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that ten applies be removed from a pile of seven apples, though it is not absurd that a bank should allow a responsible client to draw $1,000 from a credit balance of $700.

It appears then that the words positive and negative cannot well be defined separately and hence should always be defined as a pair and applied to quantities diametrically opposed to one another, equal quantities which when added serve to destroy one another.

Let us now consider roots of negative numbers. The indicated even root of a negative number was for many centuries considered an impossibility and early received the name of an imaginary number. In the strict sense of a number as used in counting, such roots are absurd in the same way as negative numbers themselves are absurd. We however have seen that, in connection with associated pairs of quantities, negative numbers have received a useful meaning. It is fair to inquire whether a reasonable meaning may not be found for roots of negative numbers.

Such a meaning was found by Caspar Wessel, a Danish surveyor, and his Memoir on this subject “On the analytical representation of direction" was read before the Royal Academy of Sciences and Letters of Denmark on March 10, 1797 and published in Volume V of the Memoirs of the Academy in 1799. It was republished in French in 1897 by the Academy on the occasion of the centennial of its presentation to the Academy. The Memoir is very interesting and an excellent presentation of the subject. It is very complete; and, had it been known to mathematicians in general, would have saved many years in the development of vector analysis and quaternions. Unfortunately it was put to sleep in the published memoirs, its slumbers not to be disturbed until other more fortunate writers had rediscovered its ideas and published them to the world. Wessel called V-I the sign of perpendicularity.

In brief Wessel recognized that minus one as an operator reverses one of a pair of directed quantities into the other associated quantity, and what was more natural than to assume the reversal takes place by means of a rotation through 180°? If so what would be more natural than to assume that V-I will produce a rotation through 90°, for a second application would be equivalent to a single application of minus one. V-I being a symbol unconnected with numbers used in counting may be endowed with such non-contradictory properties as we see fit. We therefore require that V-IV-I=-I when V-1 is used as an algebraic factor and as an oper

I ator. We thus have an operator V-1 which can operate on east to change it into north, a second application changing it to west, a third to south and a fourth back to east again.

Let us apply this new meaning of V-I to a problem. The point B is 10 units east of A, let us find a point P so located that the product of


the distances from A to P and from P to B is 34 units. Let the first distance be X and the second 10-X. We then have

10X—X—34. solving the equation and calling these two values of X, X, and X, we have

X,=5+3V-I=5 east + 3 north

X.=5—3V-I=5 east + 3 south. The diagram shows the location of the two points P1, and P, which satisfy the problem. Had the problem been stated to divide a line 10 units

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long into segments whose product is 34, we would have said that the problem is impossible for the maximum product possible is 25. The algebraic treatment of the two problems is identical.

Let us now seek for a use for 3V-I as an operator. Let us as before endow it with the property, both as an operator and as an algebraic factor, that three applications is equivalent to minus one, or

31-1V-1V-=-I Let

X='V-I or what is equivalent

Indicate the three roots as X, X, and X3, we then have

X=1/2 + 1/2 V3 V-I

X,=1/2-1/2 V3 V-1.
Let us apply 8V-I as an operator to east, we have a choice of results

SV-I east=12 east +12V 3 north
8V-1 east = west

SV-I east=12 east +12 V3 south. It is evident from the figure that the first value of V-I has rotated east 60° toward north without changing its magnitude, for 12 is the cosine and 12 V 3 the sine of 60°, and the hypothenuse is equal to unity. The second value rotates east into west through 180° without change in magnitude and the third rotates east through 300° counter clockwise or 60° clock


wise into the position shown. All on being repeated to the third time produce reversal in direction from east to west.

To avoid ambiguity in the three roots of 3-1 for which we lack distinguishing signs as for the two roots of V-1, i. e. + V-I and - V-1, we shall as a rule always understand the operator which produces the smallest counter clockwise rotation.

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figure a We may in a like way investigate the fourth root of minus one. We shall find four values:

X=72V2 +212 V-I
X=-12V2 + 12V2 V-I
, X=-1212-72V2 V-I

X=12V2—12V2V-1. As both the sine and cosine of 45° are 12 V2, we see that X, as an operator converts east into northeast without change in magnitude. The others convert east to northwest, southwest and southeast; and all on being repeated to the fourth time produce reversal. As before let us understand V-I

* as an operator rotating through the least angle 45°.

It is easily proved that all roots of minus one have this rotational property, and in general that

"V-I=COS/n + V-I sin 'n for the root of smallest rotating power. The roots are given by the general formula

( 2m +1)

( 2m + 1)T "y-1=cos

+1-i sin in which m is zero or any whole number.

It is interesting to know, though the time at our disposal does not allow of its proof, that n may be any real number or any proper or improper


fraction, or even may be a variable quantity. In the last case we have the means of representing the continuous rotation of a vector quantity.

Abbreviating the notation by writing j for V-I we have a well known formula, e being the base of the napierian logarithms:

cje = cose + jsine, which is somewhat more convenient for practical use. It is evident that if 6 is a variable, proportional to the time t, we may obtain an operator which rotates a vector at constant angular velocity p.

e pt = cos pt + j sin pt. The angle pt is the angle between the horizontal and the vector.

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An interesting simple application of this method may be made for the case of uniform circular motion. Let P be the position of a point at the distance R and direction angle pt revolving about the center of motion with angular velocity p, linear velocity V and acceleration A. We have the following relations:

P= e jpt R= (cos pt +j sin pt) R

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The relations are illustrated in the figure. In making the transformations above it may be helpful to notice that

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and also in taking the derivative of a simple harmonic quantity of the form sin pt. or cos pt, that the derivative is a/2 or 90° further in advance in phase than the quantity from which it is derived, that is,

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d sin pt;/dt =p cos pt =p sin (pt + 7/2)
d cos pt/dt =- p sin pt=p cos (pt + 5/2).

Keeping the function and its derivative in the same form has the decided advantage that change in phase is brought out more clearly than by the more usual change from sine to cosine etc. met with in the study of differential calculus.

It is clear from the equations given above that the phase of the velocity V is 90°, and the phase of the acceleration A is 180° in advance of that of the position P.

Another simple illustration of this method is found for the spiral motion of a pendulunı set in motion in what would be a circular path in a horizontal plane except for the effect of air friction which diverts the pendulum into a logarithmic, spiral gradually drawing in toward the center of the original circle. The projection of this motion on a horizontal line is a damped harmonic motion similar to the damped motion found in free pendulums swinging in a vertical plane, the vibrations of galvanometers and the oscillatory discharge of a Leyden jar. Similar though more complicated phenomena are observed in the study of the transmission of current through long telephone lines.




1. Clifford, W. K. Common Sense of the Exact Sciences. D. Appleton

& Co. New York. $1.50. Out of print. A clear and simple discussion of the ideas of number, space, quantity,

position and motion, by one of the most brilliant of recent English mathematicians.

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