Factors, coefficients and combining terms

Factors. If two or more numbers (arithmetic or literal) are multiplied the result of the multiplication is called a product. Each number that has been multiplied to arrive at¹ that product is called a factor of the product. For example, since 2*7=14, the 2 and 7 are factors of their products, 14. Similarly, the number 210 can be written as 2*3*5*7. Two, three, five and seven are called the prime factors of 210. A prime factor is a factor that is not divisible by anything other than² itself or unity. One factors³ 6 when he writes it in the form 2*3. 30 has 2, 3 and 5 as factors. Consider the number 6ab, which can be written as 2*3*a*b. Then 6ab has the following factors: 2, 3ab, a, b, 6 and so on.

When a product is broken down into its factors, it is broken down into numbers which, multiplied together, will equal the product. 2 and 2 are factors of 4. 2 and x are factors of 2x.

Coefficients. Any factor of a product may be called the coefficient of the product of the remaining factors. For example, in the expression 7xyz, 7 is the coefficient of the remaining factors xyz, or 7x is coefficient of the remaining factors yz, etc.

A coefficient which is an arithmetic number is called a numerical coefficient. Thus, 8 is the numerical coefficient in the expression 8xy. If a letter is written without a number before it, the coefficient is understood to be 1. For example, x means 1x, and ab means 1ab.

Combining Terms. An algebraic expression consists of one or more terms. If an algebraic expression consists of more than one term, as for example, 3a 2b c, the terms are separated by plus (+) or minus (—) signs.

A term or a monomial consists of numbers connected only by signs of multiplication or division. For example, 2xy and ab are terms or monomials. Thus, the algebraic expression 3x-2ab+4 has 3 terms: 3x, 2ab and 4.

The purpose of adding or subtracting numbers or objects is to find out⁴ how many of the same kind we have.

The sum of 3ab and 7ab is 10ab, because 3ab's and 7ab's more like them would yeild 10ab's. However, 2a and 3b cannot be added because these are unlike terms.

Like terms have the same literal factors. Thus, 3a and 5a are like terms, and xy and 4xy are like terms. Unlike terms do not have the same literal factors. 3d, 7x, 2y and 5xy are all unlike terms.

Adding and Subtracting Like Terms. An algebraic expression containing two or more terms can be simplified by combining like terms. Since unlike terms cannot be added or subtracted we merely indicate their addition or subtraction by signs. For example, 3x+6a-2b.

Notes:

¹ to arrive at — чтобы получить

² is not divisible by anything other than — не делится ни на что другое, кроме

³ to factor — разлагать на множители

⁴ is to find out — состоит в том, чтобы узнать

EXERCISES

I. Read the following words paying attention to the pronunciation:

combining, containing, remaining, following, addition, expression, multiplication, subtraction.

II. Form nouns and translate them into Russian:

to arrive, to multiply, to add, divisible, to consider, to subtract, to express.

III. Make up sentences of your own using the words and expressions given below:

anything other than, to be broken into numbers, to arrive at, remaining factors, cannot be added, unlike terms, to be simplified.

IV. Answer the following questions:

1. What is the result of multiplication called? 2. What numbers are called factors? 3. What coefficient is called a numerical coefficient? 4. When is the coefficient considered to be 1? 5. By what are the terms separated? 6. What is the purpose of adding or subtracting number? 7. What do like terms have? 8. How do we simplify an algebraic expression?

V. Translate into Russian:

Coefficient is a number or letter or symbol which has a fixed value; generally placed in front of a mathematical expression of letters or symbols and used as a multiplier. Common factor is a number, quantity or expression that divides exactly into two or more numbers, quantities or expressions.

VI. Translate into English

Одночлен представляет собой произведение числового множителя и букв. Числовой множитель, стоящий впереди буквенных множителей, называется коэффициентом. Так, например, число 4 является коэффициентом одночлена 4а²ху. Если произведение содержит несколько числовых множителей, то обычно перемножают их отдельно и полученное произведение ставят впереди буквенных сомножителей.

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THE FORMULA

The selling price of an article is equal to the sum of the cost of the article and the gain on the cost. If we let S. P. stand for the selling price¹, C for the cost and G for the gain, then a rule for finding the selling price can be written as the equation S.P.=C-G.

Often we have to write² the rule which is being expressed³ briefly as a formula. State the rule for the formula r = d/t. If r is the average rate, d is the distance, and t is the time, then the average rate is equal to the quotient of the distance divided by the time. Or, the average rate is equal to the distance divided by the time.

If the value of every letter but one in a formula is known, the value of the unknown letter can be found. This is known as calculating a formula for the unknown. To evaluate a formula we substitute numerical values for literal numbers, and solve the problem.

Notes:

¹ if we let S.P. stand for the selling price — если мы обозначим продажную цену буквами S. P.

² we have to write — мы должны написать

³ is being expressed — выражается

EXERCISES

I. Read the following words paying attention to the pronunciation:

rule, evaluation, calculation, expression, equal, quantity.

II. Write these words in the ing-form:

to sell, to be, to find, to gain, to write, to divide, to state, to calculate, to substitute, to provide, to call, to understand, to know, to determine, to value, to take, to vary

III. Make up sentences of your own using the words and expressions given below:

to stand for, the average rate, to calculate a formula, to solve the problem, to evaluate a formula.

IV. Translate into Russian:

Formula is a general expression for solving certain problems or cases. It is a relation established amongst quantities any one of which may be taken as the unknown if the other quantities are known or can be ascertained. In finding formulas, we are usually given the values of the related numbers to determine how each one relates to or depends upon the other. The related numbers are called variables, for their values vary. When we understand how the numbers vary, we can express in a formula the relationship between the variables.

V. Translate into English:

Часто нам приходится писать правила, которые выражаются кратко в виде формулы. Если значение всех букв в формуле за исключением одной (but one) известно, то можно найти значение неизвестной буквы. Чтобы вычислить формулу, мы заменяем буквенные выражения численными и решаем задачу.

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SYSTEMS OF TWO LINEAR EQUATIONS¹
IN TWO UNKNOWNS

Consider the equation

x — 2y = 5 (1)

In this equation x=7 and y=1, but also x=5 and y=0. There are many such pairs of values which satisfy the equation (1). To find pairs other than those given, choose a value of one letter, say y arbitrary, and then from (1) find the сcorresponding value of x. For example, let y=3. Then from (1)

x= 5 + 2y (2)

whence x=5+2*3, x— 5+6=11 and the pair of values x=11, y=3 satisfies the equation (1). The method for finding the pair of values satisfying both equations indicated above usually applies to pairs of equations of the form:

а₁x+ b₁у = c₁ (3)
а₂х + b₂у = c₂

where a_1, a_2, b_1, b₂, c_1, c₂ are known, and x and y are unknown quantities.

The equations (3) are termed linear because the unknown x and y enter to the first power only.

To solve a system of two linear equations in two unknowns, solve for one unknown in one equation and "substitute this result in the other equation, thus obtaining one equation in one unknown.

An alternative way² of solving a system of two linear equations, which is usually more convenient, is given by the following rule: multiply the two equations with numerical factors which are chosen so that³ the coefficient of one of the two unknowns have the same numerical values in both equations.

By adding or subtracting the two equations, a new equation with only one unknown quantity is obtained. Solve this equation. In order to find⁴ the second unknown quantity, substitute the value which has been found and solve for the
remaining unknown quantity. An alternative method for finding the second unknown is to repeat the above process of finding the equal coefficient for the other unknown.

Notes:

¹ equations in two unknowns — уравнения с двумя не
известными

² an alternative way — другой способ

³ so that — так что; таким образом, что

⁴ in order to find — для того, чтобы найти

EXERCISES

I.Read the following words paying attention to the pronunciation:

pair, where, there, compare, factor, letter, order, other, consider, enter, contain, obtain.

II.Underline all the suffixes and state to what part of speech the words belong:

equation, arbitrary, usually, convenient, corresponding, linear, equally, choosing, alternative, coefficient, numerical, system, factor.

III.Make up sentences of your own using the words and expressions given below:

in one unknown, in two unknowns, in three unknowns, to satisfy the equation, the method for finding, to obtain an equation, to establish.

IV. Answer the following questions:

1. What equations are termed linear? 2. What is the first operation in solving a system of two linear equations in two unknowns? 3. What do you obtain by adding or subtracting the two equations? 4. What operation do you perform to find the second unknown quantity?

V. Translate into Russian:

In order to solve two equations in two unknowns, it is necessary to eliminate one of the unknowns by combining the two equations into one equation, which only contains one of the unknowns. This simple equation is then readily solved for that unknown in the usual way. With one of the original unknowns now known, its value can be substituted for the symbol in one of the equations, and from the resulting simple equation, the other unknown can be found. There
are several methods of eliminating one of the unknowns and combining the two original equations into one.

IV.Translate into English:

Уравнением называется равенство, в котором одно или несколько чисел, обозначенных буквами, являются неизвестными.

Пусть, например, сказано, что сумма квадратов двух неизвестных чисел х и у равна 7; это можно записать при помощи следующего уравнения с двумя неизвестными

x² + у² = 7.

Уравнением первой степени с двумя неизвестными называется уравнение вида

ах + bу = с,

где х и у — неизвестные, а и b (коэффициенты при неизвестных) — данные числа, не равные оба нулю, с (свободный член — absolute term) — любое данное число.

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SQUARES AND SQUARE ROOTS

To square a number¹, you have learned, you must multiply that number by itself. The square root of a number is just the opposite. When you find the square root of a number, you are finding what number multiplied by itself gives you the number you began with². The sign for the square root is √. Thus, the square root of 25 is represented by √25. 25 is a perfect square. That is, a whole number (5) multiplied by itself will give you 25. Most numbers are not perfect

squares. In that case, to get the square root of a number we may either find it by taking an arithmetic square root or by using a table.

The process of finding a root is known as evolution; it is the inverse of involution, because by the aid of this process we try to find that which is given only when raising a number to a power (viz. the base of the power), while the data given is just what is sought for³ raising a number to a power (viz. the power itself). Therefore the accuracy of the root taken may always be checked by raising the number to the power⁴. For instance, in order to check the equality: ³√125=5, it is sufficient to cube 5; obtaining the quantity under the radical

sign, we conclude that the cube root of 125 has been found correctly.

Notes:

¹ to square a number — чтобы возвести число в квадрат

² the number you began with — зд. исходное число

³ what is sought for — искомое

⁴ by raising the number to the power — возведением числа в степень

EXERCISES

I.Read the following words paying attention to the pronunciation:

inverse, learn, perfect, order, for, opposite, not, must, number, thus.

II.Form Participles using the following verbs:

to square, to use, to raise, to multiply, to find, to check, to give, to begin, to obtain, to get, to take, to be.

III.Make up sentences of your own using the words and expressions given below:

to raise to power, to obtain the quantity, to square the number, to take an arithmetic square root, to use a table may be checked, conclude.

IV. Answer the following questions:

1. What operation should be performed to square a number? 2. What is a perfect square? 3. What do we do to get the square root of a number? 4. What is the process of finding a root called? 5. How do we check the accuracy of a root?

V. Translate into Russian:

Tables of squares are used by architects and engineers in working with squares of number. If you have a table of squares, you can find the approximate square root of any number. Sometimes it is not easy to find a square root by

inspection. If a table of squares is not at hand another method may be used.

V.Translate into English:

Чтобы возвести в квадрат число, надо умножить это число на самое себя. Извлечение квадратного корня — это действие обратное возведению в квадрат. Чтобы получить квадратный корень числа, мы можем пользоваться специальной таблицей. Правильность извлечения квадратного корня можно проверить, возведя в квадрат подкоренное

выражение; если получится данное число, то корень найден правильно.

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LOGARITHMS

An important step toward the lessening of the labour of computations was made in the seventeenth century by the discovery of logarithms. Logarithms permit us to replace long process of multiplication with simple addition; the operation of division with that of subtraction; the task of raising to any power with an easy multiplication; and extraction of any root is reduced to a single division.

The logarithm of a given number to a given base is the exponent of the power to which this base must be raised in order to obtain the given number.

Logarithms are exponents.

If a^x=b, the exponent is said to be the logarithm¹ of b to the base a, which we write x=log_ab.

The logarithm of a number to a given base is the exponent to which the base must be raised to yield the number.

Any positive number different from unity can be used as the base of a system of real logarithms.

Examples: If 10³ = 1000, then 3 = log₁₀1000

If 2³ = 8, then 3 = log₂8

If 5² = 25, then 2 = log₅25

The logarithmic system using 10 as a base is known as² the common or Griggs system and makes use of the fact that every positive number can be expressed as a power of 10. Since our number system uses 10 for a base, it is desirable for us to use 10 for the base of logarithms.

The following table shows the relationships between the exponential and logarithmic forms:

10³ = 1000 log₁₀1000 = 3.0000

10² = 100 log₁₀100 = 2.0000

10¹= 10 log₁₀10 = 1.0000

10° =1 log₁₀l = 0.0000

10^-1 = 0.l log₁₀0.1 = 1 or -1 or 9.0000-10

10^-2 = 0.01 log₁₀0.01 =2 or - 2 or 8.0000-10

From this table it is clear that any number between 100 and 1000 is a power of 10 for which the exponent is greater than 2 but less than 3, and consequently, its logarithm is between 2 and 3 (2+a decimal). Similarly, the logarithm

of 30 is (1+a decimal).

Later, when we use the table, we shall find log 30= 1.47712 which also means 10^1.47712=30. The positive decimal part of logarithm is called the mantissa and the integral part is called the characteristic.

Example. If log₁₀300=2.47712; 2 is the characteristic and 47712 is the mantissa.

Notes:

¹ the exponent is said to be the logarithm – говорят что показатель степени — это логарифм

² is known as — известна как

EXERCISES

I. Read the following words paying attention to the pronunciation:

coefficient, exponent, except, logarithmic, yield, application, multiplication, transformation.

II. Form adverbs and adjectives using the following suffixes and translate the newly formed words into English:

-ly: part, simple, easy, real, common, clear

-less: use, number, power

III. Answer the following questions:

1. When was the discovery of logarithms made? 2. Why was the discovery of logarithms an important step? 3 What operations can be replaced by logarithms? 4. What logarithmic system is known as the common or Griggs system? 5. What fact does the common logarithmic system make use of?

IV.Translate into Russian:

In giving logarithm of a number, the base must always be specified unless it is understood from the beginning that in any discussion a certain number is to be used as base for all logarithms. Any real number except 1 may be used as

base, but we shall see later that in applications of logarithms only two bases are in common use.

Suppose the logarithm of a number in one system is known and it is desired to find the logarithm of the same number in some other system. This means that the logarithm of the number is taken with respect to two bases. It is sometimes

Important to be able to calculate one logarithms when the other in known.

V.Translate into English:

Логарифм данного числа по данному основанию – это показатель степени, в которую это основание должно быть возведено для того, чтобы получить данное число. Логарифмическая система, у которой 10 — это основание, называется общей системой или системой Григса и основывается на том факте, что каждое положительное число может быть

выражено, как степень десяти.

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THE SLIDE-RULE

The slide rule (Fig. 1) presents one of the quickest and easiest ways of performing the operations of multiplication, division, raising to a power, and extracting a root, when not more than the first three or four digits of the result are necessary. Both straight and circular slide rules are in common use, but it is generally easier to learn to use the straight slide rule first. We represent here the fundamental principles of the straight (10 inches) slide rule as they are used in multiplication, division, reciprocals, proportion, squaring and extracting square roots.

Fig. 1

The straight slide rule is composed of many bodies or "frames", a slide or "slip stick", and an indicator. The slide is the main body. The indicator is the glass containing the hairline and is used to locate numbers in the various scales.

The Scales. The scales on a slide rule are graduated according to the mantissa of the positive real numbers, or, according to the logarithms of numbers from 1 to 10.

The top scale is divided into ten equal lengths (1 inch each for the 10-inch slide rule). The numbers from 1 to 10 on the bottom scale are located so that they correspond to their logarithms on the top scale. Note that the lengths be-tween the numbers on the bottom scale decreases in size as the numbers become larger. The bottom scale is called a logarithm scale.

EXERCISES

I.Read, the following words paying attention to the pronunciation:

slip, stick, ways, raise, indicating, according, corresponding, locate, scales.

II.Give the following nouns in the singular:

rules, glasses, inches, principles, reciprocals, bodies, mantissas, lengths.

III.Make up sentences of your own using the words and expressions given below:

the 10-inch slide-rule, the quickest way of performing the operation, the easiest way of performing the operation, to graduate a scale, to locate numbers, a slide.

IV.Answer the following questions:

1. What operations can we perform with the help of the slide rule? 2. What kind of slide rules do you know? 3. What is the straight slide rule composed of? 4. How are the scales on a slide rule graduated? 5. What scale is called a logarithm scale?

Geometry

Engineers, architects and people of many other professions use lines and figures in their daily work. The study of lines and closed figures made by lines is called geometry. Geometry is the branch of mathematics which investigates the relations, properties, and measurement of solids, surfaces, lines and angles.

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POINTS AND LINES

A point has no length, width or thickness. It merely indi­cates position. To represent a point in geometry we mark

a dot and label it with a capital letter. For example, A or -A would be called "point A".

A line has no width or thickness. It has length and direction. An infi­nite number of straight lines can be drawn through one point.

Since a line extends indefinite in either direction, we must with line segments, or portions in lines. The segment is represented by two capital letters, one placed at each end. The line segment AB or BA is shown in Fig. 2. It can also be represented by small letters. Hence a is line segment a.

A line joins two points. Only one straight line can be drawn between two points. There are three kinds of lines.*straight, curved and broken.

In Fig. 3 AB is a straight line; CD is a curved line; EF is a broken line. Notice that the lines are labeled by capital letters placed at the end of the line.

Lines that extend from left to right as the horizon are called horizontal lines. Examples of horizontal lines are lines on writing paper and all level lines which we find in man-made structures.¹

Note:

¹ man-made structures — постройки и сооружения, соз­данные руками человека

EXERCISES

I. Read the following words paying attention to the pronun­ciation:

length, thickness, width, thin, straight, draw, through, curve, there, that, position, representation, profession, either, or, more.

II. Form nouns adding suffixes and translate the newly formed words into Russian:

-tion: construct, represent, multiply, form;

-ment: displace, measure;

-ing: draw, study, find.

III. Make up sentences of your own using the words and ex­pressions given below:

daily work, closed figures, to be represented by, thickness to extend, indefinitely, labeled by, a broken line.

IV. Answer the following questions:

1. What is geometry? 2. What are the characteristic fea­tures of a point? 3. How do We represent a point in geometry? 4. How many lines can be drawn through one point? 5. What is a segment? 6. How many lines can be drawn between two points? 7. What kind of lines do you know?

V. Translate into Russian:

Geometry is the [branch of mathematics which investi­gates the relations, properties and measurements of solids, surfaces, lines and angles.

The two points may be at any distance apart, so a straight line may be considered as having any length.

A broken line is a line formed of successive sections, or segments, of straight lines.

A curved line, or simply a curve, is a line no portion of which is straight.

VI. Translate into English:

Через любые две точки можно провести прямую и при­том только одну. Если на плоскости взять какие-нибудь две точки и провести через них прямую линию, то все точки этой прямой будут находиться в этой плоскости.

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ANGLES

Measuring Angles. An angle is formed when two straight lines meet at a point. The lines are called the sides of an angle. The point at which the sides meet is called the vertex of the angle. The angle is read as angle BAC or CAB (Fig. 4).

The size of an angle depends upon¹ the amount one side has turned away from² the other. The length of the sides of an angle does not determine its size.

The unit of measure used in measuring an angle is the de­gree. A degree is a unit that equals 1/90 of a right angle and 1/360 of a circle. A right angle, therefore, contains 90 degrees (90°), and a circle contains 360 degrees (360°). The size of an angle is the number of degrees through which one side of the angle has turned away from the other side. Kinds of Angles.

Right Angle. If one side of an angle turns a quarter of a complete circle away from the other side, the angle that is formed is a right angle (Fig. 5). It contains 90°.

When two lines intersect at right angles, the lines are per­pendicular Each angle formed by a perpendicular line con­tains 90° (Fig. 6).

Complementary angles. When two angles put together form a right angle, and thus their sum is 90°, the angles a№ complementary. For example, angle DBC is the complemen­tary of angle ABC since their sum (60°+30°) equals 90° (Fig. 7).

Straight Angle. If one side of an angle turns half a comple­te circle away from the other side, the angle that is formed is a straight angle (Fig. 8). The sides of a straight angle lie in the same straight line. Notice _A that a straight angle is twice the size of³ a right angle since in a straight angle the side has made half a complete turn,⁴ or two quarter turns. The num­ber of degrees in a straight angle is miJ

Supplementary Angles. When the sum of two angles is 180°, the angles are said to be⁵ supplementary. For example, angle ABC is the supple­mentary angle of angle CBD since their sum (120^a+60°) is 180° (Fig. 9).

Acute Angle. If one side of an angle turns less than a quarter of a circle away from the other side, the angle formed is an acute angle (Fig. 10). An acute angle, therefore, is smaller than a right angle, or less than 90°.

Obtuse Angle. If one side of an angle turns more than a quarter of a circle but less than half a circle away from the other side, the angle formed is an obtuse angle (Fig. 11). Therefore, an obtuse angle is greater than a right angle but smaller than a straight angle. It contains more than 90º but less than 180°.

Reflex Angle. If one side of an angle turns more than half

a circle (180°) but less than a complete circle (360°) away from the other side, the angle formed is a reflex angle (Fig. 12). Therefore, a reflex angle is greater than A straight angle.

Notes:

¹to depend upon — зависеть от

²has turned away from — отклонен от

³twice the size of — вдвое больше (по величине)

⁴half a complete turn — половину полного оборота

⁵angles are said to be — про углы говорят, что они (являются)

EXERCISES

I. Read the following words paying attention to pronuncia­tion:

acute, obtuse, turn, use, unit, number, supplementary, complementary, other, but, reflex, vertex, axis, pointed, represented, straight.

II. Give words of the same root as:

Model: measure n measure v measurement v form, amount, turn, notice, determine, complement, contain.

III. Make up sentences of your own using words and expres­sions given below:

is said to be, is called, is formed, to depend upon, to turn away from, is less than, is more than, to turn more than a quarter, a degree, a circle.

IV. Answer the following questions:

1. When is an angle formed? 2. What do we call a point at which the sides of an angle meet? 3. What unit is used in measuring an angle? 4. What angles do you know? 5. How many degrees does an acute angle contain?

V. Translate into Russian:

The size of measure of an angle is determined by the amount of opening between the sides, and not by the lengths of the sides.

Two angles are said to be equal if they can be placed to­gether so that their vertexes are at the same point and the two sides of one coincide with the two sides of the other. This is a very important definition.

When several lines meet at one point to form more than one angle, any two of the angles which have one side in com­mon are said to be adjacent.

When a line is drawn through the vertex of an angle be­tween the sides it is said to divide the angle.

VI.Translate into English:

Угол образуется, когда две прямые линии встречаются в точке. Прямые линии называются сторонами угла, а точка, в которой они встречаются — вершиной угла.

Размер угла зависит от той величины (amount), на ко­торую одна сторона отклоняется от другой.

Градус — это единица измерения, используемая при измерении угла.

Углы бывают прямые, острые, тупые. Если одна сторона угла отклоняется на четверть полного круга от другой стороны, то образованный угол называется прямым углом. Прямой угол содержит 90°.

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