Mersenne numbers M521, M607, M1279, M2203 and M2281 were proved to be prime by Robinson using an early computer and the electronic age had begun. By 2003 a total of 40 Mersenne primes have been found. The largest is M20996011 which has 6320430 decimal digits.There are still many open questions (some of them dating back hundreds of years) relating to prime numbers.
Assignments:
1. Active vocabulary:
Perfect number, amicable number, composite number, infinite, contradiction, integer, speculation, to devise.
2. Give the definition of the following notions in English:
Perfect number, amicable number, composite number
3. Think of at least two examples of each type of numbers.
4. Give Russian equivalents to the following words and expressions:
Primality, proper divisor, to establish a result, Dark Ages, to prove a speculation, to demonstrate by factorizing a number, to be divisible by, decimal digits.
5. Answer the questions:
1. Who was the first to study prime numbers?
2. What were the mathematicians of Pythagoras's school mainly interested in?
3. What proof of the Fundamental Theorem of Arithmetic did Euclid give?
4. What period is called the Dark Ages?
5. What does Fermat’s Little Theorem state? Why is it so important?
6. What kind of numbers are called Mersenne numbers and why?
6. Scientific contribution of what mathematician to the prime numbers theory is described in the following passages? Arrange the passages in the chronological order.
1) He studied numbers of the form 2n – 1, which nowadays are known as numbers called after him. The largest known prime number is the number of exactly the same form.
2) This mathematician managed to show that all even perfect numbers are of such a form: 2n-1(2n - 1).
3) The proof of the Fundamental Theorem of Arithmetic together with the proof that there are infinitely many prime numbers was given by him
4) Being interested in numbers for their mystical and numerological properties, they understood the idea of primality and were occupied with the study of perfect and amicable numbers.
5) This mathematician devised a new method of factorizing large numbers.
7. Enrich each passage using the information from the text and speak on the following topics:
1) The mathematicians of Pythagoras's school and their scientific work.
2) Euclid’s speculations about prime numbers in his “Book of the Elements”.
3) Fermat’s proof of the Fundamental Theorem of Arithmetic, his Little Theorem and other works.
4) Mersenne numbers.
8. Retell the text “Prime numbers” using the statements from the previous assignment as the plan.
Text 2
Before you read:
Consider the following unsolved problems in the theory of prime numbers and give accurate translation.
Some unsolved problems
1. The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.
2. Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.
3. Are there infinitely many primes of the form n2 + 1?
(Dirichlet proved that every arithmetic progression: {a + bn | n
N} with a, b coprime contains infinitely many primes.)4. Is there always a prime between n2 and (n + 1)2?
(The fact that there is always a prime between n and 2n was called Bertrand's conjecture and was proved by Chebyshev.)
5. Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.
6. Are there infinitely many sets of 3 consecutive primes in arithmetic progression?
7. n2 - n + 41 is prime for 0
n 40. Are there infinitely many primes of this form? The same question applies to n2 - 79 n + 1601 which is prime for 0 n 79.8. Are there infinitely many primes of the form n# + 1? (where n# is the product of all primes
n.)9. Are there infinitely many primes of the form n# - 1?
10. Are there infinitely many primes of the form n! + 1?
11. Are there infinitely many primes of the form n! - 1?
12. If p is a prime, is 2p - 1 always square free? i.e. not divisible by the square of a prime.
Text 3
Before you read:
Read the following records; try to memorize the numbers and the scientists, who have announced them.
The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in November 2003) is the 40th Mersenne prime: M20996011 which has 6320430 decimal digits.
The largest known twin primes are 242206083
238880 
1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995.The largest known factorial prime (prime of the form n!
1) is 3610! - 1. It is a number of 11277 digits and was announced by Caldwell in 1993.The largest known primorial prime (prime of the form n#
1 where n# is the product of all primes n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993.Assignments:
1. Speak on the latest prime records trying to avoid peeping into the text.
Assignments for unit II:
1. Give English equivalents of the following words and expressions:
совершенное число, десятичный знак, дружественные числа, простое число, факторизация, бесконечно много, арифметическая прогрессия, целое число, метод «от противного».
2. Using the above mentioned words and expressions (English translation) make up sentences of your own.
3. Remind yourself of what you were reading in this unit and answer the questions:
a) Who is the founder of the study of prime numbers?
b) Why was the certain period in the development of the study of prime numbers called the Dark Ages?
c) What numbers do we call Mersenne numbers and why?
d) It is possible to solve the following problem: are there infinitely many primes of the form n! + 1?
e) What is the largest known prime? The largest known factorial prime?
4. Name:
a) The scientists considering the study of prime numbers;
b) the scientists who managed to pose unsolved problems in the theory of prime numbers;
c) the scientists who announced the records of prime numbers.
5. Tell the group everything you know about these mathematicians, i.e. everything found out in this unit.
Smile -
Einstein and telephone
One woman asked Einstein to remember her telephone number: 361-343.
Einstein answered:
- It’s very easy. 19 squared and 7 cubed.
New about limits
At a mathematics exam a professor asks a student to calculate the limit:
The professor is surprised:
- What is it? Why ?
The student answers:
- You explained at your lecture that
and I have used this example.
Unit III.
Game Theory
Text 1
Some Basics
Game theory is a branch of applied mathematics fashioned to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. In a typical game, decision-making “players,” who each have their own goals; try to outsmart one another by anticipating each other's decisions. (Encyclopedia Britannica)