lengths which cannot be expressed by integers or fractions. Such points represent incommensurable numbers. E.g. if we lay off on OX a line OB equal to the hypotenuse

of a right triangle whose other sides are equal to 1 unity, OB represents √2, a number which cannot be equal to an integer or a fraction.

1

For assuming that = √2, where m and n have no common factor,

n

2, which is obviously impossible, as m2 and n2 have

NOTE. While it is impossible to find integers or fractions which are exactly equal to an irrational number, we can find fractions which differ from the given surd by less than any number which we can assign. Thus, √2 differs from 1.4, 1.41, 1.414

.01, .001, etc.

...

respectively by less than .1,

Hence we may consider √2 the limit of the fraction 1.41421 Every irrational number may be regarded as the limit of a variable rational number.

Thus, every real number is represented by a point in the line XX', and, vice versa, every point in XX' represents a real number.

3. Imaginary numbers can be represented by points without XX', as may be shown by the following consideration:

If A, be taken a units to the right of O in OX, then the line OA, represents the number a, and an equal line OA, drawn in the opposite direction, represents

AY

while algebraically +a is transformed into a by multiplying

by 1. Hence it follows:

431. The multiplication of a real number by — 1, is represented. graphically by a rotation through an angle of 180°.

It is customary to rotate lines counter-clockwise, i.e. in a direction opposite to the motion of the hands of a clock.

432. To determine the algebraic meaning of a rotation through an angle less than 180°, let a rotation through an angle of 90° represent the multiplication by an unknown number x.

433. We may therefore consider a multiplication by √-1 to be represented by a revolution through a right angle counterclockwise.

434. The numbers i, 2 i, 3 i, etc., may be represented respectively by the distances 1, 2, 3, etc., laid off on OY, which is perpendicular to XX'.

Similarly, -i, -2i, 3 i, are represented on the line OY'.

The four quantities, 3, 3 i, 3,

3 i, have the same absolute value,

viz. 3, and each is represented by a line consisting of three units, but extending in different directions, viz. :

435. The line XX' is called the axis of real numbers; YY' is called the axis of imaginaries. The point O is called the origin.

436. Graphical addition. Two real numbers, OA and OB, are added graphically by drawing from A a line AC equal to

B

C

and extending in the same direction as OB. The line OC is the required sum.

E.g. To add + 5 and lay off AC3 to the left.

3, lay off OA = 5 to the right, and from A, OC or 2 is the required sum.

Y

B+31

437. Imaginaries and imaginaries, or real numbers and imaginaries, are added graphically in the same manner as real numbers. E.g. to add 4 and 3i, lay off OA=+4. From A draw AC-3 upward, i.e. equal and parallel to line 3 i, or OB. OC represents the required sum.

2i

4+31

3

NOTE. It should be noted that 4+ 3 i is represented by the length and direction of OC. but it would be erroneous to assume that 4 + 3 5 is the absolute value of 4+ 3 i, but it is not equal to 4 + 3 i.

i

The length of OC is 5, equals 5. The number

439. The absolute value or modulus of any number (i.e. real, pure imaginary, and complex) is the length of the line which represents the number. It is always taken as positive.

The absolute value of a + bi = OB, or + √a2 + b2.

440. The amplitude of OB is the angle XOB, i.e. the angle between OX and OB, measured from OX counter-clockwise.

Ex. 1. Determine the algebraic meaning of the rotation of a line through an angle of 60°.

If x is the number which, applied as a factor, produces the required

The rotation through an angle of 60° represents therefore a multiplication by 1, and line OB represents V-1. A simple geometrical deduc

tion shows that OB or √− 1 = 1 + ¿√3 • i.

Ex. 2. Construct graphically the different values of V1.

From O, draw five lines, OA, OB, OC, OD, and OE, each equal to 1, and forming angles of 72° with the two adjacent lines. If OA lies on the axis of real numbers, OA, OB, OC, OD, and OE represent the 5 values of VI. (The proof is similar to Ex. 1.) By actual measurement we find 5 values 1, .31 +.95 i, - .95 + .31 i, -.95 .31 i, .31 -.95 i.

EXERCISE 145

Represent the following numbers graphically:

B

E

ΤΑ Χ

7. Construct 5+4 i and multiply graphically by V-1.

8. Prove that the following numbers have equal absolute values: 43 i, 3-4 i, 5, and 4+3 i.

9. Which has the greater absolute value, -8 or 5-6 i? 10. Add graphically 4, 3 i, and -2.

11. Add graphically

5, -2i, +2, +5 i.

12. Solve the equation x2+2x+2=0, and represent the roots graphically.

13. Solve the equation -1=0, and represent the roots graphically.

14. Solve the equation +1=0, and represent the roots graphically.

15. Divide 2i by 1+i, and represent the quotient graphically. 16. What rotation is equivalent to a multiplication by -i? 17. Find graphically Vi.

18. Find one value of 1.