- •Grammar Revision
- •Indefinite Tenses (Active)1
- •Continuous Tenses (Active)1
- •Perfect tenses (Active)1
- •1. Read the text. To understand it better consult active vocabulary:
- •Active Vocabulary
- •4. Study the text and answer the following questions:
- •1. Read the names of the faculties. Give Ukrainian equivalents:
- •2. Read the names of the specialities. Give Ukrainian equivalents:
- •3. Look through the email and answer the following questions:
- •Active Vocabulary
- •2. Are these statements true or false? If they are false, say why. Use the following phrases:
- •3. Study the letter and answer the following questions:
- •1. Read the names of the schools. Give Ukrainian equivalents:
- •2. Read the names of the departments. Give Ukrainian equivalents:
- •3. Look through the email and answer the following questions:
- •History
- •Undergraduate Academics
- •Active Vocabulary
- •2. Are these statements true or false? If they are false, say why. Use the following phrases:
- •3. Study the letter and answer the following questions:
- •Grammar and Vocabulary Exercises
- •1. Find English equivalents to the following words and word combinations in the texts of the unit:
- •2. Translate the following words and word combinations from English into Ukrainian and use them in the sentences of your own:
- •3. Find in the texts synonyms for the following words and expressions and use them in the sentences of your own:
- •4. Complete the following sentences in the context of the above information:
- •6. Put the verb in brackets in the correct tense form:
- •8. Make the sentences from Ex. 6 negative. Conversational Practice
- •1. Learn the following expressions relating to the communication of opinions. Translate them into Ukrainian.
- •2. Discuss the following questions in the context of the topics of Unit 1, using as many of the above expressions as possible. Compare Ukraine and the usa.
- •3. Ask your friend the following questions, present the results to the whole group.
- •4. Translate the following words and word-combinations:
- •5. Interview Maksym in English. Find out what he knows about the faculty he studies at:
- •Writing
- •Extended reading
- •Grammar Revision Словотвір в текстах функціонального стилю науки
- •Основні префікси та їх значення
- •Основні суфікси іменників
- •Основні суфікси прикметників
- •Основні суфікси дієслів
- •Основні суфікси прислівників
- •Конверсія
- •Словоскладання
- •2. Learn to recognize the following international words and give their Ukrainian equivalents:
- •What is an Electronic Computer?
- •Grammar and Vocabulary Exercises
- •1. Study the Table of word-building means given in Grammar Revision.
- •3. Form the words after the model and translate them into Ukrainian:
- •9. For each definition write a word from the text:
- •Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •Conversational Practice
- •1. Suppose that the information in the statement is insufficient. Repeat the statement and add your own reasoning, thus developing the idea. Use the following phrases:
- •Computers
- •2. Answer the following questions:
- •3. Reconstruct the text “Computers” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •Writing
- •Extended reading
- •The Internet Computer.
- •Grammar Revision Passive Voice
- •Modals with the Passive Voice
- •2. Learn to recognize international words:
- •Hardware – Software – Firmware
- •Grammar and Vocabulary Exercises
- •11. Combine the words from the left-and right-hand columns to make word-combinations. Translate them into Ukrainian:
- •12. Compose sentences with the words and phrases from Ex. 11.
- •13. Write an appropriate word or phrase in the following spaces:
- •Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •Conversational Practice
- •1. Agree with the following statements, adding your own comments. Use the introductory phrases:
- •2. Suppose that the information in the statement is insufficient. Repeat the statement and add your own reasoning, thus developing the idea. Use the following phrases:
- •3. Express your personal view on the statement given below. Use the following phrases:
- •4. Give a short summary of the text.
- •1. Read and translate the text: Computer Crime
- •2. Answer the following questions:
- •3. Reconstruct the text “Computer Crime” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •Writing
- •Extended reading
- •Boolean Algebra
- •Grammar Revision Modal Verbs
- •2. Learn to recognize international words:
- •Artificial Intelligence. Is it Possible?
- •Grammar and Vocabulary Exercises
- •1. Look through the text and find sentences with Modal Verbs. Translate them into Ukrainian.
- •3. Choose the proper equivalents of the Modal Verbs:
- •4. Underline the affixes, state what part of speech they indicate and translate the following words into Ukrainian:
- •5. Give the Ukrainian equivalents of the following words and word-combinations:
- •6. Use the words from Ex. 5 to complete the following sentences:
- •Reading Comprehension
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Study the text and answer the following questions:
- •Conversational Practice
- •1. Agree or disagree with the statements given below. Use the introductory phrases and develop the idea further. Use the following phrases:
- •2. Choose the definition of artificial intelligence which, to your mind, is the correct one. Justify your choice:
- •3. Debate the given statement. It is advisable that the group be divided into two parties, each party advocating their viewpoint. Use the following introductory phrases:
- •4. Give a short summary of the text.
- •1. Read and translate the text:
- •2. Answer the following questions:
- •3. Reconstruct the text “Turing’s test” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •5. Discuss the problems. The following phrases may be helpful:
- •6. Summarize the text briefly. Writing
- •Extended reading
- •To be One with the Computer
- •Grammar Revision Sequence of Tenses
- •2. Learn to recognize international words:
- •The World of Hypotheses. Was Einstein Right?
- •Active Vocabulary
- •Grammar and Vocabulary Exercises
- •1. Look through the text and find Complex Sentences. Translate them into Ukrainian.
- •2. Join the two simple sentences to make a complex sentence. Mind the sequence of tenses rule:
- •3. Turn the following statements into indirect speech:
- •4. Define meanings of the following words by their affixes:
- •5. Study the text and give Ukrainian equivalents for the following words and word-combinations:
- •Reading Comprehension
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •Conversational Practice
- •1. Agree or disagree with the following statements. Begin your answer with the following phrases:
- •2. Discuss the statements trying to prove your point of view. Use the following phrases:
- •3. Give a short summary of the text.
- •1. Read and translate the text: Gravitation
- •2. Answer the following questions:
- •3. Reconstruct the text “Gravitation” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •Writing
- •Extended reading
- •Grammar Revision The Infinitive /інфінітив/
- •1) Як іменник інфінітив може бути:
- •2) Як дієслово інфінітив може
- •Форми інфінітива та їх комунікативні значення.
- •The Infinitive Constructions Інфінітивні звороти та їх функції у реченні. Складний додаток /Complex Object/
- •Складний підмет /Complex Subject/
- •Прийменниковий інфінітивний комплекс (The for-to-Infinitive –Construction)
- •Функції прийменникового інфінітивного комплексу
- •2. Learn to recognize international words:
- •The Theory of Equations
- •Grammar and Vocabulary Exercises
- •1. Look through the text and find sentences with the Infinitive and the Infinitive Constructions. Translate them into Ukrainian.
- •2. Translate the following sentences into Ukrainian:
- •7. Look through the text and give English equivalents for the following words and word-combinations:
- •8. Combine the words from the left-and right-hand columns to make word-combinations. Translate them into Ukrainian:
- •9. Compose sentences in English using the word-combinations from Ex. 8. Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •Conversational Practice
- •1. Choose one of the words given below and illustrate the concept:
- •2. Discuss the statements given below. Summarize the discussion. Use the following phrases:
- •3. Give a short summary of the text.
- •1. Read and translate the text: The Early Algebra Babylonian Algebra – Rhetorical Style
- •Algebra in Egypt
- •Early Greek Algebra
- •Hindu and Arabic Algebra
- •Algebra in Europe
- •2. Answer the following questions:
- •3. Reconstruct the text “The early Algebra” into a dialogue.
- •4. Agree with the statements given below and develop the idea further. Use the introductory phrases:
- •5. Annotate the text in English. Use the phrases:
- •6. Discuss the statements given below. Use the following phrases:
- •Writing
- •Extended reading
- •Grammar Revision
- •Утворення дієприкметників.
- •Функції Participle I, II в реченні
- •Дієприкметникові звороти
- •Складний додаток /Complex Object/
- •Складний підмет /Complex Subject/
- •Незалежний дієприкметниковий зворот (The Absolute Participial Construction)
- •Способи перекладу “незалежного дієприкметникового зворота” на українську мову.
- •2. Learn to recognize international words:
- •Informatics
- •Active Vocabulary
- •Grammar and Vocabulary Exercises
- •1. Study the text and find sentences with the Participle. Translate them into Ukrainian.
- •2. Translate the following sentences into Ukrainian:
- •3. Observe the time of occurrence, expressed by a Participle:
- •10. Compose sentences with the words and word-combinations from Ex.9. Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •4. Give the definitions of the terms “information” and “informatics”:
- •Conversational Practice
- •1. Agree or disagree with the statement given below. Use the introductory phrases and develop the idea further. Use the following phrases:
- •2. Discuss the following statement. Use the given phrases:
- •3. Give a short summary of the text.
- •1. Read and translate the text:
- •2. Answer the following questions:
- •3. Reconstruct the text “Cybernetics” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •5. Discuss the statements given below:
- •Writing
- •Extended reading
- •Grammar Revision.
- •1) Як іменник герундій може:
- •2) Як дієслово герундій (перехідного дієслова)2 може:
- •Форми герундія та їх комунікативні значення. Форми герундія неперехідного дієслова
- •Форми герундія перехідного дієслова
- •Порівняйте:
- •2. Learn to recognize international words:
- •Mystery of Memory
- •Active Vocabulary
- •Grammar and Vocabulary Exercises
- •1. Study the text, and find sentences with the Gerund. Translate them into Ukrainian.
- •2. Translate the following sentences into Ukrainian:
- •8. Combine the words from the left-and right-hand columns to make word-combinations. Translate them into Ukrainian:
- •9. Compose sentences in English using the word-combinations from Ex. 8. Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •Conversational Practice
- •5. Give a short summary of the text.
- •1. Read and translate the text: The Memory of the Modern Supercomputers
- •Active Vocabulary
- •2. Answer the following questions:
- •3. Reconstruct the text “The Memory of the Modern Supercomputers” into a dialogue.
- •4. Annotate the text in English. Use the phrases:
- •5. Discuss the problems trying to prove your point of view. Use the following phrases:
- •Writing
- •Extended reading
- •The Brain
- •2. Learn to recognize international words:
- •Math Concepts
- •Grammar and Vocabulary Exercises
- •1. Grammar revision.
- •1. Imperative Sentences.
- •2. Indefinite Tense-Aspect Forms.
- •3. Questions.
- •4. Negations.
- •8. Compose sentences with the words and word-combinations from Ex.7. Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following question:
- •4. Give the definitions of the terms “the real number system” and “the maths of number”:
- •Conversational Practice
- •1. Disagree with the following negative statements and keep the conversation going where possible. Begin your answer with the opening phrases:
- •2. Agree or disagree with the statements. Use the introductory phrases:
- •4. Practise problem questions and answers. Work in pairs. Change over!
- •5. What is implied in the following assertion?
- •6. Discuss the statements given below. Use the following phrases:
- •7. Give a short summary of the text.
- •1. Read and translate the text: Programming. Multiprogramming
- •2. Answer the following questions:
- •3. Reconstruct the text into a dialogue. The main rules governing a conversation in English:
- •4. Annotate the text in English. Use the phrases:
- •5. Read the statements and develop the idea further. Use the given phrases:
- •Writing
- •Extended reading
- •The Internet Programming Languages
- •2. Learn to recognize international words:
- •Automated Factory Update
- •Active Vocabulary
- •Vocabulary Exercises
- •1. Look through the text and give Ukrainian equivalents of the following words and word-combinations:
- •2. Look through the text and give Ukrainian equivalents of the following words and word-combination:
- •3. Look through the text and find words with the same meaning:
- •4. Look through the text and find words with opposite meaning:
- •5. Combine the words from the left- and right-hand columns to make word-combinations. Translate them into Ukrainian:
- •6. Compose sentences in English using the word combinations from Ex.5. Reading Comprehension
- •1. Review the whole text again. Outline the subject matter of the text, its components structure, topic sentences and main ideas. Use the following phrases:
- •2. Say whether the following statements are true or false. Justify your choice. Use the given phrases:
- •3. Answer the following questions:
- •4. Give the terms to the following definitions:
- •Conversational Practice
- •1. Clarify what we mean by the following statements:
- •2. Discuss the advantages of cim comparing traditional manufacturing and computer-integrated manufacturing. The schemes given below will be helpful.
- •3. Debate the given problem. It is advisable that the group be divided into two parties, each party advocating their viewpoint. Use the following introductory phrases:
- •4. Give a short summary of the text.
- •1. Read and translate the text:
- •Active Vocabulary
- •2. Answer the following questions:
- •3. Reconstruct the text “Planning and Justifying Factory Automation Systems” into a dialogue. The main rules governing a conversation in English:
- •4. Annotate the text in English. Use the phrases:
- •5. Express your personal view on the statement “Integrated problems require integrated solutions”. Use the following phrases:
- •Writing
- •1. Using texts a and b of Unit 10 write a composition on “My future profession”. Take into account the following outlines or give your own version.
- •Extended reading
- •Control Engineering
- •Control Engineering Practice
- •2. "Can a computer have a mind?" Provide answers to this question, discussing it with Internet community. Consult Roger Penrose's Penguin book "The Emperor's New Mind", if necessary.
- •1. Using Internet try to find out all you can about the land of Tor'Bled-Nam.
- •2. What is the very essence of mathematical visualization? Key-words: magnification, abstract mathematics, complex numbers, miracles of mathematics.
- •1. Try to find additional information about black and white holes.
- •2. Find out the names dealt with these problems/ approaches/ theories/ hypotheses.
- •3. Present your ideas on the given subject for the students' research society.
- •1. Find the film "Time Travel".
- •2. Discuss it at the students' on line conference.
- •Appendix II
- •1. Study all the texts, collect information and write two-pages-long compositions on each of the following topics:
- •2. The following questions may direct you:
- •Number Theory and its Founders
- •Pierre de Fermat
- •Leonhard Euler
- •Georg Friedrich Bernhard Riemann
- •Active Vocabulary
- •The Greek Genius
- •Natural Numbers
- •Real Numbers
- •Toward Mathematical Structure
- •Structures
- •Appendix III greek alphabet
- •Wording Mathematical Formulae
- •Giving an Oral Presentation
- •Список рекомендованої літератури
- •Contents
Список рекомендованої літератури
1. Дорожкина ВП. Английский язык для студентов-математиков: Учебник./ В.П. Дорожкина; Под общ. ред. и с предисл. В.А. Скворцова. – 3-е изд., перераб. и доп. – М.: ООО «Издательство АСТ»: «Издательство АСТ», 2001. – 496 с.
2. Дорожкина В.П. Английский язык для студентов-математиков и экономистов: Учебник/ В.П. Дорожкина; Под ред. В.А. Скворцова. – 2-е изд., перераб. и доп. – М.: ООО «Издательство Астрель»: «Издательство АСТ», 2004. – 349, [3] с.
3. Зейденберг В.К. и др. Англо-русский словарь по вычислетельной технике: Ок. 42 000 терминов / Под редакцией Е.К. Масловского. – М.: Рус. яз., 1990. – 800 с.
4. Інтерфейс. Посібник для читання англійською мовою для студентів вищих начальних закладів. Укладач – Мартинюк В.А. – Чернівці, 2006. – 176 с.
5. Комп’ютерний словник / Пер. з англ.. В.О. Соловйова. – К.: Україна, 1997. – 470 с.:іл.
6. Лисиця Н.М. та ін. Самостійне читання з англійської мови: (Гнуч. вироб. системи) : [Навч. посібник для немов. спец. вузів] / Н.М. Лисиця, С.А. Мехович, Л.П. Шевченко. – Х.: Основа, 1992. – 96 с.
7. Мюллер В.К. Великий англо-український словник. У новій редакції. 200 000 слів, словосполучень, ідіоматичних виразів, прислів’їв та приказок. – К.: Арій, 2008. – 784 с.
8. Плюхина З.А. Англичане говорят так: Учеб. пособие. – М.: Высш. шк., 1991. – 175 с.
9. Современный англо-русский политехнический словарь / Составителя В.В. Бутник. – М.: Вече, 2006. – 512 с.
10. Шахова Н.И. Курс английского языка для аспирантов: Учеб. пособие. – М.: Издательство «Наука», 1980. – 296 с.
11. Ятель Г.П. та ін. Англійська мова (поглиблений курс) для студентів технічних вузів. Підручник/ Г.П. Ятель, Б.М. Князевський, Ф.К. Кузик; За заг. Ред.. Г.П. Ятеля. – К.: Вища шк., 1995. – 254 с.
12. Dorf, Richard C. Modern control systems / Richard C. Dorf. – 5th ed. p. cm. – (Addison – Wesley series in electrical engineering). – 1990. – 603 p.
13. Penrose, Roger. The emperor’s new mind: concerning computers, minds, and the laws of physics / Roger Penrose; foreword by Martin Gardner. Printed in the United States of America, 1991. – 466 p.
14. New Webster’s dictionary and thesaurus of the English Language. Lexicon Publications, Inc. Sherman Turnpike Danbury, CT 06816. Printed in the United States of America, 1993. – 1350 p.
15. http://en.wikipedia org/wiki/Massachussets Institute of Technology
16. http://web. mit.edu
17. http://math.mit.edu/research/applied
18. MIT.edu
Contents
Вступ….....……………………………………………………….. |
3 |
Unit 1 |
4 |
1. Let’s get acquainted |
|
2. National University of Water Management and Natural Resources Use |
|
3. Massachusetts Institute of Technology |
|
4. Applied Mathematics Research |
|
5. Traditions and Student Activities |
|
6. English tense-forms |
|
Unit 2 |
39 |
1. What is an Electronic Computer? |
|
2. Computers. |
|
3. The Internet Computer. |
|
4. English Word Building. |
|
Unit 3 |
58 |
1. Hardware – Software – Firmware. |
|
2. Computer Crime. |
|
3. Boolean Algebra. |
|
4. Passive Voice. |
|
Unit 4 |
74 |
1. Artificial Intelligence. It is Possible? |
|
2. Turing’s Test. |
|
3. To be One with the Computer. |
|
4. Modal Verbs. |
|
Unit 5 |
92 |
1. The World of Hypotheses. Was Einstein Right? |
|
2. Gravitation. |
|
3. Revisiting the Big Bang. |
|
4. Sequence of Tenses. |
|
Unit 6 |
108 |
1. The Theory of Equations. |
|
2. The Early Algebra. |
|
3. Solution of Polynomial Equations of Third and Higher Degree. |
|
4. The Infinitive. The Infinitive Constructions. |
|
Unit 7 |
133 |
1. Informatics. |
|
2. Cybernetics. |
|
3. Algorithms. |
|
4. The Participle. The Participial Constructions. |
|
Unit 8 |
155 |
1. The Mystery of Memory. |
|
2. The Memory of the Modern Supercomputers. |
|
3. The Brain. |
|
4. The Gerund. Gerundial Construction. |
|
Unit 9 |
172 |
1. Math Concepts. |
|
2. Programming. Multiprogramming. |
|
3. The Internet Programming Languages. |
|
4.Grammar Revision. |
|
Unit 10 |
191 |
1. Automated Factory Update. |
|
2. Planning and Justifying Factory Automation. |
|
3. Control Engineering. |
|
4. Modern Control Systems. |
|
Professionally oriented texts for individual studying and the development of communicative language competences |
212 |
Appendix I |
212 |
Text 1. Strong AI and Searle's Chinese Room |
212 |
Text 2. The Insolubility of Hilbert's Problem |
220 |
Text 3. Construction of the Mandelbrot Set |
228 |
Text 4. Cosmology and the Big Bang |
231 |
Text 5. Black Holes |
236 |
Text 6. Hawking's box: a link with the Weyl curvature hypothesis? |
243 |
Appendix II |
252 |
Number Theory and its Founders |
252 |
Appendix III |
277 |
Greek Alphabet |
277 |
Nouns of Latin and Greek Origin |
277 |
Mathematical Symbols and Signs |
279 |
Wording Mathematical Formulae |
282 |
Giving an Oral Presentation |
285 |
Список рекомендованої літератури |
289 |
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІ). Рівне: РДТУ, 1999. – С. 17-27.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІ). Рівне: РДТУ, 1999. – С. 28-35.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІ). Рівне: РДТУ, 1999. – С. 4-10.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІІ). Рівне: РДТУ, 1999. – С. 18-20.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІІ). Рівне: РДТУ, 1999. – С. 30-31.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів 1х курсів усіх спеціальностей РДТУ (Частина ІІІ). Рівне: РДТУ, 1999. – С. 26-29.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів IIх курсів усіх спеціальностей РДТУ (Частина ІV). Рівне: РДТУ, 1999. – С. 5-18.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів IIх курсів усіх спеціальностей РДТУ (Частина ІV). Рівне: РДТУ, 1999. – С. 35-45.
1 “Складний додаток” з Participle дуже близький за значенням до “Складного додатка” з Infinitive (після дієслів, що означають сприйняття за допомогою органів чуттів). У першому випадку дія виражається як процес, а в другому- констатується факт.
1 Навчальні завдання з розвитку граматичних навичок з англійської мови (теорія, тренувальні вправи, міні та рейтингові тести) для студентів IIх курсів усіх спеціальностей РДТУ (Частина ІV). Рівне: РДТУ, 1999. – С. 24-27.
2 Перехідні дієслова мають прямий додаток (to write a letter), а неперехідні не мають (to go).
1 Recall that by the natural numbers we mean 0,1, 2, 3,4, 5, 6,. . . The reason for the ‘ ’ and ‘ ’, etc., rather than the more familiar form ( ; x, y, z > 0, w > 2) of the Fermat assertion, is that we are allowing all natural numbers for , , etc., starting with zero.
1 Recall that the prime numbers 2, 3, 5, 7, 11, 13, 17, . . are those natural numbers divisible, separately, only by themselves and by unity. Neither 0 nor 1 is considered to be a prime.
1 This is the familiar – and powerful – mathematical procedure known as reductio ad absurdum, whereby one first assumes that what one is trying to prove is false; and from that one derives a contradiction, thus establishing that the required result is actually true.
1 At the present time there is a dispute still raging as to the value of this figure, which ranges between about and years. These figures are considerably larger than the figure of years that originally seemed appropriate after Edwin Hubble's initial observations in around 1930 showed that the universe is expanding.
1 Einstein introduced the cosmological constant in 1917, but had retracted it again by 1931, referring to its earlier introduction as his ‘greatest mistake’.
1 In fact, in its final stages, the dwarf will glow dimly as a red star – but what is referred to as a 'red dwarf is a star of quite a different character!
1 This is a sample specification for one type of presentation. Other types, e.g. product presentation, marketing presentation, etc., would need a different specification.