How the use of numbers began

Arithmetic

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HOW THE USE OF NUMBERS BEGAN

Many thousands of years ago this was a world without numbers. Nobody missed them. Everyone knew just what belonged to him and what not. If a cow was missing, the owner knew it was gone, not by counting cows, but for the same reason your mother would know if you did not come home for dinner.

But some people acquired more and more property. They would count¹ one cow, two cows, three cows; one vase, two vases and three vases; always one, two, three or more of something they owned or saw.

How far we have advanced from the time of our ancestors! Today, using numbers, numerals and mathematics, man builds bridges, skyscrapers, flies off the earth like a bird, even measures the distance to the moon and the brightness of the light given off by the firefly. But just as important² thought not so exciting, is that he can tell the time, pay the grocer, count the runs in a baseball game and use the same numbers in many different ways in everyday life.

So you see, mathematics and numbers, from simple arithmetic to complex algebraic and geometric calculations, are important to life in our time.

Roman Numerals. The Romans used seven capital letters to represent numbers³. They mixed them together to form many different combinations.

The Roman system of numbers is based upon⁴ the letters, I, V, X, C, D and M. This is what each letter represents:

Roman Numeral	I	V	X	L	C	D	M
Hindu-Arabic Numeral

Notes:

¹ they would count – они обычно считали

² just as important - так

³ to represent numbers – для обозначения чисел

⁴ to base upon – основываться на

EXERCISES

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

acquire, advance, ancestors, skyscrapers, firefly, exciting, brightness, light, algebraic, geometric.

II. Form nouns of the following words:

to count, to advance, to use, to build, to fly, to pay, to represent.

III. Form adjectives of the following words by adding the suffixes -full, -less, -able and translate them into Russian:

use, need, reason, count, represent.

IV. Answer the following questions:

1. When did people begin to count? 2. For what purposes do we use numbers? 3. Why are mathematics and numbers important? 4. What letters did the Romans use to represent numbers?

V. Translate into Russian:

Primitive man knew only ten number-sounds. The reason was that he counted in the way a small child counts today, one by one, making use of his fingers. The needs and possessions of primitive man were few: he required no large numbers. When he wished to express a number greater than ten he simply combined certain of the ten sounds connected with his fingers. Thus, if he wished to express "one more than ten" he said "one-ten" and so on.

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

acquire, property, advance, measure, important, calculations, for the same reason, just as important, to make use of, in everyday life, in the same way, in a different way.

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EXERCISES

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

to separate, period, system, zero, average, digit, unequal.

II. Form nouns of the following verbs:

to read, to count, to move, to place, to contain, to find, to determine, to represent.

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

quantity, unequal, sum, to make it easier to read, to separate the figures of the number, to be determined by, ten times as great, ten times as small.

IV. Answer the following questions:

1. Why do we separate the figures of the numbers by commas? 2. How is each group of three figures called? 3. How is the system of numbers we use called? 4. How many digits does a period of a number contain? 5. How do we find the average of unequal numbers?

V. Translate into Russian:

Our present-day number-symbols are Hindu characters. It is important to notice that no symbols for zero occur in any of this early Hindu number system. They contain symbols for numbers like twenty, forty, and so on. A symbol for zero had been indented in India. The invention of this symbol for zero was very important, because its use enabled the nine Hindu symbols 1, 2, 3, 4, 5, 6, 7, 8 and 9 to suffice for the representation of any number, no matter how great. The work of a zero is to keep the other nine symbols in their proper place.

VI. Translate into English:

Десятичная система нумерации возникла в Индии. Впоследствии ее стали называть «Арабской», потому что она была перенесена в Европу арабами. Цифры, которыми мы теперь пользуемся, тоже называются арабскими.

В этой системе особо важное значение имеет десять, и поэтому система носит называние десятичной системы нумерации.

Чтобы легче читать многозначные числа, мы отделяем (separate) цифры в них запятыми по три в группе. Группу из трех цифр мы называем периодом.

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EXERCISES

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

to add, addends, adding, to subtract, subtrahend, minuend, remainder, to multiply, product, dividend, divisor, quotient.

II. Give all possible derivatives of the following verbs:

to differ, to check, to answer, to change, to obtain.

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

the numbers to be added, the exercise to be checked, the work to be done, the number to be divided, can be made, can be divided, can be checked.

IV. Answer the following questions:

1. How is the result of addition called? 2. What do we do while adding a series of numbers? 3. Why do we sometimes make mistakes in adding numbers? 4. What is the result of subtracting whole numbers called? 5. How do we check a subtraction example? 6. What is the result of multiplication called? 7. What is the result of division called?

V. Make up 6 questions to the text and answer them.

VI. Translate into Russian:

Signs of Operations Used in Arithmetic The signs most used in arithmetic to indicate operations with numbers are plus (+), minus (—), multiplication (X), and division ( : ) signs. When either of these is placed between any two numbers it indicates respectively that the sum, difference, product, or quotient of the two numbers is to be found. The equality sign (=) shows that any indicated operation or combination of numbers written before it (on the left) produces the result or number written after it.

VII. Learn by heart:

Five times five are twenty five; five times six are thirty; five times seven are thirty five; five times eight are forty; five times nine are forty five; five times ten are fifty; five times eleven are fifty five; five times twelve are sixty; six times nine are fifty four; six times ten are sixty; seven times nine are sixty three, seven times ten are seventy; eight times nine are seventy two; eight times ten are eighty; nine times nine are eighty one; nine times ten are ninety.

VIII. Translate into English:

Числа, которые нужно сложить, называются слагаемыми, а результат сложения, т.е. число, получающееся от сложения, называется суммой.

Вычитанием называется действие, посредством которого (by means of which) по данной сумме и одному данному слагаемому отыскивается другое слагаемое.

Число, которое умножают, называется множимым; число, на которое умножают, называется множителем. Результат действия, т.е. число, полученное при умножении, называется произведением.

Число, которое делят, называется делимым; число, на которое делят, называется делителем; число, которое получается в результате деления, называется частным.

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FRACTIONS AND THEIR MEANING

A fraction represents a part of one whole thing. A fraction indicates that something has been cut or divided into a number of equal parts. For example, a pie has been divided into four equal parts. If you eat one piece of the pie, you have taken one part out of four parts. This part of the pie can be represented by the fraction 1/4. The remaining portion of the pie, which consists of¹ three of the four equal parts of the pie, is represented by the fraction² 3/4.

In a fraction the upper and lower numbers are called the terms of the fraction. The horizontal line separating the two numbers in each fraction is called the fraction line. The top term of a fraction or the term above the fraction line is called numerator; the bottom term or the term below the fraction line is called the denominator.

A fraction may stand for³ part of a group. There is a group 5 apples. Each is-1/5 (one fifth) of the group. If we take away 2 apples, we say that we are removing 2/5 of the number of apples present. If we take away 3 apples, we are removing 3/5 of the apples present. In this instance, a fraction is being used to stand for a part of a group.

A fraction also indicates division. For example: one apple was divided into eight parts and the man has eaten one part. Therefore he has eaten 1/8 of the apple. How much of the apple is left? How many eighths are in the whole apple?

Principle to Remember. If in any fraction the numerator and denominator are equal, the fraction is equal to 1.

Notes:

¹ to consist of — состоять из

² is represented by (the fraction) — представлено (дробью)

³ may stand for- может означать

EXERCISES

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

sign, piece, upper, numerator, denominator, number, fraction.

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

remaining portion, equal, to apply, to consist of, represented by, may stand for, is being used, to indicate.

III. Answer the following questions:

1. What does a fraction represent? 2. What do we call "the terms of fractions"? 3. What is the numerator? (denominator?) 4. What does a fraction indicate? 5. When is the fraction equal to 1?

IV. Translate into Russian:

The horizontal line separating the two numbers in each fraction is called the fraction line. The number above the fraction line is the numerator and that below is the denominator of the fraction. The denominator names the fractional unit and the numerator indicates the number of those units contained in the fraction.

V. Translate into English:

Дробь представляет собой часть целого. Число, стоящее над чертой, называется числителем дроби. Число, стоящее под чертой, называется знаменателем дроби. Числитель и знаменатель называется членами дроби.

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TYPES OF FRACTIONS

Common Fraction. A common fraction is a number that has the numerator and the denominator represented by numbers placed the one above and the other below a horizontal line. 3/7 is a common fraction.

Proper Fraction. If the numerator of a fraction is less than denominator the fraction is called a proper fraction. The value of a proper fraction is always less than 1. 6/7, 1/5 and 9/10 are proper fractions.

Improper Fraction. If the numerator of a fraction is equal to¹ or larger than the denominator, the fraction is called a improper fraction. The value of an improper fraction is equal to or larger than 1. 5/3, 3/2, 8/8 are improper fractions.

Mixed number. A number which consists of a whole number and a fraction is called a mixed number.2¼, 5¾, 9¼ are mixed numbers.

Reducing a Fraction to Lower Terms. For convenience and clarity a fraction must always be expressed in its simplest form. That is, it must be reduced to its lowest terms. To reduce a fraction to its lowest terms², divide the numerator and the denominator by the largest number that will divide into both of them evenly.

The process of crossing all common factors out of numerator and denominator is called the redaction of a fraction to its lowest terms. The greatest (largest) quantity which is a common divisor of two or more quantities is called a greatest common divisor of these quantities. It is written G.C.D.

Notes:

1 is equal to – равно

2 to reduce a fraction to its lowest terms —здесь, для того, чтобы сократить дробь

EXERCISES

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

reduce, value, both, other, mixed, proper, improper.

II. Give all possible derivatives of the following words:

value, convenience, to represent, to express, to divide.

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

evenly, to reduce to, for the convenience, expressed in, is equal to.

IV. Answer the following questions:

1. What is a common fraction called? 2. What is a proper fraction called? 3. Is the value of a proper fraction more or less than 1? 4. What do we call mixed numbers? 5. How do you reduce a fraction to its lower terms?

V. Put 6 questions to the text.

VI. Translate into Russian:

Fractions indicate division, the numerator being a dividend, the denominator a divisor, and the value of the fraction the quotient.

A fraction can be reduced to lower terms if the numerator and the denominator are divisible by a single number, that is if they have a common divisor. In order to reduce a fraction to its lowest terms, therefore, it is seen at once that the greatest common divisor must be used.

VII. Translate Into English:

Дробь, у которой числитель меньше знаменателя, называется правильной дробью. Правильная дробь меньше единицы.

Дробь, у которой числитель равен знаменателю или больше его, называется неправильной дробью. Таким образом, неправильная дробь или равна единице, или больше ее.

Числа, которые состоят из целого числа и дроби, называются смешанными числами.

Сокращением дроби называется замена ее другой, равной ей дробью с меньшими членами, путем деления числителем и знаменателем на одно и тоже число. Это число называется наибольшим общим делителем.

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EXERCISES

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

subtract, unlike, multiply, result, change, cross, equivalent, quotient.

II. Form verbs of the following nouns:

subtraction, multiplication, division, addition, difference, equality.

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

to change to, to cross out, must be changed, can be done, divide the result by, write over, write under.

IV. Answer the following questions:

1. What should one do in order to add fractions having the same denominator? (different denominators?) 2. What should one do in order to subtract fractions having the same denominator (different denominators?) 3. How do you multiply fractions having the same (different) denominators? 4. How do you multiply a mixed number and a fraction?

V. Put 6 questions to the text and answer them.

VI. Translate into Russian:

When fractions have a common denominator, they can be added by simply adding the numerators and writing the sum over the same denominator. Any fractions with a common denominator are subtracted by subtracting the numerator of the subtrahend fraction from that of the minuend fraction, and writing the remainder over the common denominator to form the remainder fraction. Thus to add or subtract fractions, first change them into ones with the L.C.D., and then add or subtract the numerators, writing the result as the numerator of a fraction with the common denominator. This fraction is the desired sum or difference respectively.

To multiply a fraction by a whole number, multiply the numerator by that number, and write the product as the numerator of a new fraction with the same denominator. This fraction is the desired product. In order to divide a fraction by any number, multiply the denominator by that number.

VII. Translate into English:

Чтобы сложить дроби с одинаковыми знаменателями, надо сложить их числители и оставить тот же знаменатель.

Чтобы сложить дроби с разными знаменателями, нужно предварительно привести их к наименьшему общему знаменателю, сложить их числители и написать общий знаменатель.

Чтобы вычесть дробь из дроби, нужно предварительно привести дроби к наименьшему общему знаменателю, затем из числителя уменьшенной дроби вычесть числитель вычитаемой дроби и под полученной разностью написать общий знаменатель.

Чтобы умножить дробь на целое число, нужно умножить на это целое число числитель и оставить тот же знаменатель.

Чтобы разделить дробь на целое число, нужно умножить на это число знаменатель, а числитель оставить тот же.

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CHANGING FRACTIONS

The numerator and denominator of a fraction may be multiplied by the same number without changing the value of the fraction. The resulting equivalent fraction is actually the same fraction expressed in higher terms.

To change a mixed number to an improper fraction we must: 1) multiply the denominator of the fraction by the whole number; 2) add the numerator of the fraction to the product of the multiplication; 3) write the result over the denominator.

To change an improper fraction to a whole or a mixed number we must divide the numerator by the denominator. If there should be a remainder, write it over the denominator. The resulting fraction should then be reduced to its lowest terms.

To change a whole number to an improper fraction with a specific denominator: 1) multiply the specific denominator and whole number; 2) write the result over the specific denominator.

Comparing Fractions. Fractions can be compared. To compare unlike fractions we must change them to equivalent fractions so that all have like denominators.

When fractions have different numerators but the same denominator, the fraction having the largest numerator has the greatest value.

When fractions have different denominators but the same numerator, the fraction having the largest denominator has the smallest value.

EXERCISES

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

express, high, whole, compare, specific, actually.

II. Give as many derivatives as you can of the following words:

to express, actually, result, high, to remain.

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

expressed in higher terms, mixed number, to change a fraction, to change a whole number, can be compared, can be multiplied, without changing, without dividing, without comparing.

IV. Answer the following questions:

1. What is an equivalent fraction? 2. How do you change a mixed number to an improper fraction? 3. How do you change an improper fraction to a whole number or mixed number? 4. How do you change a whole number to an improper fraction with a specific denominator? 5. What must you do to compare unlike fractions? 6. How do you compare fractions?

V. Put 6 questions to the text and answer them.

VI. Translate into Russian:

When denominators and numerators of different fractions are both different, the values of the fractions cannot be compared until they are converted so as to have the same denominators.

Since fractions indicate division, all changes in the terms of a fraction (numerator and denominator) will affect its value (quotient) according to the general principles of division. These relations constitute the general principles of fractions.

VII Translate into English:

Чтобы обратить смешанное число в неправильную дробь, нужно целое число умножить на знаменатель дроби, к произведению прибавить числитель и сделать эту сумму числителем искомой (sought for) дроби, а знаменатель оставить прежним.

Чтобы обратить неправильную дробь в смешанное число, нужно числитель дроби разделить на знаменатель и найти остаток. Частное покажет число целых единиц; остаток взять в качестве числителя, а знаменатель оставить прежним.

Если числитель дроби уменьшить в несколько раз, не изменяя знаменателя, то дробь уменьшится во столько же раз.

Если числитель и знаменатель дроби увеличить в одинаковое число раз, то дробь не изменится.

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DECIMAL FRACTIONS

A decimal fraction is a special type of fraction written without a denominator (which is 10 or a power of 10) but in which the number of figures on the right-hand side of a dot, called the decimal point, indicates whether the denominator is 10 or a higher power of 10; e.g. 2/10 is written as a decimal in the form 0.2, 23/100 as 0.23, and 23/1000 as 0.023.

If any figure of the number is moved one place to the left, the value of the number is multiplied by 10.

To round off a decimal to a particular, inspect the figure to the right of the required place: if it is 5 or over, change the last required digit to the next higher figure and drop all decimals to the right of the required figures, if it is less than 5, drop all decimals to the right of the required figure.

To compare decimal fractions, annex zeroes so that the decimals have the same number of places.

A decimal fraction may be changed to a common fraction by: leaving out the decimal point, writing the decimal number as the numerator and the number shown by the name of the last decimal place as the denominator.

EXERCISES

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

special, place, particular, decimal, indicate, without, whether, higher, power, leave, greatest, figure, require, determine.

II. Form nouns of the following verbs:

to determine, to place, to indicate, to move, to inspect, to require, to compare.

III. Answer the following questions:

1. What is the decimal fraction? 2. How do we write decimal fractions? 3. How do you round off a decimal to a particular place? 4. How do you compare decimal fractions? 5. How do you change decimal fractions?

IV. Put 4 questions to the text and answer them.

V. Translate into Russian:

Since fractional parts of thing may be written either as common fractions or as decimal fractions, we should be able to change a decimal fraction to a common fraction.

To change a decimal fraction to a common fraction take out the decimal point and write the decimal number as the numerator of the fraction. For the denominator write the number as shown by the name of the last decimal place. Reduce the common fraction to lowest terms.

VI. Translate into English:

Дроби, знаменателями которых являются числа, выраженные единицей с последующими нулями (одним или несколькими), называются десятичными. Из двух десятичных дробей та больше, у которой число целых больше; при равенстве целых та дробь больше, у которой число десятых больше и т. д.

Перенесение запятой на один знак вправо увеличивает число в десять раз. Чтобы увеличить его в сто раз, нужно перенести запятую на два знака вправо и т. д.

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EXERCISES

I Read the following words paying attention to the pronunciation:

hundredth, tenths, arrange, column, subtrahend, annexed.

II Give all possible derivatives of the following words:

to arrange, to start, product, power.

III Form adverbs of the following words by adding suffixes:

power, use, part, equal.

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

in multiplying, in dividing, in subtracting, in the dividend, in the same way, mark off, since, must be written.

V. Answer the following questions:

1. How are decimal fractions added? 2. How do we write decimal fractions when we want to subtract them? 3. How do we check the answer? 4. How do we multiply (divide) decimal fractions? 5. How do we arrange numbers in adding decimal fractions?

VI. Put 5 questions to the text and answer them.

VII. Translate into Russian:

Decimals are fractions which always have a denominator of 10 or some power of 10, such as 100, 1000, etc. The denominator is usually not written down; but a dot or point called the "decimal point" is placed to the right of the digit in the numerator, which is distant from the extreme left of these digits by the number of zeroes in the denominator.

In addition or subtraction of decimals, the decimal points must be placed in a straight one under the other.

In multiplication of decimals, point off as many places in the product as there are decimal places in the multiplier and the multiplicand.

VIII. Translate into English:

При сложении десятичных дробей надо соблюдать следующий порядок: дроби записывают одну под другой так, чтобы во всех слагаемых одинаковые разряды находились друг под другом и все запятые стояли в одном и том же вертикальном столбце; справа от десятичных знаков некоторых слагаемых приписывают такое число нулей, чтобы все слагаемые после запятой имели одинаковое число цифр.

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WHAT IS PER CENT?

We have already learned two ways of writing fractional parts: common fractions and decimal fractions. Another method is by using per cents.

Per cent tells the number of parts in every hundred. This number is followed by the per cent sign (%). The word "per cent" and sign % actually refer to the denominator of a fraction expressed as hundredths.

When working with per cent, we do not write the word, but use the sign %, 20 per cent is written 20% and so on.

In working with problems involving percentage¹ we must be able to change per cents to decimals and decimals to per cents.

We can change a per cent to a decimal by dropping the per cent sign and moving the decimal point two places to the left. We can change a per cent to a common fraction with the given number as the numerator and 100 as a denominator.

One hundred per cent of quantity is the entire quantity. To find a per cent of a number, change the per cent to the equivalent decimal fraction or common fraction and multiply the number by the fraction. To find the per cent of one number from a second number, form a fraction in which the first number is the numerator and the second number is the denominator. Divide the numerator into the denominator and change the decimal fraction to a per cent. To find a number when a per cent of it is known, change the per cent to an equivalent decimal fraction or common fraction, divide given number by this fraction.

Note:

¹ problems involving percentage — задачи на нахождение процента

EXERCISES

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

entire, original, increase, per cent, percentage, method.

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

by using of, to find a per cent of, is followed by, must be able to change, must be able to use, must be able to find, be able to explain. .

III. Answer the following questions:

1. What methods are there for writing fractional parts? 2. Where do we put the per cent sign? 3. What does the sign % actually refer to? 4. How do we change a per cent to a decimal fraction?

IV. Put 5 questions to the text and answer them.

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SCALE DRAWING

When an architect makes plan¹ or blueprint of a house, his drawing is much smaller than the actual house. The plan is reduced in size to fit the paper he is using. This process of reducing in size is called drawing to scale. The reduced drawing is known as a scale drawing.

In a scale drawing each line is a definite fractional part of the line it represents. A line in the scale drawing may be one-half of the line it represents, one-forth of it, one-hundredth, one-thousandth, or any other definite part of it. In a scale drawing the scale may be written: 1´´/4=1´. This means that every ¼-inch length on the scale drawing represents a 1-foot length of the original object.

A scale drawing has the same shape as the original, but not the same size. Thus, we say that scale drawing is similar to the actual object.

The maps printed in history or geography books are also scale drawings with all distances in the same ratio to the corresponding distances in the original. For example, on a map the scale may be given as 1´´/2 = 50 miles. On another map the scale may be 1´´-1000 miles. This means that the actual distance between two points which are 1½ apart a map whose scale reading is 1´´/2=50 miles, is 150 miles.

Note:

¹ to make a plan – составлять план, зд. чертить план

EXERCISES

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

architect, actual, drawing, scale, blueprint, geography, progress, reduce.

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

to take a plan, to make a map, to make a blueprint, to fit, to draw to scale, to be similar to.

III. Put 6 questions to the text and answer them:

IV. Translate into Russian:

In choosing a scale, we always pick one that is convenient to work with—not too large for the paper we are drawing, nor too small to measure. The scale depends upon the size of the original object and how much it must be reduced.

Once a scale is chosen, the same scale must be used in drawing all parts of the same object. Whenever a scale drawing is made, the scale being used must be stated. It is usually written at the bottom of the drawing.

V. Translate into English:

Начертить план какого-либо предмета на бумаге в натуральную величину иногда бывает трудно. Если размеры предмета больше листа бумаги, на котором этот предмет нужно изобразить, то такой предмет чертят в уменьшенном виде. Если архитектор составляет план дома или чертеж « на синьке», то его чертеж должен быть намного меньше, чем действительный дом. Уменьшение размера называется черчением в масштабе. В масштабном чертеже каждая линия — это определенная часть той линии, которая существует в действительности.

TEXT

GRAPHS

A graph represents numerical relationship in visual form. By use of a graph we can show the relation between certain sets of numbers in an interesting, pictorial manner so that they can actually be seen.

The most commonly used graphs are: the pictograph, the bar graph, the line graph and the circle graph. In a pictograph, each picture or symbol represents a definite quantity. In a pictograph we use pictures of objects to represent numbers. The length of bars in a bar graph represents numerical facts. The bars are of varying length but of the same width. They are usually used to show¹ size or amount of different items or size or amount of the same item at different times. The bars of a vertical bar graph are drawn straight up and down, that is at right angles with the horizontal base line of the graph. The bars of a horizontal bar graph are drawn across the page.

The line graph shows the changes in a quantity by the rising or falling of a line. The position of the line with relation to² the horizontal and vertical scales represents numerical facts. The line connects a number of points.

An apportionment or distribution graph shows the relationship of all parts of a particular whole. The whole graph represents 100%. A chart which consists of a circle broken down into subdivision is called a circle graph. A circle graph is used to show how all the parts are related to the whole. The entire circle, which equals 360°, represents the entire thing.

Notes:

¹ are used to show – используются для того, чтобы показать

² with relation to – по отношению к

EXERCISES

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

graph, pictograph, circle, straight, right, visual, present, item, interesting, time, entire, page, change, chart.

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

can present, visual form, with relation to, is used to show, bar graph, line graph, pictograph.

III. Answer the following questions:

1. What does a graph present? 2. What can we do by using a graph? 3. What are the most commonly used graphs? 4. What is the difference between a pictograph and the bar graph? 5. How are the bars of a vertical (horizontal) graph drawn? 6. What do we call a circle or a line graph?

IV. Put 6 questions to the text.

V. Translate into Russian:

Graphs are used very frequently in newspapers, magazines, textbooks and reference books. Graphs picture facts and figures so clearly that one can understand them at a glance. Graph is the picture of mathematical equation. It is a method of showing on squared paper the changes in value of an expression containing unknown quantities when one of the unknown quantities is given various, definite values. Any other unknown quantity is dependent in some way on the value of the first unknown quantity, which is called the independent value.

VI. Translate into English:

Наиболее часто встречающиеся диаграммы — это пиктограммы, диаграммы в виде столбцов, диаграммы в виде круга и линейные диаграммы.

Распределительная диаграмма показывает соотношение всех частей одного целого.

Диаграммы обычно используются, чтобы показать размер или количество различных предметов или размер и количество одного и того же предмета в разное время.

ALGEBRA

TEXT

THE NATURE OF ALGEBRA

Algebra is a generalization of arithmetic. Each statement of arithmetic deals with¹ particular numbers: the statement (20+4)²=20²+2*20*4+4²=576 explains how the square of the sum of the two numbers, 20 and 4, may be computed².It can be shown³ that the same procedure applies if the numbers 20 and 4 are replaced by any two other numbers. In order to state the general rule, we write symbols, ordinarily letters, instead of⁴ particular numbers. Let the number 20 be replaced⁵ by the symbol a, which may denote any number, and the number 4 by the symbol b. Then the statement is true⁶ that the square of the sum of any two numbers a and b can be computed by the rule (a+b)²=a²+2a*b+b².

This is a general rule which remains true no matter what⁷ particular numbers may replace the symbols a and b. A rule of this kind is often called a formula.

Algebra is the system of rules concerning the operations with numbers. These rules can be most easily stated as formulas in terms of letters, like the rule given above for squaring the sum of two numbers.

The outstanding characteristic of algebra is the use of letters to represent numbers. Since the letters used represent numbers, all the laws of arithmetic hold for⁸ operations with letters.

In the same way, all the signs which have been introduced to denote relations between numbers and the operations with them are likewise used with letters.

For convenience⁹ the operation of multiplication is generally denoted by dot as by placing the letters adjacent to each other. For example, a*b is written simply as ab.

The operations of addition, subtraction, multiplication, division, raising to a power and extracting roots are called algebraic expressions.

Algebraic expressions may be given a simpler form by combining similar terms. Two terms are called similar, if they differ only in their numerical factor (called a coefficient).

Algebraic expressions consisting of more than one term are called multinomial’s. In particular, an expression of two terms is a binomial, an expression of three terms is a trinomial. In finding the product of multinomial’s we make use of the distributive law.

Notes:

¹ to deal with – иметь дело с; рассматривать

² may be computed – может быть вычислена

³ it can be shown – можно показать

⁴ instead of – вместо

⁵ let the number 20 be replaced – давайте заменим число 20

⁶ then the statement is true – тогда справедливо утверждение

⁷ no matter what – не зависимо от того, какой

⁸ to hold for – годится для

⁹ for convenience – для удобства

EXERCISES

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

concern, length, letters, generally, mental, check, arithmetic, width, inch, its, division, addition, which, consider, close, total, cost, only.

II. Form nouns and translate them into Russian:

add, divide, multiply, subtract, operate, state, express, represent, introduce.

III. Form adverbs of the following words by adding the suffix -ly and translate them into Russian:

general, ordinary, particular, simple, similar, different.

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

to deal with, concerning, it can be shown, may be computed, remains true, for convenience, to square, in particular, to extract a root, in terms of letters.

V. Answer the following questions:

1. What is the relationship between arithmetic and algebra? 2. In what operations in arithmetic do we use numbers? 3. What do we use in algebra to represent numbers? 4. What may a formula be considered? 5. What examples of the close relationship between arithmetic and algebra can you give?

VI. Translate into Russian:

Algebra is used in a many walks of life, from that of the philosopher to that of the manual labor. The skilled worker may use algebra to determine the location of the centre or the size of holes he must drill. Doctors, engineers, and scientists use algebra in their research.

But the use of algebra we can reduce complex problems to simple formulas. We can find the answer to problems about the universe, and problems of sewing, building, cooking, measuring, buying and selling as well.

VII. Translate into English:

Алгебра – это система правил, касающихся действий с числами. В алгебре числа обозначаются буквами, а не цифрами. Поскольку буквы обозначают числа, все законы арифметики годны для действий с буквами. Знаки, которые означают действия с цифрами, также употребляются для букв.

TEXT

SIGNS USED IN ALGEBRA

In algebra, the signs plus (+) and minus (-) have their ordinary meaning, indicating addition and subtraction and also serve to distinguish¹ between opposite kinds of numbers, positive (+) and negative (-). In such an operation as +10-10=0, the minus sign means that the minus 10 is combined with the plus 10 to give a zero result² or that 10 is subtracted from 10 to give a zero remainder.

The so-called “double sign” (±), which is read “plus-or-minus”, is sometimes used. It means that the number or symbol which it precedes may be “either plus or minus”³ or “both plus and minus”⁴.

As in arithmetic, the equality sign (=) means “equals” or “is equal to”.

The multiplication sign (×) has the same meaning as in arithmetic. In many cases, however, it is omitted. A dot (·) placed between any two numbers a little above the line (to distinguish it from a decimal point) is sometimes used as a sign of multiplication.

The division sign (÷) has the same meaning as in arithmetic. It is frequently replaced by the fraction line; thus 6/3 means the same as 6÷3 and in both cases the result or quotient is 2. The two dots above and below the line in the division sign (÷) indicate the position of the numerator and denominator in a fraction, or the dividend and divisor in division.

Parentheses ( ), brackets [ ], braces { }, and other inclosing signs are used to indicate that everything between the two signs is to be treated as⁵ a single quantity and any sign placed before it refers to everything inside as a whole and to every part of the complete expression inside.

Another sign which is sometimes useful is the sign which means “greater than” or “less than”. The sign (>) means “greater than” and the sign (<) means “less than”. Thus, a>b means that “a is greater than b”, and 3<5 means “3 is less than 5”.

The sign .·., three dots at the corners of a triangle, means “hence” or “therefore”.

Notes:

¹ serve to distinguish – служат для того, чтобы различить

² to give a zero result – дать в результате нуль

³ either plus and minus – либо плюс, либо минус

⁴ both plus and minus – как плюс, так и минус

⁵ is to be treated as – следует рассматривать как

EXERCISES

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

algebra, also, double, triangle, product, quotient, quantity, frequently, sign, minus, combine, twice, inside, sum, number, meaning, between, complete, parentheses, arithmetic, fraction, subtraction, operation.

II. Form nouns of the following words:

to indicate, to add, to operate, to subtract, to mean, to express, to divide, to place, to differ.

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

serve to distinguish, to give a zero result, to give a zero remainder, combine with, both plus and minus, either plus or minus.

IV. Answer the following questions:

1. What signs are used in algebra? 2. What do signs (+) and (-) indicate? 3. How is the sign (±) read? 4. What is the equality sign? 5 What is the meaning of the multiplication sign? 6. What is the meaning of the division sign? 7. What does the expression (a+b) mean?

V. Translate into Russian:

ab means the same as a×b and 2×c means the same as 2c, twice c. We cannot write 23, however, for 2×3 as 23 has another meaning, namely, the number twenty three. Therefore, in general, the multiplication sign (×) may be omitted between algebraic symbols or between an algebraic symbol and an ordinary arithmetical number, but not between two arithmetical numbers.

Another sign which is sometimes used is the inclined fraction line (/); thus 6/3 means the same as 6:3. This form has the advantage of being compact and also allowing both dividend and divisor (or numerator and denominator) to be written or printed on the same line.

VI. Translate into English:

В алгебре мы применяем следующие знаки: плюс, минус, знак равенства, знак умножения, знак деления, скобки круглые, квадратные и фигурные, знак «больше, чем», знак «меньше, чем» и другие. Знак три точки по углам треугольника означает «следовательно» или «поэтому».

TEXT

EQUATIONS

An equation is a statement of equality. The statement may be true for¹ all values of the letters.

The value of the letters for which the equation is true is the root or solution of the equation.

When a statement of equality of this kind is given, our interest is in finding² the value of the letter for which it is true. The following rules aid in finding the root.

1. The roots of an equation remain the same if the same expression is added to or subtracted from both sides of the equation.

2. The roots of an equation remain the same if both sides of the equation are multiplied or divided by the same expression other than zero and not involving the letter whose value is in question³.

The equation 2x = 4 where x is the unknown, is true for x=2. To illustrate the first of the above two rules, add 5x to both sides of the equation 2x = 4. We get 2x+5x = 4+5x which, like equation 2x = 4 is true for only x = 2. To illustrate the importance of the restriction in the second of the above two laws, multiply both sides of the equation by x and get (2x)x = (4)x which is true not only for

x = 2 but also for x = 0.

It is always a good plan to check the accuracy⁴ of one's work by substituting the result in the original equation to see whether the equation is true for this value.

Rule 1 is applied very frequently. It is, therefore, desirable to state it in a way which mechanizes its application.

If the equation 4x = 28-3x is given, in applying Rule 1, 3x may be added to both sides of the equation, yielding 4x+3x = 28-3x+3x = 28.

The result of the operation consists in omitting⁵ the term +3x to the left side. We call this operation transposition of the term 3x. This operation is an application of Rule 1 and may be explained in the following way:

Any term of one side of an equation may be transposed to the other side if its sign is changed.

Example. Find the value of x which satisfies 3x+ 7(4-x)+6x = 15. Clearing of parentheses and combining terms:

3x + 28 – 7x+ 6x = 15,

2x+28 =15.

Transposing +28 from the left side:

2x = 15 - 28,

2x = - 13.

Dividing each side by 2, according to Rule 2:

2x/2=-13/2; x=-13/2.

An equation which can be reduced to the form ax+b =0 (a≠0), is called a linear equation in x.

To solve an equation containing fractions, first reduce each fraction to its lowest terms. Then multiply each side of the equation by the least common denominator of all the denominators. This process is called clearing of fractions.

A quadratic equation is one which can be reduced to the form ax²+bx+c = 0 (a≠0) where a, b and c are known and x is unknown.

Notes:

¹ may be true for – может быть действительным для, пригодным для, верным, справедливым

² our interest is in finding – нам интересно найти

³ in question – искомое (которое неизвестно)

⁴ to check the accuracy – чтобы проверить точность

⁵ consists in omitting – состоит в устранении

EXERCISES

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

subtract, illustrate, result, where, which, parentheses, other, whether, always, way, statement, remain, same, equation.

II. Give the Infinitives of the following words:

is, given, finding, got, saw, known.

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

may be true for, in finding, in question, by substituting, to consist in omitting, a linear equation, the least common denominator.

IV. Answer the following questions:

1. What is an equation? 2. What are the expressions on either side of the sign of equality called? 3. What should be done to keep the balance of the equation? 4. How do we check an equation? 5. What operations must one do when solving an equation by the combination of rules?

V. Translate into Russian:

If the same number is added to each side of an equation, the equality of the two sides is not altered. If the same number is subtracted from each side of an equation, the equality of the two sides is not altered. If both sides of an equation are multiplied by the same number, the equality of the two sides is not altered. If both sides of an equation are divided by the same number, the equality of the two sides is not altered.

VI. Translate into English:

Корень уравнения остается прежним, если к обеим частям уравнения прибавить или от обеих частей уравнения вычесть одно и то же выражение. Корень уравнения остается прежним, если обе части уравнения умножить или разделить на одно и то же выражение.

Решить уравнение - значит найти те (those) значения неизвестного, при которых обе части уравнения равны одному и тому же числу (другими словами, все те значения неизвестного, при которых равенство будет верным). Говорят, что эти значения неизвестного удовлетворяют уравнению. Значения неизвестного, которые удовлетворяют уравнению, называются корнями или решениями уравнения.

TEXT

MONOMIAL AND POLYNOMIAL

Algebraic expressions are divided into two groups according to the last algebraic operation indicated.

A monomial is an algebraic expression whose last operation in point of order is neither addition nor¹ subtraction.

Consequently, a monomial is either a separate number represented by a letter or by a figure, e.g. -a, +10, or a product, e.g. ab, (a+b)c, or a quotient, e.g. (a-b)/c, or a power, e.g. b2, but it must never be either a sum or a difference.

If a monomial is a quotient, it is called a fractional monomial; all the other monomials are called integral monomials. Thus, (a-b)/c is a fractional monomial, while (x—y)ab; a(x+y)² are integral monomials.

An algebraic expression which consists of several monomials connected by the + and - signs, is known as a polynomial². Such is for instance, the expression

ab-a+b-10+(a-b)/c.

Terms of a polynomial are separate expressions which form the polynomial by the aid of the + and — signs. Usually, the terms of a polynomial are taken with the signs prefixed to them; for instance, we say: term -a, term +62, and so on. When there is no sign before the first term it is ab or +ab.

A binomial is an algebraic expression of two terms; a trinomial is an expression of three terms and so on.

Notes:

¹ neither ...nor – ни…ни

² is known as a polynomial – известен как многочлен, называется многочленом

EXERCISES

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

fractional, integral, binomial, trinomial, monomial, polynomial, divided, indicated, represented, connected.

II. Give words of the same root as:

Model : operate v; operation n; operative a

serve, express, indicate, divide, represent, connect.

III. Point out the nouns, adjectives and adverbs and write them down in three columns:

algebraic, integral, addition, last, while, point, several, separate, sign, easily, fractional, difference, term.

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

neither addition nor subtraction, neither sum nor difference, either monomial or polynomial, either multiplication or division.

V. Answer the following questions:

1. Into how many groups are algebraic expressions divided? 2. What is a monomial algebraic expression? 3. By what is a monomial represented? 4. What algebraic expression is called polynomial? 5. What are terms of a polynomial?

VI. Translate into Russian:

An algebraic expression of one term is called a monomial or simple expression. An algebraic expression of more than one term is called a polynomial; a polynomial of two terms is called a binomial.

3a+2b and x2—y2 are binomials. a+b+c is a trinomial.

VII. Translate into English:

Алгебраическое выражение, которое содержит только действия умножения, деления и возведения в степень, называется одночленом.

Алгебраическая сумма нескольких одночленов называется многочленом. Двучлен это алгебраическое выражение, состоящее из двух членов, трехчлен – алгебраическое выражение, состоящее из трех членов.

TEXT

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 множител