Основные математические понятия в английском языке: Методические указания

МИНИСТЕРСТВО ОБРАЗОВАНИЯ РОССИЙСКОЙ ФЕДЕРАЦИИ Государственное образовательное учреждение высшего профессионального образования "Оренбургский государственный университет" Кафедра английского языка гуманитарных и социальноэкономических специальностей

И.И. ПРОКОШЕВА

ОСНОВНЫЕ МАТЕМАТИЧЕСКИЕ ПОНЯТИЯ В АНГЛИЙСКОМ ЯЗЫКЕ.

МЕТОДИЧЕСКИЕ УКАЗАНИЯ

Рекомендовано к изданию Редакционно-издательским советом Государственного образовательного учреждения высшего профессионального образования "Оренбургский государственный университет"

Оренбург 2003

ББК 81.2 Англ. я 73 П - 80 УДК [802.0:51] 075

Рецензент кандидат филологических наук, доцент Т.В. Минакова

П 80

Прокошева И.И. Основные математические понятия в английском языке: Методические указания. - Оренбург: ГОУ ВПО ОГУ, 2003. - 50с.

Методические указания рекомендуются для использования на практических занятиях по английскому языку для студентов физикоматематического факультета. Цель методических указаний - научить студентов читать, понимать и употреблять основные математические понятия в англоязычной оригинальной литературе по специальности. ББК 81.2 Англ. я 73

© Прокошева И.И., 2003 © ГОУ ВПО ОГУ, 2003

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Введение Методические указания «Основные математические понятия в английском языке» предназначены для студентов физико-математического факультета. Основная цель методических указаний – познакомить студентов с математическими терминами и выражениями, принятыми в англоязычной научно-технической литературе. Методические указания представляют собой практическое руководство, в котором излагаются наиболее важные математические понятия и термины. Выбор материала обосновывается необходимостью дать представление основ понятийно-категориального аппарата студентам при их работе с профессионально-ориентированными текстами на иностранном языке. Работа состоит из 11 разделов. Каждый раздел включает в себя активную лексику, таблицу с основными понятиями и выражениями по разделам математики и практические упражнения на развитие умений и навыков чтения (перевода) текстов по специальности. В конце указаний приведены сводные упражнения с ключами к ним. К пособию прилагается справочный и вспомогательный материал, состоящий из практических заданий, задач – головоломок, латинского и греческого алфавита, глоссария, словаря математических выражений и понятий, сокращений, используемые в математике, а также ключей для тестовых заданий. С целью использования данных методических указаний при подготовке научных докладов и выступлений в них приведен список литературы и адреса сайтов в Интернете, которые можно использовать для ознакомления с современными проблемами математики на английском языке. Методические указания рекомендуются для использования на практических занятиях по английскому языку в течение второго – четвертого семестров специальностей 061800 «Математические методы в экономике», 351500 «Математическое обеспечение систем», 010200 «Прикладная математика и информатика»,в том числе при изучении языка по курсу дополнительной квалификации «Переводчик в сфере профессиональной коммуникации», а также для аспирантов.

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Unit 1.Ordinal and relation signs Look through the table and try to memorize it. signs reading examples 1. Firstly; or: in the first place 2. Secondly; or in the second place . Point; or: decimal point … And so on to 1, 2, 3… 25 (read: one, two, three and so on to twenty five And so on to infinity 1, 2, 3… (read: one, two, three and Λ∞ so on to infinity I: since, because Θ x one, or : x sub one x1 x approaches infinity or: x x→∞ tends to infinity = is equal to, or: does not equal a=b; a equals b or: a is equal to ; or: a is b a ≠ b ; a does not equal b; or: a is not is not equal to; or: does not ≠ equal equal to b; or: a is not b α,~ (is) directly proportional to p~ q approximately equals a=b; ≈ or : is approximately equal to a approximately equals b, ~ or ; a is approximately equal to b ≅ :: as (in proportions) p:q :: s:t, read p(is) to q as s (is) to t Is identical with, or is always F(x)= 0 f of x(is) identical with zero ≡ equal to ( ) Parentheses; or : round brackets [] Brackets; or: square brackets {} braces < (is) less than p (is) greater than p>q; p (is) greater than q (is) not greater than p, q stands for p=x=q ≤ >/ (is) greater than of equals The closed interval p q stands x not less than (is) not greater than of equals a ≥ b a (is) greater than or equals b ≥ a ≥ b; a ≥ b' ; a
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1.1 Assignments 1.1.1Memorize the following words and word-groups. ordinal sign простой знак [`O:dInl `saIn] relation sign знак соотношения [rI`leIS(q)n `saIn ] point; точка [poInt] decimal n. десятичная дробь, [`desIm(q)l] a десятичный decimal point точка, отделяющая [`desIm(q)l poInt] десятичную дробь от целого числа (соответствует запятой в русском языке) infinity бесконечность [In`fInItI] approximately приблизительно [q`prOks(I)mItlI] proportion пропорция, [prq`pO:Sqn] количественное соотношение, пропорциональность, часть, доля proportional пропорциональный [prq`pO:Sqnl] identical with соответствующий чему [aI`dentIk(q)l wID] либо round brackets / круглые скобки [`raund `brxkIts] parentheses [pq`renTIsI:z] square brackets / brackets [`skwFq `brxkIts] квадратные скобки braces фигурные скобки [`breIsIz] is equal to, equals равно, равняется [`I:kw(q)l] less than меньше чем [les Dxn] greater than больше чем [`greItq Dxn] since так как, [sIns] approach / tend достигать значения; [q`prouC] [tend] приближаться, стремиться 1.1.2 Read the following signs: • [] • x→∞ • p:q :: s:t • a>b • 50,150,1500Λ ∞ • s
5

1.1.3 1 2 3 4 5 6 7 8 9 10

Match the columns a) x approaches infinity or: x tends to infinity xΛ ∞ b) Parentheses; or : round brackets Θ c) is not equal to x1 d) x and so on to infinity x→∞ x=1 e) x one, or : x sub one f) X is equal to one, or: x does equal one ≠ α,~ g) (is) directly proportional to :: h) I: since, because i) Is identical with, or is always equal to ≡ ( ) j) as (in proportions)

1.1.4 • • • • • • •

Say whether the following expressions and signs are true or false: a2+b2= c2 a plus b is equal c p>q; p is greater than q a ≠ b ; a does not equal b; or: a is not equal to b; or: a is not b F(x)= 0 f of x(is) identical with zero a ≥ b a (is) greater than or equals b a ≈ b ; a approximately equals b, 1, 2, 3, …one, two, three and so on infinity

Unit 2.Operation signs and terms Look through the table and try to memorize it. Signs of operation operation examples Names of components written read addition + plus a and b are a+b=s a plus b is equal to addends; or: items; or: summands; ss sum subtraction minus l-minuend L-1=a Capital L minus 1 –subtrahend small 1 is equal to a- difference, or: remainder a a and b are factors multiplication * multiplied by a × b = a ⋅ b = ab = c a- multiplicand a times b is equal b- multiplier c; or: a multiplied c - product by b is equal to c; or: ab equals c a a - dividend division / Divided by; a :b = = a b = c b b - divisor or : over a divided by b c – quotient; or: equals c; or: a a – numerator over b is equal to c b – denominator a/b – a fraction 6

2.1 Assignments 2.1.1. Memorize the following words and word-groups: operation sign [`Opq`reIS(q)n `saIn] operation terms [`Opq`reIS(q)n `tq:mz ] addition [q`dIS(q)n] addend / item; [q`dend] [`aItem] summand [`sAmqnd] sum [sAm] subtraction [sqb`trxkS(q)n] minuend [`mInju(:)end] subtrahend [`sAbtrqhend] difference / remainder [`dIfr(q)ns] [rI`meIndq] multiplication [`mAltIplI`keIS(q)n] multiplicand [`mAltIplI`kxnd] multiplier [`mAltIplaIq] product [`prOdqkt] division [dI`vIZ(q)n] dividend / numerator [`dIvIdend] [`njHmqreItq] divisor / denominator [dI`vaIzq] [dI`nqmIneItq] quotient; [`kwouS(q)nt] a fraction [`frxkS(q)n]

знак действия выражения действий сложение слагаемое слагаемое сумма вычитание уменьшаемое вычитаемое разность умножение умножаемое множитель произведение деление делимое делитель частное дробь

2.1.2. Give the names of components of these operations: • 12 / 4 = 3 • 11 × 20 = 220 • 13-6=7 • 23+17=40 2.1.3. Match the columns 1 + a. over 2 b. minus 3 * c. plus d. multiplied by 4 ≠ 5 = e. as (in proportions) {} 6 f. Is identical with, or is always equal to 7 < g. is equal to, or: does not equal 8 :: h. is not equal to; or: does not equal 9 i. Braces ≡ 10 / j. (is) less than

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2.1.4. • • • •

Write these operations add 5 times c 2 times d multiply the sum of x and y by z subtraction 0.5 times d 0.5 times l divide v-w by r times s

2.1.5. Read these expressions • 12 × 30 = 360 d m 23 − 12 = 11 (12 + 8) / 4 = 5

• d ÷m =

• • • [(5 + 3) ∗ (12 − 7 )] = 40 • [(5 + 3) ∗ (12 − 7 )] > [(23 + 13) ÷ 4] 2.1.6. Translate into Russian A can do a piece of work in 8 days. If B can do it in 10 days, in how many days can both working together do it? Solution Let x= the required number of days, then

1 = the part of the work both can do in 1 x

day; 1 1 = the part of the work A can do in 1 day; = the part of the work B can do in 1 x x

day: 1 1 1 9 = + , or x 8 10 10

Solving, x =

8

40 4 = 4 , the required number of days. 9 9

Unit 3.Operating with fractions Look through the table and try to memorize it. Examples Rules written read 1/2 One second; or: a half 1/3 one third; a third ½; 1/3; ¼; 2/3; 1/100; and 5/16 are proper fractions. 2/3 Two thirds A proper fraction is one whose One fourth; or: a fourth; 1/4 numerator is less than denominator or: a quarter 1/100 A hundredth 5/16 Five sixteenths 23/6 and 9/9 are improper fractions. An 23/6 Twenty three six improper fraction is a fraction, whose 9/9 Nine ninths numerator is equal to or larger than the denominator Three and five seventh is a mixed 5 3 Three and five sevenths number. A mixed number is a number 7 and a fraction written together To reduce a fraction to its lowest terms, ar a ar over br equals a over divide the numerator and the = br b denominator by their highest common b factor (or: measure, or: divisor) a ar = b br

a c ad ± bc ± = b d bd

a c ac × = b d bd

a over b equals ar over br a over b, this fraction followed by plus or minus c over d equals ad plus or minus bc this sum or difference over bd a over b, this fraction multiplied by c over d equals ac over bd

To reduce a fraction to higher terms, multiply the numerator and the denominator by the same number. To find the sum (the difference) of two unlike fractions, change them to like fractions (fractions having their least common denominator) and combine the numerators. To find the product of two fractions, multiply the numerators together and the denominators together

9

5 1 5 2 * = =1 6 2 3 3 a c ad / = b d bc

Five sixths divided by a half equals one and two thirds

To find quotient of two fractions, multiply the dividend by the inverted divisor To convert an improper fraction into a mixed number, break it up into the sum of an integer and a proper fraction:

a over b, this fraction divided by c over d equals ad over bc

3.1Assignments 3.1.1. Memorize the following words and word-groups: proper fraction правильная дробь [`prOpq `frxkS(q)n] improper fraction [ Im`prOpq frxkS(q)n] неправильная дробь mixed fraction смешанная дробь [mIkst`frxkS(q)n] to reduce a fraction [rI`djHs q `frxkS(q)n] Приводить дробь a) to its lowest terms к наименьшему [` louIst `tq:mz] значению b) to its higher terms к наибольшему [`haIq ` tq:mz] значению общий делитель common factor / measure / [`kOmqn` fxktq] divisor [`meZq] [dI`vaIzq] combine сочетать [kOm`baIn] to convert преобразовывать [kOn`vq:t] an integer целое число [`IntIGq] integer solution решение в целых числах, целочисленное решение complex / Gaussian integer Комплексное целое число ratio отношение, [`reISIou] коэффициент, пропорция, соотношение inverse ratio отношение обратных [[`In`vWs `reISIou] величин 3.1.2. Give the definitions: • a proper fractions • an improper fractions • a mixed fractions 3.1.3. Do these operations: • reduce a fraction to higher terms: 10

2 3

8 12 2 3 • find the sum of two unlike fractions: + 3 4 3 5 • find the product of two fractions: × 5 12

• reduce a fraction to its lowest terms:

• convert an improper fraction into a mixed number: • find quotient of two fractions :

24 9 ÷ 8 9

45 12 * 6 2

3.1.4. Match the columns: 1 123/123 2 23/9 3 1/8 2 4 2

a. b. c. d.

A proper fractions a mixed fractions Braces An improper fractions

5 6 7 8 9

e. f. g. h. i.

Less than a mixed fractions is equal to, or: does not equal Infinity Because

5

=

{}

< Θ ∞

3.1.5. Read these operations: • • • • • •

5a a = 5b b 1 1 1 * 2 ± 2 *1 ± = 2 2 2*2 a 6a = b 6b 3 3 3*3 × = 7 5 7*5 7 7 7 *5 / = 5 5 5*7 5 1 5 2 * = =1 6 2 3 3

3.1.6. Write these operations: • a over b, this fraction followed by plus or minus c over d equals ad plus or minus bc this sum or difference over bd • a over b, this fraction divided by c over d equals ad over bc • ar over br equals a over b • Five sixths divided by a half equals one and two thirds • a over b, this fraction multiplied by c over d equals ac over bd • a over b equals 2a over 3b 11

3.1.7. Translate into Russian: • To convert an improper fraction into a mixed number, break it up into the sum of an integer and a proper fraction • To reduce a fraction to its lowest terms, divide the numerator and the denominator by their highest common factor (or: measure, or: divisor) • To find the sum (the difference) of two unlike fractions, change them to like fractions (fractions having their least common denominator) and combine the numerators • To find quotient of two fractions, multiply the dividend by the inverted divisor • To find the product of two fractions, multiply the numerators together and the denominators together • To reduce a fraction to higher terms, multiply the numerator and the denominator by the same number. Unit 4.Decimal fractions Look through the table and try to memorize it. writing reading .2

Point two

0.2

O point two, or; two tenths

0.02

O point o two, or; two hundredths

12.707

Twelve point seven o seven

0

O, or: zero, or: nought; or: cipher; or: nil (null)

4.1 TEXT (Read the text and do the tasks that follow.) FRACTIONS There are three methods of dealing with fractions: 1. by expressing each fraction as the sum of fraction as the sum of fractions with unit numerators, e.g. 5 1 1 = + 6 2 3

2. by diving the unit into sub-units which are given special names, e.g. the foot is divided into twelve parts and each twelfth part called an inch; 3. by using fractions whose denominators are powers of the same number, e.g. 3 7 5 = + 4 10 100

The Egyptians used the first method and developed a high degree of skill in it. The second method was used by the Romans, who divided their pound, their foot and one of their coins into twelfths; these fractions were called unciae, from which we get 12

the words inch and ounce. The third method dates back to the Babylonians, who used fractions with denominators which were powers of 60; the Greeks also this method in scientific work. It was not until Simon Stevin (1548-1620) published his essay on decimal fractions in 1585 that the third method, with denominators which all powers of 10, became generally adopted. Stevin was an important government official in the Netherlands in the time of the struggle against Spain. Many authors had used decimal fractions in particular problems, and decimal point first appeared in print as early as in 1492, but Stevin was the originator of general rules for the use of decimal fractions. 4.1.1Choose the best sentences ending which is true to the text: 1.The Egyptians used a. the second method of dealing with fractions. b. the first method of dealing with fractions and developed a high degree of skill in it. c. fractions called unciae. d. fractions with denominators which were powers of sixty. 2.The third method dates back to a. The Roman period. b. 1585 when Simon Stevin published his essay on decimal fractions. c. Ancient times d. The Babylonians, who used fractions with denominators which were powers of sixty 4.1.2 Choose the best reply to the sentences given. 1. What was Stevin’s contribution in mathematics? a. He used decimal fractions in particular problems. b. He was the originator of general rules for the use of decimal fractions. c. He used decimal fractions in particular problems. d. He developed the first method of dealing with fractions. 2. What times does the third method date back to? a. It dates back to ancient Rome. b. It dates back to the fourteenth century c. It dates back to Babylonians. d. It dates back to the twentieth century. 4.1.3 Rearrange the words to make sentences. 1.with three are methods of dealing There fractions. 2. method The used Egyptians the first. 3. in print decimal as early in as 1492 point appeared The first. 4.2 Assignments 4.2.1. Memorize the following words and word-groups: Десятичная дробь decimal fraction [`desIm(q)l`frxkS(q)n] 13

Zero / nought / cipher / nil / null

[`zIqrou] [nO:t] [`saIfq] [nIl] [nAl]

4.2.2. Match the columns 1 .2 a) 2 23/6 b) 3 2/2 c) 5 d) 4 3 7

ноль

Two second Twenty three sixth Point two Twelve point seven nought seven

5

3.57

e)

Three and five sevenths

6 7 8 9 10

0.02 12.707 0

f) g) h) i) j)

naught; or: cipher; Point; or: decimal point nought point nought two, And so on to three point fifty seven

. …

Unit 5.Roots Look through the table and try to memorize it. symbols reading n

c =b т

n c b x 3

n

x

c = C mn m

L= R 2 ± x 2

14

(the) n-th root of c is equal to b radical root sign, or: radical sign index of a root radicand value of a root square root of x cube root of x The n-th root of c to the m-th power equals c to the power of m over n Capital L equals the square root of (out of) capital R squared plus or minus x squared

additional word combinations To extract a root To express a radical by a power with a fractional exponent

5.1 Assignments 5.1.1. Memorize the following words and word-groups: To extract a root извлекать корень из [Iks`trxkt q `rHt] To express выражать [Iks`pres ] a fractional exponent дробный показатель [`frxkSqnl степени eks`pounqnt] radical знак корня, радикал [`rxdIk(q)l] root sign, or: radical sign [`rHt `saIn] Знак корня (обычно с чертой вверху, в отличие от радикала) index of a root показатель корня [`Indeks Ov q `rHt] (pl indexes) pl [`IndeksIz]} radicand подкоренное число, [`rxdIkqnd] выражение value of a root величина, значение [`vxljH Ov q `rHt] корня square root of корень квадратный из [`skwFq `rHt Ov] cube root of корень кубический из [`kjHb `rHt Ov] 5.1.2. Give the names to the components of these operations: • • т c • т • a=z 5.1.3. Read these roots: • 3 9*3 = 3 •

n

• • •

6

(a + b )m = C mn (a + b )4 = (a + b )2 64 = 2

a2 + b4 = y

5.1.4. Match the columns 1

2

2 3

¼ () 4 5 6

a)

2 5

{} n

c =b т

radical

b) cube root of x c) braces d) (the) n-th root of c is equal to b e) a proper fraction f) a mixed fractions 15

7

3

x

g)

Round brackets Unit 6.Powers.

Look through the table and try to memorize it. symbols Reading bn = c

b 2 , b3 b −n am/n = n am

b to the n-th power is equal to c b to the n-this equal to c the n-th power of b is equal to c b to the power of n is equal to c b to the power n is equal to c b –base n –power exponent c – value of a power bn – power b squared, b cubed b to the power of minus n The m over n-th power of a equals the square root of a to the m-th power

Additional word combinations To raise to a power to square a number to cube number

6.1 Assignments 6.1.1. Memorize the following words and word-groups: power степень [`pauq] to raise to a power to возводить в степень [`reIz to q `pauq to] base основание [bqIs] power exponent показатель степени, [`pauq] value of a power величина степени [`vxljH Ov q `pauq ] involution

[`invq`lHSqn]

6.1.2. Read the following expressions. • a 3 = log c d •

δ 2u =0 δt 2

• a4 = n am • 6 = 216 3

16

возведение в степень

6.1.3. Write the following expressions • bn = c • a 3 = 12 + 15 • b3 + a 2 = c 4 • b −(n+1) = a • 6 = 216 3

6.1.4. Give the names to the components Model: a+b=c a and b mean addends; c means a sum • bn = c • т • •

d n b×с = а

Unit 7.Logarithms Look through the table and try to memorize it. symbols reading log b c = n log b c

B C N a 3 = log c d

the logarithm of c to the base b is equal to n logarithmic expression

additional word combinations natural logarithm of a number (ln c) (common) logarithm of a number (log c; or: lg c)

base antilogarithm value of a logarithm a cubed equals the logarithm of d to the base c

7.1 Assignments 7.1.1. Memorize the following words and word-groups: logarithm логарифм [`lOgqrITqm] natural logarithm натуральный логарифм [`nxtSrql`lOgqrITqm] common logarithm общий логарифм [`kOmqn `lOgqrITqm] logarithmic expression логарифмическое [`lOgq`rITmIk выражение Iks`preSqn] antilogarithm антилогарифм [`xntI` lOgqrITqm] value of a logarithm значение (величина) [`vxljH Ov q ` логарифма lOgqrITqm] 17

lOgqrITqm] 7.1.2. Read the following symbols • log a x = m • a 2 = log b d • log a b = y 7.1.3. What does B, C, N mean in logarithmic expressions? Answer using the following pattern: B means the base

7.1.4. Choose the correct definitions of the term “logarithm” a) One of a series of numbers set out in tables which make it possible to work out problems in multiplication and division by adding and subtracting. b) Branch of mathematics that deals with the relations between the sides and angles of triangles c) Variable quantity, dependent in value on another. 7.1.5. Match the columns. n 1 a) c =b log b c 2 b) c) 3 lnc 4 5 6 7

b2 b −n b

radical Natural logarithm Common logarithm

d)

(the) n-th root of c is equal to b

e) f) g)

Square root of b b to the power of minus n b squared,

Unit 8.Some algebraic expressions and formulas Look through the table and try to memorize it. symbols reading a prime a’ a second prime; or; a double prime a’’ a triple prime a’’’ ′ a1 a first prime a 2′′ a second second prime a c′′ a second prime sub c; or: a c-th second prime modulus M or µ first derivative of z Z& & & second derivative of z Z dz the first derivative of z with respect to x dx 18

the second derivative of y with respect to x

d 2z dx 2 d nz dx n y = f (x )

the n-th derivative of y with respect to x

→ x→a x → x0

∫

m

n

⋅⋅⋅

∫ 2 xdx = x

2

x

d lxdx dx x∫0

m = R1 x − P1 ( x − a1 ) − P2 ( x − a 2 )

∑ n

∑x l =n

y is a function of x ; or: y equals f of x approaches; or: approaches the limit; or: tends to x approaches the limit a x approaches x nought; or: x tends to x naught integral of … from n to m; or: integral of … between limits n and m the integral of 2x dx is x2 d over dx of the integral from x sub 0 (or: from x zeroth) to x of capital Lxdx m equals 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; or: m equals R first x minus P first multiplied by the difference between x and a sub one minus P second multiplied by the difference between x and a sub two the sum Summing over x sub i from one to n

i

8.1 Assignments 8.1.1. Memorize the following words and word-groups: a prime штрих, со штрихом [praIm] a double prime двойной штрих; два [`dAbl `praIm] штриха a triple prime тройной штрих; три [`trIpl `praIm] штриха a first prime [`fq:st `praIm] modulus абсолютное значение [`mOdjulqs] pl moduli (числа), модуль [`mOdjulaI] derivative of

[dI`rIvqtIv qv]

with respect to approach

[wID rIs`pekt tq] [q`prouC]YUU

integral of

[`IntIgrql ]

производная; производное число от что касается, по приближаться, подходить; достигать значения интеграл от 19

[`lImIt]

limit

предел

8.1.2. Read the symbols. • a’ • y = f (x ) • → • x→a • a 2′′ • ac′′ • ∑ •

∫

m

n

⋅⋅⋅

8.1.3. Read the following equations. • ∫ 2 xdx = x 2 y = f (x ) m = R1 x − P1 ( x − a1 )

•

•

− P2 ( x − a 2 )

8.1.4. Match the columns. 1 x→a a) a double prime b) second derivative of z 2 ∑ 3 a 2′′ c) the sum 4

& Z&

d)

a second second prime

5

a ′′

e)

x approaches the limit a

8.1.5. Write the symbols that are implied by the following definitions. A. Line or point that may not or cannot be passed; greatest or smallest amount, degree, etc. of what is possible. B. Of, denoted by, an integer; made up of integers. C. Total obtained by adding together items, numbers or amounts.

20

Unit 9.Fundamental symbols and expressions concerning the theory of sets Look through the table and try to memorize it. symbols Reading a is an element of M; or: a belongs to M α ∈Μ a is not element of M; or: a does not belong to M α ∉Μ Μ = {2,4,6} M is the set with the elements 2,4,6 M is an empty set (or: a null set) Μ = 0/ Μ⊆Β M is a subset of B M is proper subset of B Μ⊂Β ΑΥΒ the union of A and B ΑΙ Β the intersection of A and B the Cartesian product A and B are equivalent to each other Α×Β 9.1 Assignments 9.1.1. Memorize the following words and word-groups: fundamental symbols основные символы [`fAndq`mentl `sImbqlz] основные выражения fundamental expressions [`fAndq`mentl ` Iks`preSqnz] the theory of sets теория множеств [`TIqrI qv `sets] the set ряд; совокупность; [set] множество; семейство (кривых) an empty set пустое множество [` emptI `set] a subset подмножество [`sAb `set] a proper subset собственное [`prOpq `sAb `set] подмножество union of sets объединение множеств [`jHnjqn ] the intersection of sets пересечение множеств [`Intq`sekSqn] the Cartesian product прямое произведение [kR`tJzjqn `prOdqkt] to be equivalent to Быть эквивалентным [I`kwIvqlqnt] (равным, соответствующим) чему -либо 9.1.2. Translate into Russian: • a is an element of M; or: a belongs to M • a is not element of M; or: a does not belong to M • M is the set with the elements 2,4,6 • M is an empty set (or: a null set) • M is a subset of B • M is proper subset of B 21

• the union of A and B • the intersection of A and B • the Cartesian product A and B are equivalent to each other 9.1.3. Read these symbols: • ∈ • ⊆ • Ι • × 9.1.4. Match the columns 1 2 3

α ∉Μ Μ = {2,4,6} n

∑A l =n

4 5 6

a does not equal m a) b) 2,4,6, and so on infinity c) the union of A and B

i

2,4,6 Λ ∞ a≠m ΑΥΒ

d) M is the set with the elements 2,4,6 a does not belong to M e) f) Summing over A sub i from one to n

9.1.5. Insert a word instead of a symbol. a)is an element; b) is a subset c) is proper subset of 1. α ∈ Μ a)is an element; b) is a subset of c) is proper subset of 2. Μ ⊆ Β a) is an empty; b) does not belong to c) is proper subset of 3. Μ = 0/ a) the union of; b) are equivalent to each other; c) is proper 4. Α Υ Β subset of a) the union of; b) the intersection of; c) is proper subset of 5. Α Ι Β

22

Unit 10.Classification of the elementary functions Look through the table and try to memorize it. functions reading 4 rational integral y equals the sum of a (sub) K, x of the y = ∑ ax k functions power of k, taken k equal to zero to k equal k =a 4 7+x rational fractional y equal the fraction with the numerator 7 y= 2 functions plus x and the denominator 2 minus x 2+ x squared y equals the negative square root of the irrational y = − z2 − x2 difference z squared minus x squared functions y = ln x exponential y equals “l” “n” “x” functions trigonometric Y=sin x y equals the sine of x functions Y=cos x y equals the cosine of x Y=tg x y equals the tangent of x Y=ctg x y equals the cotangent of x Y=sec x y equals the secant of x Y=csc x y equals the cosecant of x −1 y equals: the inverse sine of x, antitrigonometric y = sin x or: the arcsine of x functions; or: or: the angle whose sine is x inverse trigonometric functions k

23

10.1 Assignments 10.1.1. Memorize the following words and word-groups: rational integral function [`rxSqnl `IntIgrql рациональная интегральная функция `fANkSqn] rational fractional рациональная дробная [`rxSqnl `frxkSqnl functions функция `fANkSqn] irrational function иррациональная [I`rxSqnl `fANkSqn] функция exponential function показательная функция [`ekspou`nenSql `fANkSqn] trigonometric function тригонометрическая [ ` trIgqnq`metrikc c функция `fANkSqn] antitrigonometric [` xntI`trIgqnq`metrikc c обратная function тригонометрическая `fANkSqn] функция the sine синус [saIn] the cosine косинус [`kousain] the tangent тангенс [`txnGqnt] the cotangent котангенс [`kou`txnGqnt] the secant секанс [`sJkqnt] the cosecant косеканс [`kousJkqnt] the arc sine арксинус [`Rk`sain] the arc cosine арккосинус [`Rk`kousain] the angle угол [`xNgl] the inverse обратный, [`In`vWs] противоположный, инверсный 10.1.2. Read the function •

y=

4

∑ ax

k

k =a k

24

7+x 2 + x2

•

y=

• • • • • • • • •

y = − z2 − x2 y = ln x

Y=sin x Y=cos x Y=tg x Y=ctg x Y=sec x Y=csc x y = sin −1 x

10.1.3. Match the columns ΥΙ Χ 1 a) 2 2 ∫ 2 xdx = x b) 3 4

т

y=

11 + 7 2 + 22

5 y = − z2 − x2 6 7

y = ln x y=

4

∑ ax

c)

y equals the negative square root of the difference z squared minus x squared

d)

the intersection of Y and X

e)

y equals the sum of a (sub) K, x of the power of k, taken k equal to zero to k equal 4 rational fractional functions radical

f) g)

k

the integral of 2x dx is x2 exponential functions

k =a k

10.1.4. Give the examples of the functions: Model: Trigonometric function is Y=sin x • rational integral functions • rational fractional functions • irrational functions • exponential functions • trigonometric functions • inverse trigonometric functions Unit 11.Expressions concerning intervals and limits Look through the table and try to memorize it. Signs reading open interval a b (a, b) [a, b] closed interval a b (a, b] half – open interval a b, open on the left and closed on the right Capital x equals the open interval minus infinite plus X = (− ∞;+∞ ) infinite x approaches x nought; or x tends to x nought X → x0 lim f ( x) = L the limit of f x as x tends to x one is capital L x→ x 1

lim f ( x) ≠ f ( x0 )

x → x0

lim a n = 0 x →∞

the limit of f of x tends to x nought is not equal to f of x nought the limit of a sub n is zero as n tends to infinity

25

11.1 Assignments 11.1.1. Memorize the following words and word-groups: open interval открытый интервал [`oupqn `Intqvql] closed interval закрытый интервал, [`klouzd `Intqvql] half – open interval наполовину открытый [`hRf `oupqn `Intqvql] интервал infinite бесконечный, [`InfInIt] бесконечно большой infinity бесконечность [In`fInItI] 11.1.2. Read the signs half – open interval a b, open on the left and closed on the • (a, b] right • X = (− ∞;+10) Capital x equals the open interval minus infinite plus ten y equals the tangent of x • Y=tg x the limit of f x as x tends to x one is capital L f ( x) = L • xlim →x 1

•

lim f ( x) ≠ f ( x0 the limit of f of x tends to x nought is not equal to f of x

x → x0

• lim a n = 0 x →∞

nought

the limit of a sub n is zero as n tends to infinity

11.1.3. Match the columns 1 23/6

a)

2 [a, b] 3 (a, b) 4 x= (x0 ;+∞ )

b) c) d)

5 X → x0 f ( x) = L 6 xlim →x

e) f)

f ( x) ≠ f ( x0 ) 7 xlim →x

g)

an = 0 8 lim x →∞

h)

1

0

x approaches x nought; or x tends to x nought improper fraction closed interval x equals the open interval x nought plus infinite open interval a b the limit of f x as x tends to x one is capital L the limit of f of x tends to x nought is not equal to f of x nought the limit of a sub n is zero as n tends to infinity

11.1.4. Translate into Russian • m equals 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 • a is directly proportional to b • The limit for delta x tending to zero, of the sum of small f of x sub k 26

delta x taken from x sub k equal to a to x sub k equal to b minus delta x equals the integral from a to b of small f of xdx equals capital F of x between limits a and b equals capital F of minus capital F of a equals capital A • x approaches x nought; or x tends to x nought • the logarithm of c to the base b is equal to n • the limit of f of x tends to x nought is not equal to f of x nought 11.1.5. Insert a proper word 1. Open interval can be shown by . . . . a) parentheses b) brace c) point 2. Closed interval can be shown by . . . . a) square brackets b) brace c) round brackets 3. Half open interval can be shown by . . . . a) brackets b) round brackets on the left and square brackets on the right a) infinity Tasks. 1. Analyse and memorize m / ( m −1)   z  ϕ ( z ) = b  2 +  − 1 cm   

a) ϕ of z is equal to b, square brackets, parenthesis, z divided by c sub m plus two, close parenthesis, to the power of m over m minus 1, minus 1, close square brackets. b) ϕ   оf z is equal to b, multiplied by the whole quantity: the quantity two plus z over c sub m, to the power m over minus 1, minus 1. 

ϕ i (t1 ) − ϕ i (t2 ) ≤  t1 − 

β

 β  − Μ  t2 −  j j 

The absolute value of the quantity ϕ   sub j of t sub one, minus ϕ sub j of t sub two is less than or equal to the absolute value of the quantity M of t sub one minus β over j, minus M of t sub two minus β over j n

k = max ∑ aij (t ) ; j

t ∈ [a, b];

j = 1,2....n

i =1

27

K is equal to the maximum over j of the sum from i equals one to i equals n of the modulus of a sub i j of t, where t lies in the closed interval a, b and where j from one to n. t

t

τ

τ

lim ∫ { f [sϕ n (s )] + ∆ n (s )} ds = ∫ f [sϕ ⋅ (s )]⋅ ds n →∞

The limit as n tends to infinite of the integral of f of s and ϕn of s plus delta sub n of s, with respect to s, from τ to t, is equal to the integral of f of s and ϕ of s, with respect to s, from τ to t. Ψn − rs +1 (t ) = e

t λq + s

pn−rs +1

Ψsub n minus r sub s plus one of t is equal to p sub n minus r sub s plus 1 times e to the power t times λ sub q plus s.

L

+ n

g = (− 1) (a0 g ) n

(n )

+ (− 1)

( n −1)

(a g

n +1

+ ... + an g

)

L sub n adjoint of g is equal to the minus one to the n, times the n-th derivative of a sub zero conjugate times g, plus, minus one to the n minus one, times the n minus first derivative of a sub one conjugate times g plus and so on to plus a sub n conjugate times g.     dF λ i (t ), t1  dF λ i (t ), t1    λ i, (t )+   =0 dλ dt

The partial derivative of F of lambda sub i of t and t with respect to lambda, multiplied by lambda sub i prime of t plus the partial derivative of F with arguments lambda sub i of t and t, with respect to t, is equal to zero. d2y + [1 + b(s )] ⋅ y = 0 ds 2

The second derivative of y with respect to s plus y times the quantity 1 plus b of s is equal to zero

( ); ( z → ∞; arg z = γ )

f ( z ) = ϕ€mk + 0 z

−1

f of z is equal to ϕ sub mk hat plus big O of one over the absolute value of z, as absolute z becomes infinite, with the argument of z equals gamma. 28

n

(

Dn−1 ( x ) = Π 1 − x s2 s =0

)

ε −1

D sub n minus one prime of x is equal to the product from s equal zero to n of, parenthesis, one minus x sub s squared, close parentheses, to the power epsilon minus one. Κ (t , x ) =

1 2 Пi

∫

1 w− = ρ 2

K (t , z ) dw w − w(k )

K of t and x is equal to one over two πi times the integral of K of t and z, over w minus w of x, with respect to w along curve of the modulus of w minus one half, is equal to rho. d 2u + a 4 ∆∆u = 0; (a > 0) dt 2

the second partial (derivative) of u with respect to t plus a to the fourth power, times the Laplacian of the Laplacian of u, is equal to zero, where a is positive c + i∞

1 Dk ( x ) = ζ 2 Пi c −∫i∞

k

xn (w) dw; (c > 1) w

D sub k of x is equal to one over two πι times integral from c minus ι infinity to c plus i infinity of dzeta to the k of w, x to the w over (or: divided by) w, with respect to w, where c is greater than one. 2. Practice reading the following expressions by yourself, check your answer using the keys a.

2 2 3 1  15 + 7  + 3(x − 2)  = 5 + x 7 3 4 2 

b.

lim f (x ) = L x →1

c. f ' (x ) = lim s →0

d. S =

f ( x + S )x − f ( x ) Sx

ds dt 29

e.

∫ f (x )dx = F (x ) + c

f. x = (− ∞;+∞ ) g.

b− x

lim ∑

x → 0 xk = a

h. lg a

3n

i. M =

a

f ( x k )Sx = ∫ f (x )dx = F ( x ) b = F k (b ) − F (a ) = A b

a

a 2n = lg x = lg c p P, g ; g ≠ 0 g

Check your answer a. Two seventh times – brace open – fifteen, plus seven times – bracket open – one half plus three times – parenthesis open – x minus two parenthesis, bracket, brace closed – equals five and two thirds, plus three quarters x b. The limit of f of x as x tends to x sub one is capital l c. f prime (or: α as h) of x is limit of f of x plus delta x minus f of x over delta x as delta tends to zero d. S dot equals ds by dt e. The integral of small f of x dx equals capital F of x plus capital O if capital F prime is equal to small f of x f. Capital X equals the open interval minus infinity; plus infinity g. The limit for delta x tending to zero, of the sum of small f of x sub k delta x taken from x sub k equal to a to x sub k equal to b minus delta x equals the integral from a to b of small f of xdx equals capital F of x between limits a and b equals capital F of minus capital F of a equals capital A h. The logarithm of a to the power of three n equals the logarithm of the square root of x minus logarithm of the nth power of the fraction a squared over c i. Set of all fractions p/g where p and g are elements of the set of integers and g cannot be zero

30

APPENDIX. AGE PROBLEM Once there lived a clever king. One day a scientist came to him and surprised him with his knowledge. He amused the king by telling him many things that were new and interesting. Finally, the king wanted to find out the age of the scientist but the did not want to ask him. “Professor,” said the king, “I have an interesting problem for you. Think of the number of the month of your birth, but don’t tell me.” The scientist was 60 years old, and his birthday was in December. So he thought of 12. “All right,” he said. “Multiply it by two,” said the king. “I have done it.” “Add 5.” “I have”. “Now multiply by 50.” “Yes.” “Subtract 365.” “Yes.” “Add your age.” “Yes.” “Add 115.” “Yes.” “And now,” said the king, “tell me result.” “Twelve hundred and sixty,” answered the scientist. “Thank you,” said the king. “You were born in December and now you are sixty years old.” “How do you know that?” asked the scientist. “From your answer,” replied the king. “Oh!” the professor said laughing, “That is a polite way to find out one’s age.” Can you find out your friends’ age in the same way? FIND THE NUMBERS Can you find two numbers composed only of ones which give the same result when you add them and when you multiply them? I and 11 are very close, but they are not the numbers which you must find, because if you add them, you will have 12, and if you multiply them, you will have only 11. HUNDREDS AND HUNDREDS. 1. Arrange 4 eights so that they will exactly equal 1. 2. Arrange 4 fives in one line to make 100. 31

3. Arrange the numbers 1,2,3, 4, 5, 6, 7, 8and 9 so that they will make exactly 100. STRANGE FIGURES. The father asked his three sons to write four numbers using five odd figures so that they would equal fourteen. Can you do it? AT THE ZOO. One day a man went to the Zoo with a bag of nuts. He stopped near three cages of monkeys and decided to give them all the nuts in the bag. “If I divide the nuts equally among the eleven monkeys in the first cage,” he thought, “one nut will remain. If I divide equally among the thirteen monkeys in the second cage, eight nuts will remain. If I divide them among the seventeen monkeys in the third cage , three nuts will remain. And if I divide the nuts equally among the forty-one monkeys in all three cages or among the monkeys in any two cages, some nuts will remain too. How can I I divide them so that none will remain?” Could you help the man to divide his nuts among the monkeys? MEASURE THEWATER. A girl was sent to the well with two jugs. One of them contained 7 litres and the other 11 litres. She had to bring back exactly 2 litres of water. How could she do it? What is the smallest number of times she had to fill and empty her jugs?

32

KEYS. 1.Find the Numbers The two numbers are 11 and 1.1(one point one)/ 2. Hundreds and hundreds 8 8 8 8 2. (5 + 5) × (5 + 5) = 100

1. × = 1

3. 1 + 2 + 3 + 4 + 5 + 6 + 7 + (8 × 9) = 100 4. Strange Figures Write the following four numbers composed of five odd figures 11, 1, 1, 1. After addition you will have 14. 5. At the Zoo. The smallest number of nuts is 2.179. The best way to solve this problem is to deal first with the first two cages and find that 34 will satisfy the case for 11 and 13 monkeys. Then you must find the smallest number which will satisfy the condition for the 17 monkeys. 6.Measure the Water The smallest number of times the girl had to fill and empty her jugs is 14. Here is how she could do it: 7-litre jug: 7; 0; 7; 3;3;0;7; 6; 6; 0; 7; 2. 11-litre jug: 0; 7; 7; 11; 0; 3; 3; 10; 10; 11; 0; 6; 6; 11. Keys 1. Learn to read ordinal and relation signs 1.1.3. Match the columns 1 2 3 4 5 6 7 8 9 10 D h e a f c j g i b 2.Operation signs and terms 2.1.3. Match the columns 1 2 3 4 c b d h

5 g

6 i

7 j

8 e

9 f

10 a

8 h

9 i

10 j

3. Operating with fractions 3.1.4. Match the columns 1 2 3 4 A b c d

5 e

6 f

7 g

4. Decimal fractions 4.1.2.Choose the best sentences ending which is true to the text: 1 2

33

4.1.3. Choose the best reply to the sentences given. 1 2 4.2.2. Match the columns 1 2 3 4 c b a e

5 j

6 h

7 d

8 f

9 g

10 i

5. Roots 5.1.4. Match the columns 1 2 3 4 f e g c

5 d

6 a

7 b

6. Powers 6.1.3Write the following expressions • b to the n-this equals to c • a to the cubed power is equal to 12hlus 15 • b cubed power plus a squared power is equal to c forth power • b to the power of minus parentheses n plus one equals to a • six cubed power is equal to two hundred sixteen 7.Logarithms 7.1.4. Choose the correct definitions of the term “logarithm”- a 7.1.5. Match the columns 1 2 3 4 d c b a

5 g

6 f

7 e

8.Some algebraic expressions and formulas 8.1.4. Match the columns 1 2 3 4 5 e c d b a 8.1.5. Write the symbols that are implied by the following definitions A B C →

∫

∑

9. Fundamental symbols and expressions concerning the theory of sets 9.1.4. Match the columns 1 2 3 4 5 6 E d f b a c 9.1.5. Match the columns 1 2 3 4 5 A b a a b

34

10. Classification of the elementary functions 10.1.3. Match the columns 1 2 3 4 5 6 7 d a g f c b e 11. Expressions concerning intervals and limits 11.1.3. Match the columns 1 2 3 4 5 6 7 8 9 a b c d e f g h i 11.1.5. Insert a proper word 1 2 3 a a b

10 j

35

Список использованной литературы. 1. Чистик М.Я. Учебник английского языка для политехнических вузов. Учебник. М., Высшая школа, 1975-350с. 2. Пронина Р.Ф. Пособие по переводу английской научно-технической литературы М., Высшая школа, 1973-197с. 3. Носенко И.А., Горбунова Е.В. Пособие по переводу научнотехнической литературы с английского языка на русский М., Высшая школа, 1974-152с 4. Глушко М.М. Русско-английский математический словарь. – М.: Изд МГУ, 1988 5. Курашвили Е.И., Медведева Т.Г., Михалкова Е.С. Лабораторные работы по переводу английской научно-технической литературы М., Высшая школа, 1976-108с 6. Кабо П.Д., Фомичева С.Н. Сборник научно-популярных статей М., Просвещение, 1983-143с. 7. Дорожкина В.П. Английский язык для студентов математиков Учебник, М. ООО «Издательство Астрель», 2001-496с. 8. Шаншиева С.А. Английский язык для математиков. – М.: Изд. МГУ, 1976 9. Басс Э.М. Научная и деловая корреспонденция. (Английский язык) – М.: Наука, 1991 10. Wise, GaryL., ; Hall, Eric B. “Counterexamples in probability and real analysis”, Oxford University Press, New York, 1993 11. Longman Essential Activator, Addison Wesley Longman Limited, 1999, 997p. 12. Alison Pohl, Test Your Professional English: Accounting, Penguin English, 2002, 106 p. 13. Ресурсы Интернета: www.math-atlas.org www.awl-elt.com www.penguinenenglish.com www.cambridge.org www.oxford.org

36

Приложение А (справочное)

Capital letter

Latin alphabet used in mathematics Small letter Reading

A

a

[ei]

B

b

[bi:]

C

c

[si:]

D

d

[di:]

E

e

[i:]

F

f

[ef]

G

g

[dZI:]

H

h

[eiC ]

I

i

[ai]

J

j

[dZei]

K

k

[kei]

L

l

[el]

M

m

[em]

N

n

[en]

O

o

[ou]

P

p

[pi:]

Q

q

[kju:]

R

r

[a:]

S

s

[es]

T

t

[ti:]

U

u

[ju:]

V

v

[vi:]

W

w

[`dAblju:]

X

x

[eks]

Y

y

[wai]

Z

z

[zed] 37

Приложение Б (справочное) Greek alphabet used in mathematics. Capital letter

Small letter

Name of the letter

Reading of the letter

Α

α

alpha

[`xlfq]

Β

β

beta

[`bi:tq US : `beItq]

Γ

γ

gamma

[`gxmq]

∆

δ

delta

[`deltq]

Ε

ε

epsilon

[ep`saIlqn US: `epsIlPn]

Ζ

ζ

zeta

[`zi:tq US : `zeItq]

Η

η

eta

[`i:tq US : `eItq]

Θ

θ

theta

[`Ti:tq US : `TeItq]

Ι

ι

iota

[aI`qVtq]

Κ

κ

kappa

[`kxpq]

Λ

λ

lambda

[`lxmdq]

Μ

µ

mu

[mju:]

Ν

ν

nu

[nju: US : `nH]

Ξ

ξ

xi

[ksaI]

Ο

ο

omicron

[qV`maIkrqn US : `PmIkrPn]

Π

π

pi

[paI]

Ρ

ρ

rho

[rqV]

Σ

σ

sigma

[`sIgmq]

Τ

τ

tau

[taV]

Υ

υ

upsilon

[ju:p`saIlqn US : `jHpsIlPn]

Φ

φ

phi

[faI]

Χ

χ

chi

[kaI]

Ψ

ψ

psi

[saI]

Ω

ω

omega

[`qVmIgq]

38

Приложение В (справочное) addition quantity unknown quantity sum item total subtraction subtract difference multiplication multiply factor product division divide (into, by) dividend divisor quotient number natural numeral involution power raise to power exponent square cube

Glossary process of adding; smth added or joined amount, sum or number symbol (usually x) representing an unknown quantity in an equations total obtained by adding together items, numbers or amounts single article or unit in a list complete, entire; amount process of subtracting ; instance of this take (a number, quantity) a way from (another number, etc ) amount, degree, manner, in which things are unlike multiplying or being multiplied; instance of this add (a given quantity or number) a given number of times whole number (except 1) by which a larger number can be divided exactly quantity obtained by multiplication dividing or being divided separate, find out how often one number is contained in another number to be divided by another number by which another number is divided Number obtained by dividing one number by another 3.13.33 and 103 are numbers; (pl) arithmetic Ordinary, normal; simple (word, figure or sign ) standing for a number; of number Arabic numerals ,1,2,3, etc, roman numerals, I, II, III, etc. anything internally complex or intricate result obtained by multiplying a number or quantity by itself a certain number of time move from a lower to a higher level symbol that indicates what power of a factor is to be taken of a number multiplied by itself; result when a number or quantity is multiplied by itself; multiply a number by itself product of a number multiplied by itself twice; multiply a number by itself twice 39

radical sign, radical root extract index (indices) fractions numerator denominator integer integral decimal decimal point

ratio proportion

constant term extreme limit equations identity formula, formulas (formulae)

40

relating to the root of number of quantity quantity which, when multiplied by itself a certain number of times, produces another quantity that which has been extracted; take or get out something that points to or indicates; pointer showing measurement number that is not a whole number number above the line in a vulgar fractions number or quantity below the line in a fraction whole number (contrasted with fraction ) of, denoted by, an integer; made up of integers of tens or one-tenths the point in eg 15.61. NOTES: in most continental countries a comma is used in places of the GB/US decimal point. Thus 15.61 is written 15,61 in Russia. relation between two amounts determined by the number of times one contains the other equality of relationship between to sets of numbers; statement that two ratios are equal; relation of one thing to another in quantity, size, etc; relation of a part to the whole; number or quantity that does not vary part of an expression joined to the rest by +or either end of anything; highest degree; (pl) qualities Line or point that may not or cannot be passed; greatest or smallest amount, degree, etc of what is possible statement of equality between two expressions by the sign = as in : 2x +5 =11 state of being identical; absolute sameness statement of a rule, fact, etc especially one in signs or numbers, as in mathematics

Приложение Г (обязательное) Vocabulary accuracy n - точность addition n – сложение, прибавление adjacent a – смежный, соседний algebra n - алгебра alter v – изменять, переделывать, изменяться altitude n - высота amount n – сумма, количество, объем angle n - угол apply v – прилагать, применять, употреблять appropriate v – присваивать, предназначать, a подходящий, соответствующий approximately adv – приблизительно area n – площадь, зона arrange v – приводить в порядок arrangement n- устройство associate v – соединять, связывать, соединяться average a - средний axis n - ось balance n – равновесие, v уравновешивать base n - основание binary a – бинарный, двойной brackets n - скобки calculation n - вычисление calculus n - исчисление capability n - способность capable a - способный cell n – элемент ячейки chord n - хорда cipher n –шифр v шифровать circle n - круг circumference n - окружность coefficient n - коэффициент combine with v - соединяться comparatively adv – сравнительно, относительно complete a – полный, завершенный; v завершать component n – компонент, составная часть compose v - составлять composition n - состав cone n - конус 41

conclude v – делать вывод condition n – состояние, условие conical a – конический, конусообразный consequently adv – следовательно, поэтому constitute v - составлять contiguous a - смежный contract v – сокращать уменьшать(ся) conversely adv – обратно. наоборот count v -считать cube n -куб curve n - кривая decimal a - десятичный decode v - расшифровывать decomposition n - разложение decrease n – уменьшение, убывание, v – убывать, уменьшаться deduce v – выводить (заключение), проследить definite a - определенный degree n - градус denominator n - знаменатель destination n - назначение determine n – определять, устанавливать, побуждать, заставлять diameter n - диаметр difference n - разность different a – различный, отличный(от других) differentiate v - различать digit n – цифра, разряд direction n - направление discontinuous a - прерывистый discover v – открывать, обнаруживать divide v – производить деление division n - деление domain n – область, сфера dominant a – преобладающий, господствующий due a – должный, обусловленный enlarge м – увеличивать(ся), расширять(ся) equal a –равный, v – равняться, уравнивать equation n - уравнение equivalent n – эквивалент, a равносильный even a - четный exaggerate v - преувеличивать exceed v – превышать, превосходить exception n - исключение excess n – избыток, излишек expand v - расширяться 42

extent n – протяжение, объем, предел extremely adv - крайне familiar a - близкий figure n - цифра fit v - соответствовать foundation n – основание, фундамент, основа, базис fraction n - дробь frequency n -частота function n - функция geometry n - геометрия give v – вызывать, давать height n - высота horizontal n - горизонталь identical a - одинаковый; подобный; идентичный identity v - опознавать, идентифицировать imbalance n - неустойчивость imply - означать; подразумевать increase n - увеличение; v - увеличивать, увеличиваться indicate v - указывать, показывать infinity п - бесконечность instability n - неустойчивость, непостоянство instance n - пример, отдельный случай intensity n - напряженность, интенсивность interaction n - взаимодействие intermediate a - промежуточный introduce v - вводить invariably adv - неизменно, всегда inversely adv - 1) обратно 2) обратно пропорционально involve v - включать irregular a - беспорядочный irrespective adv - независимо length n - длина level n - уровень limit n – граница, предел, v - ограничивать line n – линия, черта literal a - буквенный majority n - большинство make up v – составлять; образовывать mathematics n - математика 43

maximum n – максимум; наибольшее значение mean n – среднее число, середина measure n - мера v - измерять measurement n - измерение medium n – среда a - средний middle 1) a – средний 2) n - середина minimum n – минимум, наименьшее значение minute a - маленький multiplication n – мат. умножение multiply v – мат. умножать mutual a – взаимный, общий natural a – естественно; натурально negative n – отрицание a – отрицательный v - отрицать negligible a - незначительный notation n – система обозначений number n - число numeration n – исчисление, нумерация numerator n – мат. числитель дроби numerical a - числовой obvious а - очевидный occur v – иметь место; случаться odd a - нечетный opposite a - противоположный order n - порядок ordinary a - обычный otherwise adv – 1) иначе 2) в противном случае outcome n - результат owing to prep – вследствие; благодаря pattern n - образец peak n – высшая точка, пик percentage n – процент, процентное соотношение permanent a – вечный, бесконечный phenomenon n – явление, феномен positive a – положительный possible a - возможный power n - степень precisely adv - точно prism n - призма probability n - вероятность probably adv - возможно proceed v – 1) продолжать 2) происходить 3) переходить (к) produce v – предъявлять, предоставлять; производить 44

product n - произведение property n - свойство proportion n - пропорция protect v – защищать, охранять protractor n - транспортир purely adv - исключительно purpose n - цель quantity n - количество quarter n - четверть random a - случайный at random беспорядочно range n - 1)ряд; линия 2) диапазон 3) перен. область распространения rate n – норма v - оценивать reaction n - реакция reason n – причина, довод, основание reduce v – уменьшать, сокращать reflect v - отражать reflection n – отражение, отображение relatively adv - относительно reorganization n – реорганизация, преобразование resemble v – быть похожим residual n – остаток, разность; а - остаточный result n – результат; v – проистекать; приводить к чему-либо right-angled a - прямоугольный ruler n - линейка scan v – внимательно рассматривать score n - счет separate a – отдельный, особый; v – отделять, разделять set-square n - угольник several a - несколько shape n - форма share n – часть, доля signal n – сигнал; знак signify v - обозначать similar a – похожий, подобный significance n – 1) значение, смысл 2) важность size n - размер strengthen v – усиливать (ся) structure n – структура; устройство; строение subject n - предмет subtend v – мат. Стягивать (о дуге); противолежать (о стороне треуголькика) subtraction n - вычитание 45

sufficiently adv - достаточно supremacy n - превосходство systematically adv - систематически tangent n – 1) касательная 2) тангенс temporary a - временный tend v – направляться; стремиться; иметь склонность к tensile a - растяжимый thence adv – отсюда; из этого следует theorem n - теорема theoretical a - теоретический theory n - теория therefore adv – поэтому, следовательно thus adv – таким образом; итак tiny a - крошечный total a – полный; общий transfer n – передача, перенос v – переносить; перемещать; передавать transmit v - передавать treble a – тройной; v - утраивать triangle n - треугольник trigonometry n - тригонометрия twice adv - дважды undoubtedly adv - несомненно uneven a – неровный, нечетный uniformly adv - равномерно unit n - единица value n – значение, величина variant n – разновидность; а – отличный от других variation n – изменение, отклонение vary v – изменять(ся), менять(ся) vertex n - вершина virtually adv - фактически

46

Приложение Д (справочное) Abbreviations used in mathematics a 1. [absolute] абсолютный (о величине) 2. [altitude] высота 3. [angle] угол 4. [apothem] апофема 5. [area] площадь AA [arithmetic average] среднее арифметическое abs [absolute] абсолютный (о величине) absc [abscissa] абсцисса Abs. E [absolute error] абсолютная ошибка ad inf [ad infinitum] без конца, до бесконечности AGP [arithmetic-geometrical progression] арифметико-геометрическая прогрессия alt [alternation] чередование AM [arithmetic mean] среднее арифметическое amt [amount] количество, величина anal [analysis] анализ, расчёт ans [answer] ответ AP [arithmetic progression] арифметическая прогрессия appr. [approximate] приближённый, приблизительный asymp [asymptotic] асимптотический avg [average] среднее // средний B [base] основание BCD [binary-coded decimal] десятичное число в двоичном коде BFS [basic feasible solution] базисное возможное решение BV [basic variable] основная переменная c 1. [circumference] окружность 2. [correction] исправление CDF [cumulative distribution function] (интегральная) функция распределения char 1. [character] характер 2. [characteristic] характеристика cir [circular] круговой ckw [clockwise] по часовой стрелке cl [closure] замыкание clopen [closed and open] замкнутый и открытый coeff [coefficient] коэффициент col [column] столбец CS [cumulative sum] общая сумма cyl [cylinder] цилиндр D 1. [data] данные 2. [decimal] десятичное число // десятичный 3. [diameter] диаметр 4. [discriminator] определитель, дискриминант 5. [distance] расстояние 6. [double]двойной; дублированный DE [differential equation] дифференциальное уравнение def [definition] определение deg [degree] степень 47

diag [diagonal] диагональ diam [diameter] диаметр diff [difference] разность dim [dimension] размерность dom [domain] область doz [dozen] дюжина DR [derived rule] выведенное правило E, e [error] ошибка EC [error correcting] исправление ошибки, внесение поправки eff [efficiency] эффективность EG [Euclidean geometry] евклидова геометрия elim [elimination] исключение eq 1. [equal] равный 2. [equation] уравнение est 1. [estimate] оценка 2. [estimated] расчетный est [estimate] оценка ETR [exponential function of trace] экспоненциальная функция следа (матрицы) ex 1. [exercise] упражнение 2. [exponential] экспоненциальная функция // экспоненциальный ext [extension] расширение f [function] функция fr [frontier] граница (множества) GCD [greatest common divisor] наибольший общий делитель GM [geometric mean] среднее геометрическое gp [group] группа grad [gradient] градиент h [height] высота hex 1. [hexagon] шестиугольник 2. [hexation] гексация hor [horizontal] горизонтальный hyp [hypothesis] гипотеза i 1. [imaginary] мнимый 2. [intrinsic] собственный, внутренний iff [if and only if] тогда и только тогда, когда im [image] образ 2. [imaginary] мнимый inc [inclusively] включительно int [interior] внутренность (множества) l 1. [length] длина 2. [lower] нижний LA [linear algebra] линейная алгебра LC [log-convex] логарифмически выпуклый LCD [lowest common denominator] общий наименьший знаменатель LCM [lowest common multiple] наименьшее общее кратное lev [level] уровень LHS [left-hand side] левая часть lim [limit] предел mat [matrix] матрица math 1. [mathematical] математический 2. [mathematics] математика max [maximum] максимум 48

med [median] медиана mpy [multiply] умножать MS 1. [mathematical system] математическая система 2. [mean square] средний квадрат 3. [more significant] более значащий, старший (о разряде) 4. [multiplicative system] мультипликативная система NC [necessary condition] необходимое условие neg [negative] отрицательный nhood [neighbourhood] окрестность No [number] 1. число, количество 2. номер NSE [number of solutions of equation] число решений уравнения ob 1. [object] объект 2. [observation] наблюдение; измерение ODE [ordinary differential equation] обыкновенное дифференциальное уравнение ord [order] порядок OTE [other things equal] при прочих равных условиях P 1. [perimeter] периметр 2. [point] точка 3. [probability] вероятность PCS [probability] of correct selection] вероятность правильного выбора p. ct [per cent] процент PE 1. [permissible error] допустимая погрешность 2. [probable error] вероятная ошибка pop [population] совокупность pr 1. [probability] вероятность 2. [proven] доказанный PS [proportion of successes] доля Q [quantity] количество QED [quod erat demonstrandum] что и требовалось доказать QEI [quod erat inveniendum] что и требовалось найти QT [quadratic transformation] квадратичное преобразование R 1. [radius] радиус 2. [rate] скорость, интенсивность rad [radical] радикал rect [rectangle] прямоугольник res [residue] вычет resp [respectively] соответственно RF [random function] случайная функция RHP [right half-plane] правая полуплоскость RHS [right-hand side] правая сторона rot [rotation] вращение S 1. [side] сторона 2. [significant] значимый 3. [surface] поверхность sd [subdivision] подразделение SE. [simultaneous equations] система уравнений seq [sequential] последовательный sgn [signum] сигнум SHM [simple harmonic motion] простое гармоническое движение sk [skew ness] асимметрия SM 1. [scatter matrix] матрица рассеяния 2. [simplex method] симплексный метод, симплекс-метод 3. [stepwise maximization] многошаговая максимизация S of S [sum of squares] сумма квадратов 49

sp 1. [space] пространство 2. [spur] след T [total] сумма thm [theorem] теорема tran [translation] параллельный перенос TS [trigonometric series] тригонометрический ряд U 1. [unit] 1. единица // единичный 2. элемент (выборки) 2. [upper] верхний V [volume] объём VAR [variable] переменная (величина) // переменный w [width] 1. ширина 2. широта WF [well-formed] правильный; корректный WRT [with respect to] относительно Z 1. [zero] нуль 2. [zone] зона; область

50