Plane trigonometry, by S.L. Loney.

COMPLEX QUANTITIES.

473

The effect of prefixing - to a is therefore (Fig.
Art. 48) to rotate OA in the positive direction through
two right angles. The operation -1 performed on a
therefore means turning a through two right angles.
402. Now V/-1 x V/-1 =-1; hence whatever meaning we give to the operation V- 1 it must be such that
performing that operation twice shall be the same thing as
performing the operation - 1.
Let us therefore assign to the operation V/-1 the
turning any length through one right angle in the
positive direction. Performing the operation V/-1 on a
twice will therefore, as it should do, turn a through two
right angles.
Hence, with this interpretation, V/-1 a means a line
drawn at right angles to the line denoted by a.
403. We can now shew what is denoted by
x + - ly.
Draw OX and 0 Y two lines at right angles. Measure
along OX a distance OM equal to x and y
then draw MP parallel to 0 Y and equal
to y, so that MP represents V- 1 y.
Then P is the point that represents the P
quantity x + /- 1 y, or, again, we may o M X
say that OP is the line representing this quantity.
We have OP = VOM2 + MP2 = V/ 2 + y',
MP
and z MOP = tan-' O = tan —
OML x