THE POSITIVE AND NEGATIVE SIGNS.

7. Quantity, which can be expressed by any number, whole, fractional, or decimal, is designated in algebra by some letter of the alphabet or other easily written symbol. There are, furthermore, two signs which can be prefixed to the letter, one the positive, +, read 'plus,' another the negative, —, read 'minus,' and the power of these signs is that as one or other of them is prefixed to a representation of quantity, the subject whose magnitude is so represented has a contrary property or affection in some defined respect.

If a denotes a sum of money received, -a can denote the same sum of money paid away, the quantity represented having these contrary properties or affections in respect of being given or received.

If +a denotes a number of yards, as the distance which a person walks in one direction along an unlimited straight line, -a can denote the same number of yards as the distance which he walks in the contrary direction.

If a denotes a number of years that are past, -a may denote the same number of years yet to come.

If +a means a number of feet which is the height of a point above the ground, -a can mean the same number of feet as the depth of a point below the ground.

So, to extend this conception to apply to different algebraic symbols of quantity, if a denotes a sum of money received, -b can denote a sum in magnitude b, paid away.

If +a denotes an advance through a certain number of yards, -b denotes a retreat through the number of yards expressed by b.

The positive and negative signs have thus an antagonistic and reversing power on the meaning of the symbols before which they stand. Of the two opposite characters which magnitude may have in some defined respect, it is immaterial which of the two signs is taken to represent one of those characters, the other sign representing the opposite

character. Thus, a ora, may designate a height a, and then a or a will accordingly designate the same depth; +a or a being taken to mean a sum of money a received, then -b or +b will respectively mean another sum of money b paid away.

8. Obs. When a symbol stands without either algebraic sign prefixed to it, it is understood to bear the positive sign. Thus +a+b+c is written a+b+c.

9. Def.-Quantities with the positive sign prefixed are called positive quantities. Quantities with the negative sign prefixed are called negative quantities.

10. Def.-By quantities of the same kind are meant such as can be referred to the same unit, quantities, for instance, of length, of weight, sums of money.

11. When the sign of a quantity is changed from positive to negative, or from negative to positive, the sign is said to be reversed.

12. The following signs are used as a kind of shorthand, to save writing

Thus, 54 means the statement that 5 is greater than 4.

< means 'less than.'

Thus, 9<11 means that 9 is less than 11.

ADDITION AND SUBTRACTION OF ALGEBRAICAL QUANTITIES.

13. In this view of the meaning of the two algebraical signs, the positive and the negative, it will be seen that quantities of the same kind (10) bearing the same sign, be it positive or negative, can be added one to another. The result of that addition is expressed by writing the algebraic symbols which express these quantities one after another in a orizontal line. Then if a and +b mean separate

amounts of money received, +a+b or a+b (8) means the amount received altogether, the sum, as the word is understood in arithmetic. If -c and -d mean, in consistence with the former representations, separate amounts of money paid away, -c-d means the amount paid away altogether. For example, if a be 32l., b 261., c 34l., d 20l., when the positive sign is taken to designate money received, a+b means 587. received, and -c-d means 547. paid away.

If a, b, c, d continue to denote the same numbers, and these are now understood to mean numbers of miles, if +a be the distance a man travels one day directly towards the east, + the distance he travels the next day in the same direction, a+b will represent the 58 miles of his journeying towards the east, and consistently -c-d can represent 54 miles of the journeying of some other man towards the

west.

Again, quantities of the same kind bearing opposite signs can in virtue of the reversing power of these signs be subtracted one from another. The result of this subtraction is expressed by writing the symbols one after another with their proper algebraic signs in a horizontal line. Thus +a denoting money which a person owes, and -b consistently denoting money due to him, +a-b or a-b will mean what is called the balance of his account, or the difference between the debts due to him and the debts which he owes. If a is a greater quantity than b, +a so absorbs -b as to leave a positive excess, and a-b means an excess of debt owing over money expected. On the contrary, if b exceeds a, —b so absorbs +a as to leave a negative excess, meaning that a balance remains of money expected.

14. Obs. The order in which the symbols of quantity are written is immaterial. Thus a+b−c, b+a−c, b−c+a, a−c+b, −c+a+b, −c+b+a, all mean the same thing, namely, that after quantities a and b have been added together, the difference is taken between their sum and

another quantity c, and this difference is positive or negative, as a+b is greater or less in magnitude than c.

In writing such an expression as this it is usual to place a positive quantity first, and the writing of its sign is dispensed with (8). Also when other reasons do not interfere, it is customary to write the algebraical symbols in the order in which they stand in the alphabet. Neither of these arrangements however is in any wise obligatory.

15. Hence a-a or —a+a has no magnitude, the one term meaning a magnitude operating with a reversing or destructive effect on another equal magnitude.

—a=o.

16. The sign, read 'equals,' denoting equality of the quantities between which it stands, and the sign o, read 'zero' or 'naught,' denoting nullity or absence of all magnitude, are used in Algebra as in Arithmetic. Thus a-a All quantities which admit of their magnitude being numerically expressed, being neither zero nor infinitely great, are called finite quantities.

17. Since magnitude is contemplated as in only two opposite states, two negative signs amount to the positive sign. For ―a expresses a quantity in a contrary condition to the same quantity +a. The reversing then of -a throws it into the same condition with +a.

18. If a means 20 and b means 10 in the following questions,

1. What will be expressed when it is stated that a traveller has journeyed a-b miles to the east? And what if he is said to have journeyed a+b miles towards the east?

2. If a man's expenses in one day are a£, what can -b£

consistently represent? What will be meant by stating that b-a is his day's expenses?

3. If the boys added to a school after the holidays are a in number, what would be meant by -a new boys? Would the school be made larger or smaller by receiving -a+b new boys?

4. If a denote degrees in a thermometer above freezing point, what will -b degrees mean?

5. If a boy wins a games over a playfellow, how may the latter be said to win -a games? What would be meant by stating that one of them won b-a games?

6. If a denotes the number of days a man works, would -b denote a number of days when he was idle? Or what would be meant by stating that he worked -b days?

MULTIPLICATION OF ALGEBRAIC QUANTITIES.

19. The expression a+a means the addition of two quantities of the same magnitude and of the same sign. The result, therefore, is double of either of them, and is written. 2a. Similarly a+a+a would be 3a, and if b means the number of times that a is repeated,

a+a+a+... would be ba.

On this suggestion the result of multiplying together two quantities a and b, whether they mean whole numbers or fractions, is written ba or ab. Sometimes the sign of multiplication (x) or a dot (.) is interposed between the letters, thus, a.b or axb. The order of the letters, be it ab or ba, is immaterial, as we know in Arithmetic that when two quantities are multiplied together, it is immaterial which is regarded as the multiplier.

If ab is now viewed as a single quantity, the result of multiplying it by another quantity will be written abc or cab, with or without the sign × or. above mentioned. This result being that of multiplying a, b, c together, and the order which the multiplication is performed being