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OPERATIONS IN MATHEMATICS Addition (Сложение) a + b + c is read: a plus b equals c; a and b is equal to c; a added to b makes c; a plus b is c. a, b are called “addends” or “summands” (слагаемые); c is the “sum”. Subtraction (Вычитание) 4 – 3 = 1 is read: three from four is one; four minus three is one; four minus three is equal to one; four minus three makes one; the difference between four and three is one; three from four leave(s) one. 4 is called “a minuend” (уменьшаемое); 3 is “a subtrahend” (вычитаемое); 1 is “a difference” (разность). Multiplication (Умножение) 2 х 3 = 6; 2 ∙ 3 = 6 is read: two multiplied by three is six; twice three is six; three times two is six; two times three make(s) six. 5 ∙ 3 = 15 is read: five threes is (are) fifteen. 2, 5 are “multiplicands” (множимое); 3 is “a multiplier” or “factor” (множитель); 6, 15 are “products” (результат). Division (Деление) 35 ÷ 5  7 is read: thirty five divided by five is 7; five into thirty five goes seven times; 35 divided by 5 equals 7. 35 is “a dividend” (делимое); 5 is “a divisor (делитель); 7 is “a quotient” (частное). Involution or Raise to power (Возведение в степень) 3^{2 }is read: three to the second power; 3 squared. 5^{3 }is read: five cubed; 5 to the third power; 5 to power three. x^{2} – x is called the “base of the power”; 2 is called “an exponent or index of the power”. Evolution (Извлечение из корня) is read: the square root of nine is three. ^{ }is read: the cube root of twenty seven is three. is called “the radical sign” or “the sign of the root”. To extract the root of …  извлекать корень из… FRACTIONS (ДРОБИ) Common Fractions (Простые дроби) Common (simple, vulgar) fractions nowadays more often than not are written on one line: 1/2, 3/5, 4/7, 1/3 in printing. But there are printed works where traditional writing is used: Common fractions are read in the same way as we, Russians do, i.e.: the numerator is read as a cardinal number and the denominator as an ordinal number. If the numerator is greater than one the nominator takes the plural ending –s: 1/9_{ } a ninth, one ninth 3/7 – three sevenths 5/8 – five eighths,  two one hundred and twentythirds  three quarters, three fourths  thirtyfour seventyeighths  twothirds, etc. In mixed numbers the integer is read as a cardinal number and fraction must be added with “and”. E.g.: 3 2/5 – three and two fifths 10 2/7 – ten and two sevenths 5 1/2  five and a half 7 1/3  seven and a third 247 86/93  two hundred and fortyseven and eightysix ninetythirds 347/1000  three hundred and fortyseven thousandths The reading of small fractions is often simplified: 1/2  a half, one half 1/3  a third, one third 1/4  a quarter, one quarter, a fourth, one fourth instead of : one the second, one the third, one the fourth. Decimal Fractions In decimal fractions the ;point (.) is used after the whole number in distinction from Russian, where comma (,) is used and where this sign is not read. But in Russian we must always say – десятых, сотых, тысячных и т.д., in English it is suffice to write (.) and to say “point”. After the point (.) all numbers are read separately. Nought, O may often be omitted but the point (.) is never omitted because it shows that the number is a decimal fraction. In the USA “O” is preferred to be read as “zero”. The point may be written in the upper, middle or down part of the decimal fraction: 2.5; 2∙5; 2˙5. 0.5 1) o [ou] point five 2) nought point five 3) zero point five .5 point five 0.05 1) o [ou] point o [ou] five 2) nought point nought five 3) zero point zero five .05 1) point o [ou] five 2) point nought five 3) point zero five 0.005 1) o [ou] point o [ou] o [ou] five 2) o [ou] point two oes [ouz] five 3) nought point nought nought five 4) zero point two zeros five 5) point OO five 6) point nought nought five 7) point two noughts five 8) point two Oes five .005 1) point nought nought five 2) point zero zero five 3) point two oes [ouz] five 0.75 1) nought (o [ou], zero) seventyfive (seven five) 2) point seventyfive (seven five) 1.3 one point three 4.7 four point seven 10.35 ten point three five 247.864 two hundred and fortyseven point eight hundred and sixtyfour .6 = .6666 It is customary abbreviate recurring decimals by placing dots over the integers which are at the beginning and the end of the recurring part. Thus we write . = .2727… . = .2313131… . = 62371371… and so on. Ratio (Отношение) a : b is read: the ratio of a to b; 10 : 5 is read: the ratio of ten to five 4 : 2 = 2 is read: the ratio of four to two is two is read: the ratio of twenty to five equals the ratio of sixteen to twenty four; twenty is to five as sixteen is to four. Proportion (Пропорция) In proportion we have two equal ratios. The equality is expressed by the sign :: which may be substituted by the international sign of equality =. a : b :: c : d or a : b = c : d is read a is to b as c is to d 2 : 3 :: 4 : 6 or 2 : 3 = 4 : 6 is read two is to three as four is to six. The extreme terms of proportion are called “extremes”, the mean terms are called “means”. The proportion can vary directly (изменяться прямо пропорционально) and it can vary inversely (изменяться обратно пропорционально): x y : x varies directly as y; x is directly proportional to y; x = k/y : x varies inversely as y; x is inversely proportional to y. Equations and Identities (Уравнения и тождества) There are different kinds of equations. In general the equation is an equality with one or several unknown variable(s). The reading of equations is the same as in Russian: 30 + 15 + x^{2} + x^{3} = 90 is read: thirty plus fifteen plus x squared plus x cubed is equal to ninety. 2 + b + + b^{4} = 160 is read: two plus b plus the square root of six plus b to the fourth power is equal one hundred and sixty. The identity is an equality, valid at all admissible values of its variables. The identities are read: a + b = b + a  a plus b equals b plus a: sin^{2}x + cos^{2}x = 1  sine squared x plus cosine squared x is equal to one. Arithmetical and Geometrical Progressions Арифметическая и геометрическая прогрессии An arithmetical progression is a sequence much as 3, 5, 7, 9 … in which each member differs from the one in front of it by the same amount. A geometrical progression is a sequence such as 3, 6, 12, 24 … in which each member differs from the one in the same ratio. “The number of families holidaying abroad grew now in geometrical progression”. Mathematicians more often use now the expressions arithmetic sequence and geometric sequence. READING FORMULAS (Чтение формул) a b = c a divided by b is equal to c 2 x 2 = 4 twice two is four c x d = b c multiplied by d equals b dx differential of x = a plus b over a minus b is equal to c plus d over c minus b y_{ab} ∙ x_{bc} = 0 y sub a minus b multiplied by x sub b minus c is equal to zero + [1 + b(s)]y = 0 the second derivative of y with respect to s plus y times open bracket one plus b of s in parentheses, close bracket is equal to zero ∫f(x)dx the integral of f(x) with respect to x the definite integral of f(x) with respect to x from a to b (between limits a and b) c(s) = K_{ab}_{ }c of s is equal to K sub ab x_{ab} = c x sub a minus b is equal to c a b a varies directly as b a : b :: c : d; a is to b as (equals) c is to d a : b = c : d x 6 + 42 1) x times six is forty two; 2) x multiplied by six is forty two 10 ÷ 2 = 5 1) ten divided by two is equal to five; 2) ten over two is five a squared over c equals b 1) a raised to the fifth power is c; 2) a to the fifth degree is equal to c a^{3} = log_{c}b a cubed is equal to the logarithm of b to the base c the logarithm of b to the base a is equal to c x sub a minus b is equal to c = 0 the second partial derivative of u w with respect to t equals zero c : d = e : 1 c is to d as e is to 1 15 : 3 = 45 : 9 1) fifteen is to three as forty five is to nine; 2) the ratio of fifteen to three is equal to the ratio of forty five to nine p T p is approximately equal to the sum of x sub i delta x sub i and it changes from zero to n minus one the square root of a squared plus b squared minus the square root of a squared plus b sub one squared by absolute value is less or equal to b minus b sub one by absolute value (by modulus) _{ }a to the power z sub n is less or equal to the limit a to the power z sub n where n tends (approaches) the infinity the sum of n terms a sub j, where j runs from l to n = 3 the fourth root of 81 is equal to three c d c varies directly as d sin = a sine angle is equal to a integral of dx divided by (over) the square root out of a square minus x square d over dx of the integral from x sub 0 to x of capital x_{dx} 2 : 5 1) two is to five; 2) the ratio of two to five 5:4::20:16 1) five is to four as twenty is to sixteen; 2) the ratio of five to four equals the ratio of twenty to sixteen; 3) five has the same ratio to four as twenty has to sixteen ab 1) a is not equal to b; 2) a differs from b; 3) a is different from b 1) a approximately equals b; 2) a is approximately equal to b p plus (or) minus q m > n m is greater than n m < n m is less than n 1) a is greater than or equal to b; 2) a is greater than or equals to b 1) a is less than or equals to b; 2) a is less than or is equal to b y → r y approaches r y r y approaches r Note: Some authors use the notation . The symbol signifies an approach to a limit. Thus may be read “x approaches 2” (as a limit). If the words “as a limit” are not expressed, they must be always understood. two times thee to the nth (power) minus 2 is not less than five hundred Note: Just the symbol means “not equal to”, so the symbol means “not less than”. B = capital B is equal to infinity 1) the modulus of a; 2) the absolute value of a; 3) the numerical value of a the modulus of the quantity x minus b is greater than zero and less than or equal to capital C the interval a to b 28^{o} 28 degrees (angular measure and temperature measure) 56' 1) 56 minutes (angular measure); 2) 56 feet (linear measure) 45'' 1) 45 seconds (angular measure); 2) 45 inches (linear measure) the square root (out) of 7 5% 5 per cent 2/9 % 1) two ninths per cent; 2) two ninths of one per cent ½ % 1) a half per cent; 2) a half of one per cent 0.47% 1) point four seven per cent; 2) zero point fortyseven per cent; 3) nought point fortyseven per cent; 4) o point four seven of one per cent 7 %_{0} 7%_{0} seven per mille c is equal to (dash, line of division) a over (divided by, by) b Note: The words dash and line of division are often omitted. c(a+b) 1) c parenthesis a plus b close parenthesis; 2) c round brackets opened a plus b round brackets closed; 3) c times (multiplied by) the quantity a plus b 3[(4+5)620] 1) three, square brackets, parenthesis, four plus five, close parenthesis, times (multiplied by) six minus twenty, close square brackets 2) three, square and round brackets opened, four plus five, round brackets closed, six, minus twenty, square brackets closed; 3) three times (multiplied by) the whole quantity: the quantity four plus five, times six, minus twenty 2{703[(4+5)620]} 1) two, braces, seventy minus… close braces 2) two, braces opened, seventy minus … braces closed ABC EDF the triangle ABC is congruent to the triangle EDF BDAC BD is perpendicular to AC AB is parallel to CD the angle A is equal to the angle B 3! = 1 x 2 x 3 = 6 factorial three is equal to one times two times three is equal to six ^{*} the fourth binomial coefficient the number of combinations of seven things taken two at a time is equal to seven times six over factorial two is equal to twentyone the number of permutations of seven things taken five at a time 1) the derivative of y with respect to x; 2) d over (by) dx of y the derivative of the quantity two x square plus five with respect to x 1) second derivative of y with respect to x; 2) d two over (by) dx of y d over (by) dx of the integral from a to b of f of x dx x is equal to the logarithm of capital N to the base q arc sin a 1) the angle whose sine is a; 2) the inverse sine of a; 3) the antisine of a; 4) the arc sine of a Note: The symbol arc sin a is sometimes written sin^{1}a. sin (arc sin a) the sine of the angle whose sine is a sin 23º the sine of 23º cos 47º the cosine of 47º sec 80º the secant of 80º the tangent of a (one) half (of) A sin α the sine of (the angle) α the cosine of the angle of one half A minus B (the difference of A and B) sin(αβ) the sine of (the angle) α minus β cot(α+β) the cotangent of (the angle) alpha plus β sin ABC the sine of the angle ABC cos ABC the cosine of the angle ABC tan β the tangent of (the angle) β 
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