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Text 4 algebraic language

Use of letters. Letters are used to express the general properties of numbers. Suppose that we want to express briefly in a written form that the product of two numbers remains unaltered when we interchange the position of the multiplicand and the multiplier. Then, representing one of the numbers by the letter a and the other by the letter b, we shall be able to write the equality: a*b=b*a, or, shortly: ab=ba, having agreed once and for all that the multiplication sign is understood between any two letters, written side by side, if no other sign is indicated. Con­sequently, letters are always used to express that a certain property is peculiar to numbers in general and not to any particular numbers.

Letters of the Latin alphabet are generally used to represent numbers.

Algebraic expression. An algebraic expression is an expression in which several numbers represented by letters (or by letters and figures) are connected by means of signs indicating the operations to which the numbers must be subjected and the order of these operations.

Such are, for instance, the expressions:

a/100*p; ab; 2x+l.

For the sake of brevity we shall often simply say «expression» instead of «algebraic ex­pression».

To evaluate an expression when the numerical values of the letters are given, is to substitute the numerical equivalents for the letters and perform the operations indicated in the expression The number obtained is known as the numerical value of the algebraic expression for the giver. numerical equivalents of the letters. Hence, the numerical value of the expression a/100*p when

p=3 and a = 520 is

520/100*3=5.2*3=15.6=5.2Ч3=15.6.

Order of operations. With regard to the order in which the operations indicated in an

algebraic expression should be performed, it was agreed upon to perform the operations of the

higher order first, i. е., involution and evolution, then multiplication and division, and, finally,

addition and subtraction.

Thus, if we have the expression 3a2b-b/c+d we must, when evaluating it, first perform the с involution (square the number a and cube the number b), then the multiplication and division

(multiply 3 by a and the result obtained by b; divide b by c) and, finallv, the subtraction and

addition (subtract from 3a b and add d to the result).

Notion of Values which may be taken in two opposite senses. Problem. — At midnight a thermometer read 2° and at noon 5°. How many degrees did the temperature change between midnight and noon?

The conditions of this problem are not sufficiently clear; we must know whether the reading at midnight was 2° below or above 0°, the same must be mentioned for the noon reading. If e.g. both at midnight and at noon the temperature was above 0°, then during the given period of time the temperature rose from 2° to 5°, i. e. 3°; while if the temperature at midnight was 2° below zero and at noon it was 5° above zero, the temperature rose 2 + 5, i. e. 7°, and so on.

In this problem we deal with a quantity having a direction: the number of degrees may be read either upwards or downwards from zero. The temperature above 0° (heat) is known as positive and is recorded as the number of degrees preceded by the + sign, and the temperature below 0° I cold) is known as negative and is recorded as the number of degrees preceded by the - sign (there will be no misunderstanding if the first reading is taken without any sign at all).

Now let us formulate our problem, for instance, as follows: At midnight a thermometer read -2° and at noon it read +5°. What is the change in temperature between midnight and noon? As it is, the problem has a definite answer: The temperature rose 2 + 5, i. e. 7°.

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