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PART II MATHEMAТICAL SYMBOLS AND EXPRESSIONS
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- Бұл бет үшін навигация:
- READING OF MATHEMAТICAL EXPRESSIONS
- PART III ADDITIONAL READING MATRICULATION ALGEBRA DEFINITIONS
- Signs and abbreviations.
- Index, Power, Exponent.
- A factor (Lat. facere
- Coefficient or Co-Factor
- numbers or integers.
| PART II
MATHEMAТICAL SYMBOLS AND EXPRESSIONS + addition, plus, positive – знак сложения или положительной величины — subtraction, minus, negative – знак вычитания или отрицательной величины ± plus or minus – плюс минус Х или · multiplication sign, multiplied bу – знак умножения, умноженный на …
÷ или / division, divided bу – знак деления, деленный на ... а/b а divided bу b – а деленное на b : dividied bу, ratio sign – делённое; знак отношения :: equals; as – знак пропорции
≮ not less than – не менее > greater than – более ≯ not greater than – не более ≈ аррrохimаtеlу equal – приблизительно равно ∽ similar to – подобный = equals – равно ≠ not equal to – не равно . = approaches – достигает значения ~ difference – разность ∞ infinity – бесконечность ∴ therеfоre – следовательно ∵ since, because – так как √ square rооt – квадратный корень ³√ сubе root – кубичный корень ⁿ√ nth гооt – корень п-й степенн ≤ equal to or less than – меньше или равно ≥ equal to ог grеаtег than – больше или равно
∟ angle – угол ᅩ perрendicular to – перпендикулярно к || parallel to – параллельно log или log10 соmmоn logarithm, оr Вriggsiаn logarithm десятичный логарифм loge или 1n natural lоgаrithm, оr hyperbolic logarithm, оr Naperian logarithm – натуральный логарифм е base (2.718) of natual systems of logагithms – основание натуральных логарифмов sin sinе – синус (sin) cos cosine – косинус (cos) tan tangent – тангенс (tg) ctn или cot cotangent – котангенс (ctg sec secant – секанс (sec) csc cosecant – косеканс (cosec) vers vеrsinе, versed sine – синус-верзус covers coversine, coversed sine – косинус-верзус sin ־¹ antisine – арксинус (arcsin) cos־¹ anticosine – арккосинус (arccos) sinh hyperboliс sine – синус гиперболический (sh) cosh hyperbolic cosine – косинус гиперболический (сh) tanh hyperbolic tangent – тангенс гиперболический (th) f(x) или (x) function of х – функция от х f' f primed – производная ∆х increment of х – приращение х ∑ summatiоn of – знак суммирования ∫ integral of – интеграл от a ∫ integral between the limits а and b – интеграл в пределах от а до b b ○ или ⊙ circ1e; сirсumfеrеnсе – круг; окружность
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AB length of 1iпe from А to В – длина отрезка АВ μ micron = 0,001 mm – микрон (10־³ мм) mμ millimiсrоn = 0,001μ – миллимикрон (10־⁷ см) º degree – градус ´ minute – минута ´´ second – секунда ＃ 1. № (номер), если знак предшествует числу; 2. англ. фунт, если знак поставлен после числа ⍧ centre line – центральная линия, линия центров 1st first – первый 2nd second – второй 3rd third – третий 4th fоurth – четвертый (все однозначные порядковые от 4 до 9 имеют окончание th) 5' 1. пять футов; 2. угол в 5 мин 9" 1. девять дюймов; 2. угол в 9 сек .5 (англичане и американцы иногда не пишут нуль uелых) 1.5 (англичане и американцы отделяют знаки десятичных дробей не запятой, а точкой, ставя ее вверху, в середине или внизу строки) 7,568 = 7568; 1,000,000= 10⁶ (англичане и американцы в многозначных числах отделяют каждые три цифры запятой) .0⁵103 = 00000103 = 0,00000103 (англичане и американцы иногда записывают, таким образом, для краткости малые дроби, впрочем, в большинстве случаев они пользуются общепринятой записью 10З x 10־⁵ ) 2/0, 3/0 и т. д. означают номера размеров проводов 00, 000 и т.д, согласно британскому стандартному калибру проводов (SWG) READING OF MATHEMAТICAL EXPRESSIONS
e «e to the power π »
a
b
t
d²y
ds² plus у times thе quantity l plus b of s, is equal to zеrо» tl
d²u
dt² equal to zero» PART III ADDITIONAL READING
DEFINITIONS
5. Signs and abbreviations. — The following signs and abbreviations are used in Algebra :— + plus, the sign of addition. − minus, the sign of subtraction. × into, or multiplied by, the sign of multiplication. ÷ by, or divided by, the sign of division. ~ the sign of difference ; thus, a~6 means the difference between a and b, whichever is the larger. = is, or are, equal to. .•. therefore. 6. The sign of multiplication is often expressed by a dot placed between the two quantities which are to be multiplied together. Thus, 2.3 means 2×3; and a. b means a × b. This dot should be placed low down, in order to distinguish it from the decimal point in numbers. Thus 3.4 means 3×4; but 3·4 means 3 decimal point 4, that is 3 + ·4. More often between letters, or between a number and a letter, no sign of multiplication is placed. Thus 3a means 3 × a; and bcd means b × c × d.
a Thus — means a ÷ b. For convenience in printing this line is sometimes b a written in a slanting direction between the terms ; thus a/b = —. b The words sum, difference, multiplier, multiplicand, product, divisor, dividend, and quotient are used in Algebra with the same meanings as in Arithmetic. 8. Expressions and terms. — Quantities in Algebra are represented by figures and by letters. The letters may have any values attached to them, provided the same letter always has the same value in the same question. The letters at the beginning of the alphabet are generally used to denote known quantities, and the letters at the end of the alphabet are used to denote quantities whose values are unknown. For example, in the expression ax + by — c, it is generally considered that a, b, and c denote known values, but x and y denote unknown values. An algebraical expression is a collection of one or more signs, figures, and letters, which are used to denote one quantity.
A simple expression consists of only one term. A compound expression consists of two or more terms. Thus a, bc, and 3d are simple expressions; and x + 3yz — 2xy is a compound expression denoting one quantity ; and x, 3yz, and 2xy are terms of the expression. A binomial expression is a compound expression consisting of only two terms; e.g., a+b is a binomial expression. A trinomial expression is a compound expression consisting of only three terms; e.g., a — b + c is a trinomial expression. A multinomial expression is a compound expression consisting of more than three terms.
When a term is preceded by no sign, the sign + is to be understood. The first term in an expression is generally positive, and therefore has no sign written before it. Thus, in a + 2b — 3c, a and 2b are positive terms, and 3c is a negative term.
Arithmetic that the operations of multiplication and division are to be performed before operations of addition and subtraction. For example — 2 × 3 + 4 ÷ 2 — 5 = 6 + 2 — 5 = 3. A similar sum in Algebra would be ab + — e. From the way in which this is written, the student would expect that he must multiply a by b, and divide c by d, before performing the operations of addition and subtraction. 10. Index, Power, Exponent.—When several like terms have to be multiplied together, it is usual to write the term only once, and to indicate the number of terms that have to be multiplied together by a small figure or letter placed at the right-hand top corner of the term. Thus:—
a² means a.a, or a × a. a3 means a.a. a, or a×a×a. a4 means a.a.a.a, or a× a× a × a. a2 is read a square; a3 is read a cube ; a4 is read a to the fourth power, or, more briefly, a to the fourth; a7 is read a to the seventh power, or a to the seventh; and so on. Similarly, (3a)4 = 3a × 3a × 3a × 3a= 81a4; and ab means that b a's are to be multiplied together. 11. Instead of having several like terms to multiply together, we may have a number of like expressions to multiply together. Thus, (b + c)3 means that b + c is to be multiplied by b + (b + c). This will be explained more fully when the use of brackets has been c, and the product multiplied again by b+c; i.e., (b + c)3 = (b + c) × (b + c) × explained. 12. The small figure or letter placed at the right-hand top corner of a quantity to indicate how many of the quantities are to be multiplied together is called an index, or exponent. This index or exponent, instead of being a number or letter, may also be a compound expression, or, in fact, any quantity; but we, at first, restricts ourselves to positive integral indices. We say, therefore, that an index or exponent is an integral quantity, usually expressed in small characters, and placed at the right-hand top corner of another quantity, to express how many of this latter quantity are to be multiplied together. A power is a product obtained by multiplying some quantity by itself a certain number of times. 13. Notice carefully that an index or an exponent expresses how many of a given quantity are to be multiplied together. For example, a5 means that five a's are to be multiplied together. In other words, the index expresses how many factors are to be used. The index, if a whole number, is always greater by one than the number of times that the given quantity has to be multiplied by itself. For example, the 5 in a5 expresses the fact that five factors, each equal to a, are to be multiplied together; or, in other words, that a is to be multiplied by itself four times. Thus, a5 = a × a × a ×a × a. This fact is often overlooked by beginners. 14. Factor, Coefficient, Co-Factor. — A term or expression may consist of a number of symbols, either numbers or letters, which are multiplied together. For example, the term 15a2bc consists of the numbers 3 and 5 and the letters a, a, b, c all multiplied together. A factor (Lat. facere, to make) of an expression is a quantity which, when multiplied by another quantity, makes, or produces, the given expression. In the above example 3, 5, a, b, c, and also 15, ab, ac, &c., are all factors of 15a2bc. For we may consider that 15a2bc = 3 × 5 a × a × b × c; or that 15a2bc = 15 × ab × ac; or that 15a2bc = 15 × a2 bc; or that 15a2 bc = ab × 15ac ; &c. 15. It is evident that the term 15a2 bc may be broken up into factors in several ways. Sometimes the factors of a quantity may be broken up again into simpler factors. Thus the factors 15 and a2 bc may be broken up again into 5 and 3 and into ab and ac; and ab and ac may be broken up again into a and b, and into a and c. When a quantity has been broken up into its simplest factors, these factors are called the simple or prime factors of the quantity. In whatever way we begin to break up a given integral quantity into factors, if we continue to break each factor into simpler factors as long as this is possible, we shall always arrive at the same set of simple factors from the same integral quantity. There is therefore only one set of simple or prime factors for the same integral quantity. In the above example the simple factors of 15a2bc are 3, 5, a, a, b, c. 16. When a quantity is broker up into only two factors, either of these factors may be called the Coefficient or Co-Factor of the other factor. For example, in 15a2bc we may call 15 the coefficient of a2bc, or 15a2 the coefficient of bc, or 3ab the coefficient of 5ac, &c. It is convenient, however, to use the word coefficient in the sense of numerical coefficient, and to speak of 15 as the coefficient of a2 bc in 15a2bc. In this sense the coefficient of a quantity is the numerical factor of the quantity.
19. Roots. — We have seen that a×a = a2. Here we multiply the quantity a by itself and so get a2. Suppose we reverse this process; that is, we have a quantity given us, and we try to find some quantity which, when multiplied by itself, will produce the given quantity. For example, what quantity multiplied by itself will give a2? Evidently, a is the required answer. Again, what number multiplied by itself will produce 16? Here 4 is the answer. In these cases we are said to find a root of a2, and of 16. A root of a given quantity is a quantity which, when multiplied by itself a certain number of times, will produce the given quantity.
The square root of a quantity is indicated by the sign √, which was originally the first letter in the word radix, the Latin for root. Thus, √16 = 4 ; √a2 = a.
roots of a quantity. Thus, √64 = 2, because 2×2×2×2×2×2 = 64. Similarly, ⁵√a5 = a; 7 √a7 = a ; ³√x3= x ; ³√y3 = y.
24. Brackets. — In Arithmetic each number, as, for example, 13, is thought of as one number. It is true that 13 is equal to the sum of certain other numbers; e.g., 6 + 4 + 3 = 13; but we do not necessarily consider 13 as made up of these numbers, 6, 4, and 3. So also in Algebra each expression must be considered as expressing one quantity, e.g., a + b — c represents the one quantity which is obtained by adding 5 to a and then subtracting c from the sum of a and b. So also each of the expressions in Exercises I a and I b represents one quantity. The answer to each of these examples is the numerical value of the example when the letters a, b, c, d, e, and f have the numerical values mentioned.
3 + 8 ÷ 2 — 2×3 = 3 + 4—6 = 1. Exactly the same rule applies in Algebra. For example, consider the expression a + bc — d÷ e + f. Here we must first multiply b by c, and divide d by e. Then we add the product to a, then subtract the quotient, and finally add f to get the result.
For example, in Arithmetic, (3 + 7) × 2 = 10 × 2 = 20. Here we treat 3 + 7 as one term, and therefore we place it in brackets. If we leave out the brackets, 3 + 7×2 = 3 + 14 = 17. Exactly the same thing is done in Algebra. For example, (a + b) ×c means that the sum of a and b is to be multiplied by c; whereas a+b×c means that first of all b is to be multiplied by c, and then the product is to be added to a.
30. In both these examples a negative answer can be expressed as a positive answer by altering the form of the answer. This can always be done with concrete quantities, and, in Arithmetic, whenever we arrive at a negative result, we transpose the form of the answer and express the result as a positive answer. In Algebra, however, it is convenient to leave a negative result and even to speak of a negative quantity without expressing any positive quantity. Thus we speak of —a, or of —3b, &c., as well as of +a or of +3b, &c. 31. The signs, therefore, + and — are used to distinguish quantities of opposite kinds. Every term in an algebraical expression and also every factor in every term must be thought of as being preceded by either + or —. If no sign is expressed, the sign + is understood. This use of the signs + and — is so constant and so important that + and — are often spoken of as the signs in an expression, and to change the signs in an expression means to change all + signs to —, and all — signs to +. For example, a + b—c is the same expression as -a—b+c with the signs changed. This use of + and — before each term must not be confused with the use of the same signs to mark operations of addition and subtraction. жүктеу/скачать 1.41 Mb. Достарыңызбен бөлісу:
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