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CHAPTER II

POSITIVE AND NEGATIVE NUMBERS, PAREN

THESIS

12. We frequently find quantities 120

For FEVER

HEAT

SUM R
HEAT

60

40

that are opposed to each other.
instance, a rise in temperature is of
opposite kind to a fall in tempera- 100
ture; a gain in business is opposed
to a loss; travelling a certain dis-
tance north is opposed to a distance
going south. These quantities, which 80-
oppose each other and which may be
called of opposite kind, are not con-
sidered in Arithmetic, but must be
cared for in Algebra. We distinguish
these by calling one quantity plus or
positive and its opposite, minus or
negative. Thus, on a thermometer,
degrees of temperature above the
zero point are called plus or positive
and degrees below zero are called 20-
minus or negative; + 10° means 10
degrees above zero and 8° means
8 degrees below zero. Either of two
opposite quantities may be called the
positive and the other the negative;
but it is usual to call that quantity
positive which implies an increase or
gain. For instance, if we were climb-
ing a steep mountain side and occa-
sionally slipping back a certain dis-
tance, those distances that bring us

FREEZING

110

BLOOD
HEAT

90

70

TEMPERATE

50

30

10

ZERO

10

20

-30

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RENÉ DESCARTES DU PERRON

(From an Old Engraving of the Painting by Franz Hals)

Born at La Haye, in Touraine, 1596. Died at Stockholm, 1650. René Descartes was philosopher, physicist and mathematician. The interpretation and systematic use of negative quantities began with Descartes.

He discarded Vieta's algebraic notation and devised the symbolism now used.

He is also the discoverer of analytic geometry.

nearer the top of the mountain would be called plus or positive and the distances we slip backwards would be minus or negative.

Only one kind of quantities are considered in Arithmetic and they are measured by stating the number of units they contain. For this purpose, we use the Natural Series of Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and so on, numbers starting with zero and increasing by one as far as we please. This series of numbers is no longer sufficient, when we consider positive and negative quantities. Suppose the thermometer indicates 10° and there is a drop in temperature amounting to 15°. Counting backwards from 10, in the natural series of numbers, we reach zero but still have 5 degrees drop to count. To enable us to count this negative temperature, we extend the natural series of numbers, beginning at zerg and counting negative units in the opposite direction, that is, to the left. We then have the Algebraic Series of Numbers.

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0, + 1, + 2, +3, + 4, + 5, + 6, +7, + 8,. . .

Using this algebraic series of numbers, it is evident that a drop of 15° from 10° brings us to -5° or 5 degrees below zero.

Similarly, if a man has a capital of $5000 and gains $1000, he will have $6000, but if he then loses $8000, his capital would become equal to $2000 less than zero, or -$2000, which means that he is $2000 in debt.

We have here one instance of the extension of arithmetical methods, as the algebraic series of numbers enables us to subtract a larger from a smaller number, which we were unable to do in Arithmetic.

13. The Absolute Value of a number is its value without regard to its sign. Thus +5 and -5 have the same absolute value of 5.

It should be noted that, when no sign is written before a number, plus is understood. The minus sign, indicating a negative number, must always be written.

14. Double Meanings of the + and Signs. From the foregoing explanations, it should now be clear that the plus and minus signs not only indicate an addition or subtraction as in Arithmetic, but also show that the numbers or quantities are of opposite kinds. These two meanings of the signs do not contradict each other. If we write 8 5, indicating that 5 should be subtracted from 8, the result of the subtraction, 3, is the same, as if we had combined 8 positive units with 5 negative units, that is 8 + (-5). In the subtraction, the 5 units destroy 5 of the 8 units and leave 3 units. Similarly, since 5 is of opposite kind to 8, five of the +8 units are cancelled or balanced by the five negative units and we have 3 positive units left. Two numbers of opposite kind, that is, one plus and the other minus, are said to have unlike signs. Thus, xy and ab, have unlike signs.

PARENTHESIS

15. Parenthesis Preceded by + Sign.-If a man, in an automobile, travelled 100 miles the first day, and on the second day, went 40 miles in the morning and 30 miles in the afternoon, the total distance travelled in the two days can be found, either by successively adding to the first day's distance, the distance he went in the morning and then the distance he went in the afternoon or by adding to the first day's distance, the distance he travelled the second day. The two methods may be written,

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Similarly, a + (b + c) = a + b + c.

If a man in an automobile travels 100 miles the first !day, 40 miles in the morning of the second day but travels

back 30 miles in the afternoon, then, at the end of the

second day, the distance from his starting point can be found, either by adding to the first day's distance, (the distance he travelled in the morning of the second day and then subtracting the distance he returned in the afternoon or by adding to the first day's distance, the difference between the distances travelled in the morning and afternoon of the second day.

By the first method, we get, 10040 - 30 = 110 By the second method, we get, 100+ (4030) = 110

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We have, from the above, the general

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Rule for Removal of Parenthesis Preceded by + Sign.A Parenthesis, preceded by the sign +, may be removed without making any change in the signs of the terms of the expression.

16. Parenthesis Preceded by - Sign.-If a man, starting at a certain place, travels 100 miles the first day but returns, 40 miles in the morning of the second day and 30 miles in the afternoon, his distance from the starting point at the end of the second day, may be found either by successively subtracting from the first day's distance, the distances he returned in the morning and afternoon of the second day, or by subtracting from the first day's distance, the sum of the distances he returned on the second day. This may be written,

First method 100 40 30 30.

Second method 100

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=

(40 + 30)

=

30.

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Similarly, a (b + c) = a − b c.

If a man, starting at a certain place, travels 100 miles the first day, returns 40 miles in the morning of the second day and then, in the afternoon, finds it necessary again to travel 30 miles away from his starting point, the distance from his starting point at the end of the second day, may be found either by subtracting from the distance he travelled the first day, the distance he returned on the morning of

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