Mathematics − the language of science

1. 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.

2. 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.

3. It is generally true that algebra in its development has passed successively through three stages: 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.

4. 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.

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

6. Within half a century of Viete'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 Vietan one, but has survived to this day.

7. 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.

8. It is convenient because the literal notation is free from ail 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.

9. 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.

9. Look through the list of words and phrases and check if you know their Ukrainian equivalents. Use the Mini-Dictionary (UNIT 7) if necessary.

   precise and concise statements	shorthand
   to conceal the meaning	essay
   spoken language	vowels and consonants
   the sum is independent of the order of the terms	to deal with
   in this capacity	literal notation
   to permeate	calculus symbol
   to pervade all sciences	to grasp
   abbreviated	to be due to
   verbal	insoluble by other methods
   to denote	ambiguity
   duly	susceptible
   Dy (derivative of y)	to superscribe

10. Explain the meaning of the words and phrases.

common language, universal, a similar metamorphosis, abbreviated algebra, ambiguity, the given quantities, literal notation, to permeate, magnitudes

11. Cross the odd word out.

1. abase, abbreviate, shorten,compress;

2. concise, compact, brief, conclusive;

3. equivocacy, ambivalence, alternative, ambiguity;

4. permeate, perpetrate, spread through, pass through;

5. metaphor, metamorphosis, transfiguration, conversion.

12. Match the words with their definitions.

    1	quantity	a	to spread through or throughout
    2	universal	b	a voiced speech sound whose articulation is characterized by the absence of friction-causing obstruction in the vocal tract, allowing the breath stream free passage
    3	pervade	c	something that serves as the object of an operation.
    4	vowels	d	a speech sound produced by a partial or complete obstruction of the air stream by any of various constrictions of the speech organs, such as (p), (f), (r), (w), and (h)
    5	consonants	f	to indicate or specify, point out
    6	designate	g	applicable or common to all purposes, conditions, or situations

13. Fill in the word from the list below. Use each word only once. Translate the phrases into Ukrainian.

magnitudes, gradually, pervade, use, designed, insoluble, ideas, symbolic, ambiguities, literal

1	...... by other methods	6	the ...... notation
2	the unknown .......	7	to ...... all sciences
3	words of frequent .....	8	mathematical in its .....
4	the ..... sense	9	to be ..... abbreviated
5	carefully and purposefully .....	10	to be free from all .....

14. Find the words in the texts to which the following ones are the antonyms. The first letter is given to make the task easier.

vowels (c), definite (a), occasional (f), minus (p), known (u), verbal (m), presence (a), dependent (i), different (s)

15. Pick up from the text “Mathematics − the language of science” all the adjectives to the following words.

........... algebra	......... statements
.......... language	........... sense
............ example	........... notation
............ symbol	............ mathematics

16. Match the words and phrases with their Ukrainian equivalents.

    1	a similar metamorphosis	a	набирати вигляду
    2	vowels and consonants	b	продукт суспільного розвитку
    3	to take the form (of)	c	цілеспрямовано розроблений
    4	a product of social development	d	аналогічні метаморфози
    5	a powerful technique	e	невідомі величини
    6	purposefully designed	f	голосні та приголосні
    7	unknown magnitudes	g	могутній спосіб (метод)

17. Fill the cells in the table with the words derived from the given nouns.

    Noun	Adjective
    symbol
    mathematics
    precision
    concision
    science
    verb
    frequency
    similarity
    convenience
    power
    effect

18. Make up adverbs adding “-ly” to the given words. Translate these words into Ukrainian.

careful, purposefil, practical, mathematical, general, successive, verbal, independent, symbolic, typical, certain, frequent, effective, convenient, literal, powerful, basical, qualitative, physical, social

19. Fill the gaps in the text about mathematics with the missing word(s) from the list below.

massive systems; operations; lasting understanding; mundane; measurements; discoveries; records; galaxies; quadratic equations; calculations; complex theories; real-world applications; universe; predictions; abstract; real-world application

Mathematics is the science of numbers and how they interrelate. It is how we perform 1) ……with numbers to make 2) ….... We use mathematics in everything from 3) …… activities like balancing a checkbook to theorizing how 4) …… such as entire 5) ….... move through the cosmos. And mathematics can be 6) …… − pure mathematics is a pursuit without a goal for application in the real world, though the results of 7) ..….. in pure mathematics can lead to 8) ….....

The concept of numbers dates further back than writing. We have 9) …..... of ancient people recording numbers in various ways dating back to 30,000 B.C. By 1950 B.C., humans were inventing and solving 10) …….. Ancient Babylonians developed 11) …...... for geometry and algebra. These weren't just philosophies − ancient civilizations used mathematics the way we do today to explain the way things work.

Without mathematics, it would be virtually impossible for us to build a deep, 12) …..... of the 13) ...….. Mathematics allows us to take 14) ..…. and make 15) ...…. based upon observations.

20. Fill in the correct word derived from the word in bold.

Like Thales, Pythagoras is rather known for 1) _______ (mathematical) than for 2) _________ (philosophical). Anyone who can recall math classes will remember the first lessons of plane 3) _______ (geometrically) that usually start with the Pythagorean theorem about right-angled 4) ______ (triangular): a²+b²=c². In spite of its name, the Pythagorean theorem was not discovered by Pythagoras. The earliest known 5) _______ (formulate) of the theorem was written down by the Indian 6) _____ (mathematics) Baudhāyana in 800BC. The principle was also known to the earlier 7) _______ (Egypt) and the Babylonian master builders. However, Pythagoras may have proved the theorem and 8) ________ (popularization) it in the Greek world. With it, his name and his philosophy have survived the 9) _________ (turbulent) of history.

21. Match 1-8 with a-h to make sentences.

    1	Maths can solve	a	signs and symbols.
    2	Maths developed from	b	the civilized world.
    3	Where there is no difficultyin maths there is no	c	abstract quantities.
    4	Algebra consists of	d	problem.
    4	The language of maths is one and the same	e	counting, calculating, measurement.
    6	Mathematicians deal with	f	a lot of problems.
    7	Prehistoric people knew how to count	g	abstract models.
    8	Mathematics is concerned	h	beautibul theories.

22. Work in small groups. Arrange the following words and phrases in the correct order to make the sentences. The first word is underlined.

1. "key / the sciences / Mathematics / to / the door and / is." Roger Bacon

2. "rules / on / certain simple / Mathematics / meaningless marks / played / is / game / according / a / with / paper / to." David Hilbert

3. “Uglification, and Derision, / Arithmetic / The / of / different / are / Ambition, Distraction / branches.” Lewis Caroll

23. Learn the wording of some mathematical equations.

1. 

"Two plus x plus the square root of four plus x squared is equal to ten."

2. 

“M” is equal to “R” sub one multiplied by “x” minus “P” sub one, round brackets opened, “x” minus “a” sub one, round brackets closed, minus “P” sub two, round brackets opened, “x” minus “a” sub two, round brackets closed."

24. Do the wording of the following:

• numerals: 30; 13; 2000; 101,101; 12,020,987;

• fractions: 2.5 ; 3 ; 10.4 ; 0.7 ; 2;

• mathematical expressions: ; ; ; ;

• dates: Dec. 5 ; Jan. 1, 2012 ; 1990 ; 1905 ; Sept. 3, 2000.

25. Form the ordinal numerals from the cardinal ones: 22; 203; 44; 100; 1001; 100000.

26. Give examples of equations or formulae that you had to solve at your lessons of mathematics.

27. Find and correct mistakes.

1. My birthday is twenty – seven March.

2. My phone number is naught sixty-seven, double two, three, eighty-nine, twenty-four.

3. The population of Kyiv is about three millions.

4. I got twenty-five from forty in my test.

5. It was a second typical example of the rule.

6. We have done at least 70 per cents of the work.

7. Seven, five, three are even numbers.

28. What will probably these people do in an hour?

a)

b)

c)

d)

29. This time you have to write three sentences about the past, present and future. Alex always solves mathematical problems in the morning. It always takes him an hour, from 8.30 until 9.30 a.m. So:

1. At 9 o'clock yesterday morning Alex ...............

2. It's 9 o'clock now. He ...........................

3. At 9 o'clock tomorrow morning he ...............................

30. You want to ask your friend to do something for you. Use the prompts below to make questions, as in the example. Use the Future Continuous.

1. You want your friend to solve this mathematical problem.

Will you be solving this mathematical problem?

2. You want your friend to explain the difference between the words "equation" and "expression".

3. You want your friend to translate this sentence.

4. You want your friend to prove the theorem.

31. Choose the correct answer.

1. By the end of May, Tom will have studied/will have been studying English for a year.

2. By June, Kate will have been finishing/will have finished studying at the University.

3. Hopefully, they will have learned/will have been learning everything by the time they sit the exam.

4. By 7 o'clock, I will have been studying/will have studied mathematics for three hours.

5. By Monday, we will have written/will have been writing an essay about mathematics.

32. Put the verbs into the correct tense.

1. The people in each country ........................ (to translate) algebra into their own spoken language.

2. Algebra .................... (to pass) three stages in its development.

3. Common language .................. (to be) a product of social development.

4. The given magnitudes ....................... (to design) by consonants.

5. Power of transformation ....................... (to lift) algebra above the level of a convenient shorthand.

6. The history of symbols “+” and “−“ ...................... (to illustrate) the point.

33. Put articles where necessary.

1. … common language is … product of … social development.

2. … algebra, … language of … mathematics, consists mostly of … signs and symbols.

3. … algebra is one and … same throughout … civilized world.

4. … algebra in … broad sense of… term, deals with … operations upon … symbolic forms.

5. … Cartesian notation not only displace … Vietan one, but has survived to this day.

6. It is … symbolic language that is one of … basic characteristics of … modern mathematics.

34. Translate into English.

1. Алгебра – це точна, стисла та універсальна наука.

2. Самі слова використовують у їх символічному змісті.

3. У середньовічній Європі “мінус” та “плюс” позначалися повними словами.

4. Скорочення перетворилося у символ.

5. В своєму розвитку алгебра пройшла декілька ступенів.

6. Сучасна алгебра об’єднує велику кількість самостійних дисциплін.

7. Метод аналізу математичних моделей посідає провідне місце серед інших методів дослідження.