a whole thus a-b-c is equivalent to a- (b-c), and Ja+b is equivalent to (a+b). It should be noticed that where no vinculum or bracket is used, a radical sign refers only to the number or letter which immediately follows it: thus √2a means that the square root of 2 is to be multiplied by a, whereas √2a means the square root of 2a; also a +x means that x is to be added to the square root of a, whereas √a+x means that x is to be added to a and that the square root of the whole is to be taken. a+b The line between the numerator and denominator of a fraction acts as a vinculum, for is the same as (a+b). NOTE. It is important for the student to notice that every term of an algebraical expression must be added or subtracted as a whole, as if it were enclosed in brackets. Thus, in the expression a + bc − d ÷ e +ƒ, b must be multiplied by c before addition, and d must be divided by e before subtraction, just as if the expression were written a + (bc) − (d ÷ e) + ƒ. EXAMPLES. 1. Find the numerical values of the following expressions in each of which a=1, b=2, c=3, and d=4. (i 2. If a=3, b=1 and c=2, find the numerical values of (iii) c3 63, (iv) a3+3ac2 - 3a2c – c3, and (v) 2ab2c3b4c2a - 2c4a2b. Ans. 19, 6, 0, 1, 0. 3. Find the values of the following expressions in each of which a=3, b=2, c=1 and d=0. √a2 – b2, √5ab+c, √√(bac2 +b2c1) and 3/ a2+4b2+ 4c2, when a=5, b=4, c=3. Ans. 3, 13, 60, 5. 5. Shew that a2 – b2 and (a+b) (a − b) are equal to one another (i) when a=2, b=1; (ii) when a=5, b=3; and (iii) when a=12, b=5. and 6. Shew that the expressions a3 – b3, (a - b) (a2+ab+b2), (a - b)3 + 3ab (a − b), (a+b)3 - 3ab (a+b) – 2b3 are all equal to one another (i) when a =3, b=2; (ii) when a=5, b=1; and (iii) when a=6, b=3. CHAPTER II. FUNDAMENTAL LAWS. 17. WE have said that all concrete quantities must be measured by the number of times each contains some unit of its own kind. Now a sum of money may be either a receipt or a payment, it may be either a gain or a loss; motion along a given straight line may be in either of two opposite directions; time may be either before or after some particular epoch; and so in very many other cases. Thus many concrete magnitudes are capable of existing in two diametrically opposite states: the question then arises whether these magnitudes can be conveniently distinguished from one another by special signs. 18. Now whatever kind of quantity we are considering +4 will stand for what increases that quantity by 4 units, and -4 will stand for whatever decreases the quantity by 4 units. If we are calculating the amount of a man's property (estimated in pounds), +4 will stand for whatever increases his property by £4, that is +4 stands for £4 that he possesses, or that is owing to him; so also - 4 will stand for whatever decreases his property by £4, that is, -4 will stand for £4 that he owes. If, on the other hand, we are calculating the amount of a man's debts, +4 will stand for whatever increases his debts, that is, +4 will now stand for a debt of £4; so also -4 will now stand for whatever decreases his debts, that is, 4 will stand for £4 that he has, or that is owing to him. If we are considering the amount of a man's gains, +4 will stand for what increases his total gain, that is, +4 will stand for a gain of 4; so also - 4 will stand for what decreases his total gain, that is, -4 will stand for a loss of 4. If however we are calculating the amount of a man's losses, +4 will stand for a loss of 4, and 4 will stand for a gain of 4. Again, if the magnitude to be increased or diminished. is the distance from any particular place, measured in any particular direction, +4 will stand for a distance of 4 units in that direction, and 4 will stand for a distance of 4 units in the opposite direction. 19. From the last article it will be seen that it is not necessary to invent any new signs to distinguish between quantities of directly opposite kinds, for this can be done by means of the old signs + and The signs and are therefore used in Algebra with two entirely different meanings. In addition to their original meaning as signs of the operations of addition and subtraction respectively, they are also used as marks of distinction between magnitudes of diametrically opposite kinds. The signs and are sometimes called signs of affection when they are thus used to indicate a quality of the quantities before whose symbols they are placed. The sign+, as a sign of affection, is frequently omitted; and when neither the + nor the sign is prefixed to a term the sign is to be understood. 20. A quantity to which the sign is prefixed is called a positive quantity, and a quantity to which the sign is prefixed is called a negative quantity. The signs and are called respectively the positive and negative signs. NOTE. Although there are many signs used in algebra, the name sign is often used to denote the two signs + and exclusively. Thus, when the sign of a quantity is spoken of, it means the + or sign which is prefixed to it; and when we are directed to change the signs of an expression, it means that we are to change + and wherever they occur, into and + respectively. 21. The magnitude of a quantity considered independently of its quality, or of its sign, is called its absolute magnitude. Thus a rise of 4 feet and a fall of 4 feet are equal in absolute magnitude; so also +4 and -4 are equal in absolute magnitude, whatever the unit may be. ADDITION. 22. The process of finding the result when two or more quantities are taken together is called addition, and the result is called the sum. Since a positive quantity produces an increase, and a negative quantity produces a decrease, to add a positive quantity we must add its absolute value, and to add a negative quantity we must subtract its absolute value. Thus, when we add +4 to +6, we get + 6 + 4 ; and when we add − 4 to +10, we get +10 – 4. and Hence So also, when we add +b to +a, we get +a+b; and when we add -b to +a, we get +a-b. Hence and +a+(+b)=+a+b, We therefore have the following rule for the addition of any term to add any term affix it to the expression to which it is to be added, with its sign unchanged. When numerical values are given to a and to b, the numerical values of a+b and a-b can be found; but |