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