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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 times, with a remainder , this remainder to contain the tenth part of the unit line times, with a remainder , to contain the hundredth part of the unit line times, with a remainder , etc.
The sequence of rational numbers thus constructed, viz., (adopting the decimal notation) is regular; for the difference between its th term and each succeeding term is less than , a fraction which may be made less than any assignable number by taking 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.
() 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 denote the given number, when thus expressed, to find the corresponding point we have only to lay off the th part of the unit segment times along the line, from the null-point to the right, if is positive, from the null-point to the left, if is negative.
() If the number is irrational, we usually cannot construct the point, or even prove that it exists.
But let a denote the number, and any regular sequence of rationals which defines it, so that will approach a as limit when is indefinitely increased.
Then, by (), there is a sequence of points on the line corresponding to this sequence of rationals. Call this sequence of points . It has the property that the length of the segment will approach 0 as limit when is indefinitely increased.
When is made to run through the sequence of values , the corresponding point will run through the sequence of positions . And we assume that just as there is in the real system a definite number a which is approaching as a limit, so also is there on the line a definite point A which approaches as limit. It is this point A which we make correspond to a.
Of course there are infinitely many regular sequences of rationals defining a, and as many sequences of corresponding points . 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.
Suppose the point to be moving continuously from to along the line ; and let a, b, and x denote the lengths of the segments , , and respectively, being the null-point.
It will then follow from our assumption that the segment contains a point for every number between a and b, that as moves continuously from to , 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 and be drawn intersecting at right angles at the point .
Fig. 1.
Make the “axis” of real numbers, using its points to represent real numbers, after the manner described in § 40, and make the axis of pure imaginaries, representing by the point of whose distance from is when is positive, and by the corresponding point of when is negative.
The point taken to represent the complex number is , constructed by drawing through and , the points which represent and , parallels to and , 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 be made to move along any curve in its plane, the corresponding number 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 (Fig. 1), i. e. , is called the modulus of . Let it be represented by .
44. Argument. The angle made by with the positive half of the axis of real numbers is called the angle of , or its argument. Let its numerical measure be represented by .
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 , the perpendicular from to the axis of real numbers, to , i. e. , is called the sine of , written .
is by this definition positive when lies above the axis of real numbers, negative when lies below this line.
46. Cosine. The ratio of , the perpendicular from to the axis of imaginaries, to , i. e. , is called the cosine of , written .
is positive or negative according as lies to the right or the left of the axis of imaginaries.
47. Theorem. The expression of in terms of its modulus and angle is .
The factor 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 , 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 and , representing and respectively; required the point representing .
The point required is , the intersection of the parallel to through with the parallel to through .
For completing the construction indicated by the figure, we have , and therefore ; and similarly .
Cor. I. To get the point corresponding to , produce to , making , and complete the parallelogram , .
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 is less than and, therefore, than , unless , , are in the same straight line, when . Similarly, , which is equal to the modulus of the difference of the numbers represented by and , is less than, at greatest equal to, .
49. Problem II. Given and , representing and respectively; required the point representing .
Therefore ; 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 a line making an angle with , and laying off on this line the length .
Cor. I. Similarly the product of numbers having moduli , , , respectively, and arguments , , , , is the number
In particular, therefore, by supposing the numbers equal, we may infer the theorem
which is known as Demoivre’s Theorem.Cor. II. From the definition of division and the preceding demonstration it follows that
the construction for the point representing 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 ; where is its modulus and the circular measure of its angle.
It has already been proven that a complex number may be written in the form , where and have the meanings just given them. The theorem will be demonstrated, therefore, when it shall have been shown that
If be any positive integer, we have, by § 36 and the binomial theorem,
Let 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
Therefore is the limit of as approaches .
To construct the point representing :
Fig. 3.
On the axis of real numbers lay off .
Draw equal to and parallel to , and divide it into equal parts. Let be one of these parts. Then is the point .
Through draw at right angles to and construct the triangle similar to .
is then the point .
In like manner construct to represent , for .
Let be indefinitely increased. The broken line will approach as limit an arc of length of the circle of radius and, therefore, its extremity, , will approach as limit the point representing (§ 47).
Therefore the limit of as is indefinitely increased is .
But this same limit has already been proved to be .
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