Unit 4

Grammar:

1. Perfect Tenses (Active, Passive)

2. Numerals

3. Cleft sentences: it is (was)…that (who)

Reading Exercises

1. Practise reading the two – syllable words:

concise, because, conceal, consist, design, translate, pre­cise, upon, pervade, include, belief, become, idea, com­plete, themselves, denote, itself, appear, displace, notation, survive, supply, extreme, permit, consider, involve, effect, subject, attach, transform.

2. Read observing the correct pronunciation of vowels in stressed and unstressed syllables:

language, algebra, alphabet, practical, capable, par­ticular, common, consonant, level, general, even, latter, ex­cept, develop.

3. Read the following many-syllable words:

influence, reasoning, capacity, typical, gradual, even­tual, superscript, century, magnitude, consonant, alphabet, geometry, quantity, difficult, literal, ambiguity, paraphrase, equivalent.

4. Memorize the spelling and pronunciation of the following words: permeate ['pq:mIeIt] - проникати, throughout[Tru:'aut] - повсюди, pervade [рq:'veid] - наповнювати, пронизувати, though ['Dou[- хоча, certain ['sq:tn[- певний, enough [I'nAf[– достатньо, доволі, medieval ["medi'Jvql[- середньовічний, technique [tek'ni:k]- техніка, метод, ambiguity ["xmbI'gjHItI[- двозначність, невизначеність, metamorphosis ["metq'mLfqsIs[ - метаморфоза.

1. Read observing the correct pronunciation of the - ed ending:

abbreviated, illustrated, denoted, designated, calculated, generalized, symbolized, civilized, concealed, enabled, sup­plied, involved, solved, passed, expressed, developed, dis­placed.

2. Read paying attention to the shift of stress in the following numerals:

thirteen — thirteen years thirteen — thirty

fourteen — fourteen languages fourteen — forty

fifteen — fifteen students fifteen — fifty

sixteen — sixteen centuries sixteen — sixty

seventeen — seventeen words seventeen — seventy

eighteen — eighteen sentences eighteen — eighty

nineteen — nineteen countries nineteen — ninety.

Text 4 a mathematics — the language of science

Human language is capable of precise statements because it is a system of symbols. But common language is a product of social development, customs and traditions. Even by the most careful choice of words the meaning concealed in them may influence our reasoning. Algebra, the language of mathe­matics, consists mostly of signs and symbols and is carefully and purposefully designed. It is precise, concise and univer­sal, i. e. one and the same throughout the civilized world, though the people in each country translate it into their own spoken language.

Algebra in the broad sense of the term, deals with oper­ations upon symbolic forms.

In this capacity it not only per­meates all of mathematics, but pervades practically all

sciences including formal logic, philosophy, and even lin­guistics, poetry and music.

In our scientific age there is a general belief that all science, as it grows to perfection, be­comes mathematical in its ideas.

It is generally true that in the development of algebra three stages have been passed successively: verbal, abbrevia­ted and symbolic. Verbal algebra is characterized by the com­plete absence of any symbols, except, of course, that the words themselves are used in their symbolic sense. To this day verbal algebra is used in such a statement as "the sum is independent of the order of the terms", which in symbols is designated by a + b = b + a.

Abbreviated algebra of which the Egyptian is a typical example, is a further development of verbal one. Certain words of frequent use are gradually abbreviated. The history of the symbols "+" and "-" may illustrate the point. In medieval Europe the latter was denoted by the full word "minus", then by the first letter "m" duly superscribed. Eventually the let­ter itself was dropped, leaving the superscript only. The sign "plus" has passed through a similar metamorphosis. The ab­breviation has become a symbol.

The turning point in the history of algebra was an essay written late in the sixteenth century by a Frenchman; it was Viète who denoted the unknown magnitudes by vowels. The given magnitudes were designated by consonants.

Within half a century of Viète's death there appeared Descartes's Geometry. In it, the first letters of the alphabet were used for the given quantities, the last - for those unknown.

The Cartesian notation not only displaced the Viètan one, but has survived to this day.

It is symbols that permit of concise, clear representation of ideas which are sometimes quite complex. Consider, for example, how much is involved in the calculus symbol "Dy". Once we have grasped the meaning and use of a symbol there is no need to think through the origin and development of the idea symbolized, each time it is used. It is due to a powerful technique based upon the use of symbols that mathe­matics is so effective in problems which are insoluble by other methods.

It is convenient because the literal notation is free from all ambiguities of words. The letter is susceptible of oper­ations and this enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is this power of transformation that lifts algebra above the level of a convenient shorthand.

It is symbolic language that is one of the basic charac­teristics of modern mathematics. And modern mathematics supplies a language for the treatment of the qualitative prob­lems of physical and social sciences.

NOTES

1. ...is capable of (precise) statement - здатний (точно) передавати висловлювання

2. a product of social development - продукт суспільного розвитку

3. throughout the (civilized) world - у всьому (цивілізованому) світі.

4. spoken language - розмовна мова

5. in this capacity - в цій якості

6. calculus symbol - символ обчислення

7. Dy (derivative of у) - похідна від у

8. to superscribe - робити напис зверху

9. late in the sixteenth century - наприкінці XVI століття

10. to think through - додумувати до кінця, проникати в суть справи

11. a powerful technique - могутній спосіб (метод), засіб

12. literal notation — буквене позначення, буквений запис

13. susceptible (of) — який (що) допускає, піддається чому-н,

14. to take the form (of) — набирати вигляду