Chapter 6
Graphical Representation of Numbers. the Variable

40. Correspondence between the Real Number-System and the Points of a Line. Let a right line be chosen, and on it a fixed point, to be called the null-point; also a fixed unit for the measurement of lengths.

Lengths may be measured on this line either from left to right or from right to left, and equal lengths measured in opposite directions, when added, annul each other; opposite algebraic signs may, therefore, be properly attached to them. Let the sign + be attached to lengths measured to the right, the sign to lengths measured to the left.

The entire system of real numbers may be represented by the points of the line, by taking to correspond to each number that point whose distance from the null-point is represented by the number. For, as we proceed to demonstrate, the distance of every point of the line from the null-point, measured in terms of the fixed unit, is a real number; and we may assume that for each real number there is such a point.

1. The distance of any point on the line from the null-point is a real number.

Let any point on the line be taken, and suppose the segment of the line lying between this point and the null-point to contain the unit line α times, with a remainder d1, this remainder to contain the tenth part of the unit line β times, with a remainder d2, d2 to contain the hundredth part of the unit line γ times, with a remainder d3, etc.

The sequence of rational numbers thus constructed, viz., α,α.β,α.βγ, (adopting the decimal notation) is regular; for the difference between its μth term and each succeeding term is less than 1 10μ1, a fraction which may be made less than any assignable number by taking μ great enough; and, by construction, this number represents the distance of the point under consideration from the null-point.

By the convention made respecting the algebraic signs of lengths this number will be positive when the point lies to the right of the null-point, negative when it lies to the left.

2. Corresponding to every real number there is a point on the line, whose distance and direction from the null-point are indicated by the number.

(a) If the number is rational, we can construct the point.

For every rational number can be reduced to the form of a simple fraction. And if α β denote the given number, when thus expressed, to find the corresponding point we have only to lay off the βth part of the unit segment α times along the line, from the null-point to the right, if α β is positive, from the null-point to the left, if α β is negative.

(b) If the number is irrational, we usually cannot construct the point, or even prove that it exists.

But let a denote the number, and α1,α2,,αn, any regular sequence of rationals which defines it, so that αn will approach a as limit when n is indefinitely increased.

Then, by (a), there is a sequence of points on the line corresponding to this sequence of rationals. Call this sequence of points A1,A2,,An,. It has the property that the length of the segment AnAn+m will approach 0 as limit when n is indefinitely increased.

When αn is made to run through the sequence of values α1,α2,, the corresponding point An will run through the sequence of positions A1,A2,. And we assume that just as there is in the real system a definite number a which αn is approaching as a limit, so also is there on the line a definite point A which An approaches as limit. It is this point A which we make correspond to a.

Of course there are infinitely many regular sequences of rationals α1,α2, defining a, and as many sequences of corresponding points A1,A2,. We assume that the limit point A is the same for all these sequences.

41. The Continuous Variable. The relation of one-to-one correspondence between the system of real numbers and the points of a line is of great importance both to geometry and to algebra. It enables us, on the one hand, to express geometrical relations numerically, on the other, to picture complicated numerical relations geometrically. In particular, algebra is indebted to it for the very useful notion of the continuous variable.

One of our most familiar intuitions is that of continuous motion.


pict


Suppose the point P to be moving continuously from A to B along the line OAB; and let a, b, and x denote the lengths of the segments OA, OB, and OP respectively, O being the null-point.

It will then follow from our assumption that the segment AB contains a point for every number between a and b, that as P moves continuously from A to B, x may be regarded as increasing from the value a to the value b through all intermediate values. To indicate this we call x a continuous variable.

42. Correspondence between the Complex Number-System and the Points of a Plane. The entire system of complex numbers may be represented by the points of a plane, as follows:

In the plane let two right lines XOX and Y OY be drawn intersecting at right angles at the point O.


pict
Fig. 1.


Make XOX the “axis” of real numbers, using its points to represent real numbers, after the manner described in § 40, and make Y OY the axis of pure imaginaries, representing ib by the point of OY whose distance from O is b when b is positive, and by the corresponding point of OY when b is negative.

The point taken to represent the complex number a + ib is P, constructed by drawing through A and B, the points which represent a and ib, parallels to Y OY and XOX, respectively.

The correspondence between the complex numbers and the points of the plane is a one-to-one correspondence. To every point of the plane there is a complex number corresponding, and but one, while to each number there corresponds a single point of the plane.19

If the point P be made to move along any curve in its plane, the corresponding number x may be regarded as changing through a continuous system of complex values, and is called a continuous complex variable. (Compare § 41.)

43. Modulus. The length of the line OP (Fig. 1), i. e. a2 + b2, is called the modulus of a + ib. Let it be represented by ρ.

44. Argument. The angle XOP made by OP with the positive half of the axis of real numbers is called the angle of a + ib, or its argument. Let its numerical measure be represented by 𝜃.

The angle is always to be measured “counter-clockwise” from the positive half of the axis of real numbers to the modulus line.

45. Sine. The ratio of PA, the perpendicular from P to the axis of real numbers, to OP, i. e. b ρ, is called the sine of 𝜃, written sin𝜃.

sin𝜃 is by this definition positive when P lies above the axis of real numbers, negative when P lies below this line.

46. Cosine. The ratio of PB, the perpendicular from P to the axis of imaginaries, to OP, i. e. a ρ, is called the cosine of theta, written cos𝜃.

cos𝜃 is positive or negative according as P lies to the right or the left of the axis of imaginaries.

47. Theorem. The expression of a + ib in terms of its modulus and angle is ρ(cos𝜃 + isin𝜃).

   For by § 46 a ρ = cos𝜃, a = ρcos𝜃;  and by § 45,  b ρ = sin𝜃, b = ρsin𝜃.    Therefore  a + ib = ρ(cos𝜃 + isin𝜃).

The factor cos𝜃 + isin𝜃 has the same sort of geometrical meaning as the algebraic signs + and , which are indeed but particular cases of it: it indicates the direction of the point which represents the number from the null-point.

It is the other factor, the modulus ρ, the distance from the null-point of the point which corresponds to the number, which indicates the “absolute value” of the number, and may represent it when compared numerically with other numbers (§ 37),—that one of two numbers being numerically the greater whose corresponding point is the more distant from the null-point.

48. Problem I. Given the points P and P, representing a + ib and a + ib respectively; required the point representing a + a + i(b + b).

The point required is P, the intersection of the parallel to OP through P with the parallel to OP through P.

For completing the construction indicated by the figure, we have OD = PE = DD, and therefore OD = OD + OD; and similarly PD = PD + PD.

Cor. I. To get the point corresponding to a a + i(b b), produce OP to P, making OP = OP, and complete the parallelogram OP, OP.


pict
Fig. 2.


Cor. II. The modulus of the sum or difference of two complex numbers is less than (at greatest equal to) the sum of their moduli.

For OP is less than OP + PP and, therefore, than OP + OP, unless O, P, P are in the same straight line, when OP = OP + OP. Similarly, PP, which is equal to the modulus of the difference of the numbers represented by P and P, is less than, at greatest equal to, OP + OP.

49. Problem II. Given P and P, representing a + ib and a + ib respectively; required the point representing (a + ib)(a + ib).

     Let a + ib = ρ(cos𝜃 + isin𝜃), §47  and a + ib = ρ(cos𝜃 + isin𝜃);  then (a + ib)(a + ib) = ρρ(cos𝜃 + isin𝜃)(cos𝜃 + isin𝜃) = ρρ[(cos𝜃cos𝜃 sin𝜃sin𝜃) + i(sin𝜃cos𝜃 + cos𝜃sin𝜃)].      But cos𝜃cos𝜃 sin𝜃sin𝜃 = cos(𝜃 + 𝜃),20  and sin𝜃cos𝜃 + cos𝜃sin𝜃 = sin(𝜃 + 𝜃).21

Therefore (a + ib)(a + ib) = ρρ[cos(𝜃 + 𝜃) + isin(𝜃 + 𝜃)]; or, The modulus of the product of two complex numbers is the product of their moduli, its argument the sum of their arguments.

The required construction is, therefore, made by drawing through O a line making an angle 𝜃 + 𝜃 with OX, and laying off on this line the length ρρ.

Cor. I. Similarly the product of n numbers having moduli ρ, ρ, ρ, ρ(n) respectively, and arguments 𝜃, 𝜃, 𝜃, theta(n), is the number

ρρρρ(n)[cos(𝜃 + 𝜃 + 𝜃 + 𝜃(n)) + isin(𝜃 + 𝜃 + 𝜃 + 𝜃(n))].

In particular, therefore, by supposing the n numbers equal, we may infer the theorem

[ρ(cos𝜃 + isin𝜃)]n = ρn(cosn𝜃 + isinn𝜃),
which is known as Demoivre’s Theorem.

Cor. II. From the definition of division and the preceding demonstration it follows that

a + ib a + ib = ρ ρ[cos(𝜃 𝜃) + isin(𝜃 𝜃)];
the construction for the point representing a + ib a + ib is, therefore, obvious.

50. Circular Measure of Angle. Let a circle of unit radius be constructed with the vertex of any angle for centre. The length of the arc of this circle which is intercepted between the legs of the angle is called the circular measure of the angle.

51. Theorem. Any complex number may be expressed in the form ρei𝜃; where ρ is its modulus and 𝜃 the circular measure of its angle.

It has already been proven that a complex number may be written in the form ρ(cos𝜃 + isin𝜃), where ρ and 𝜃 have the meanings just given them. The theorem will be demonstrated, therefore, when it shall have been shown that

ei𝜃 = cos𝜃 + isin𝜃.

If n be any positive integer, we have, by § 36 and the binomial theorem,

1 + i𝜃 n n = 1 + ni𝜃 n + n(n 1) 2! (i𝜃)2 n2 + n(n 1)(n 2) 3! (i𝜃)3 n3 + = 1 + i𝜃 + 1 1 n 2! (i𝜃)2 + 1 1 n 1 2 n 3! (i𝜃)3 + .

Let n be indefinitely increased; the limit of the right side of this equation will be the same as that of the left.

But the limit of the right side is

1 + i𝜃 + (i𝜃)2 2! + (i𝜃)3 3! + ; i. e.ei𝜃.22

Therefore ei𝜃 is the limit of 1 + i𝜃 n n as n approaches .

To construct the point representing 1 + i𝜃 n n:


pict
Fig. 3.


On the axis of real numbers lay off OA = 1.

Draw AP equal to 𝜃 and parallel to OB, and divide it into n equal parts. Let AA1 be one of these parts. Then A1 is the point 1 + i𝜃 n .

Through A1 draw A1A2 at right angles to OA1 and construct the triangle OA1A2 similar to OAA1.

A2 is then the point 1 + i𝜃 n 2.

 For AOA2 = 2AOA1;  and since OA2 : OA1 :: OA1 : OA, andOA = 1,  the length OA2 =  the square of lengthOA1.(see§49)

In like manner construct A3 to represent 1 + i𝜃 n 3, A4 for 1 + i𝜃 n 4,An for 1 + i𝜃 n n.

Let n be indefinitely increased. The broken line AA1A2An will approach as limit an arc of length 𝜃 of the circle of radius OA and, therefore, its extremity, An, will approach as limit the point representing cos𝜃 + isin𝜃 (§ 47).

Therefore the limit of 1 + i𝜃 n n as n is indefinitely increased is cos𝜃 + isin𝜃.

But this same limit has already been proved to be ei𝜃.

 Hence ei𝜃 = cos𝜃 + isin𝜃.23