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ON THE USE OF THE SIGNS + AND
THE student is probably aware that, in the application of Algebra to Problems concerning distance, we sometimes find that the solution of an equation gives the measure of a distance with the sign - before it.
Example. Let M, N, O be places in a straight line; let the distance from M to N be 3 miles, and the distance from N to 0, 3 miles.
One man A starting from M, rides towards O at the rate of 10 miles an hour, while another man B starting simultaneously from N, walks towards O at the rate of 4 miles an hour;
If Q be the point at which they meet, how far is Q beyond 0?
Let P be any point beyond 0, and let x be the number of miles in OP. We wish to find x, i. e. the measure of OP, so that P may coincide with Q, the point at which A overtakes B.
When A arrives at P, he has ridden 6+x miles. The time
occupied at the rate of 10 miles an hour is
6 + x
When B arrives at P, he has walked 3+ miles. The time
occupied at the rate of 4 miles an hour is
When P is the point at which they meet, these times are equal, so that
Thus the required number of miles has the sign and we have failed to find a point beyond O at which A overtakes B.
121. Such a result can generally be interpreted by altering the statement of the problem, thus:
The line whose measure appears with the sign - before it, must be considered as drawn in the direction opposite to that in which it was drawn in the first statement of the problem.
Example. Taking the former example, let us alter the question as follows:
If Q be the point at which A overtakes B, how far is Q to the left of O?
Let P be any point to the left of O, and let x be the number of miles in OP.
We wish to find x (i.e. the measure of OP), so that P may coincide with Q, the point at which A overtakes B.
When A arrives at P, he has ridden 6-x miles.
When B arrives at P, he has walked 3
Proceeding as before, we get
; whence = +1.
Therefore if P is to coincide with (the point at which A overtakes B), OP must be one mile to the left of 0.
122. The consideration of such examples as the above has suggested, that the sign may be made use of, in the application of Algebra to Geometry, to represent a direction exactly opposite to that represented by the sign +. Accordingly the following Rule, or Convention, has been
Any straight line AB being given, then
lines drawn parallel to AB in one direction shall be positive; that is, shall be represented algebraically by their measures with the sign + before them:
lines drawn parallel to BA in the opposite direction shall be negative; that is, shall be represented algebraically by their measures with the sign - before them.
123. We may choose for the positive direction in each case that direction which is most convenient.
Example. Let LR be a straight line parallel to the printed lines in the page,
and let lines drawn in the direction from L to R in the figure that is, from the left-hand towards the right, be considered positive. Then by the above rule, lines drawn in the direction from R to L, that is, from right to left, must be negative.
124. In naming a line by the letters at its extremities, we can indicate by the order of the letters, the direction in which the line is supposed to be drawn.
Example. Let O and P be two points in the line LR as in the figure, and let the measure of the distance between them be a.
Then OP, i.e. the line drawn from 0 to P, which is in the positive direction, is represented algebraically by + a.
While PO, i.e. the line drawn from P to 0, which is in the negative direction, is represented algebraically by -a.
Hence in using the two letters at its extremities to represent a line, the student will find it advantageous always to pay careful attention to the order of the letters.
Example. Let LR be a straight line parallel to the printed lines in the page.
Let A, B, C, D, E be points in LR, such that the measures of AB, BC, CD, DE, are 1, 2, 3, 4 respectively.
Find the algebraical representation of
(i) AC+ CB
(ii) AD+DC - BC.
(i) The algebraical representation of AC is +3,
(ii) The algebraical representation of AD is +6, that of DC is -3, and that of BC is +2.
Therefore that of AD+ DC-BC=6-3-2 = +1.
This is equivalent to that of AB.
In the above figure, find the algebraical representation of
(1) AB+BC + CD.
(3) BC +CD+DE+ EC.
(2) AB+BC+ CA.
(8) CD-BD+BA + AC + CE.
* By AC+CB (attention being paid to direction), we mean 'Go from A to C and from C to B.' The result is equivalent to starting from A and stopping at B, i.e. equivalent to AB.
ON THE USE OF THE SIGNS + AND IN TRIGONOMETRY.
126. IN Trigonometry in order conveniently to treat of angles of any magnitude, we proceed as follows.
We take a fixed point 0, called the origin; and a fixed straight line OR, called the initial line.
The angle of which we wish to treat is described by a line OP, called the revolving line. This line OP starts from the initial line OR, and turns about 0 through an angle ROP of any proposed magnitude into the position OP.
127. We have already said in Art. 41
(i) that, when an angle ROP is described by OP turning about in the direction contrary to that of the hands of a watch, the angle ROP is said to be positive; that is, is represented algebraically by its measure with the sign + before it.
(ii) that, when an angle ROP is described by OP turning about O in the same direction as the hands of a watch, the angle is said to be negative; that is, is represented algebraically by its measure with the sign - before it.