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Example. Find the root between 4 and 5 of

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= = 0.

16 and ƒ(5) = 55; hence 4 is nearer to the

We then have a = 4 and b = 5.

Substituting in (1), x=4+

Since f(4.2)=

=

the root than 4.2.

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- 4.072 and ƒ(4.3) = 2.297, 4.3 is nearer to

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Continuing in this way, the approximate value of the root may be found to any desired degree of accuracy.

Note This method of approximation has the advantage of being applicable to any form of equation. It may, therefore, be applied to the solution of exponential equations, and others not in the algebraic form.

APPENDIX I.

DEMONSTRATION OF THE FUNDAMENTAL LAWS OF ALGEBRA FOR PURE IMAGINARY AND COMPLEX NUMBERS (Art. 327).

Note. It will be understood throughout the following discussion that every letter represents a positive real number (Art. 318), unless the contrary is expressly stated.

728. Let XX' be a fixed straight

line, and O a fixed point on the line.

A' -a 0 + a A

We may suppose any positive real number, a, to be represented by a line OA, the point A being taken a units to the right of O in the line OX.

Then with the notation of Art. 28, any negative real number, -a, may be represented by a line OA', the point A' being taken a units to the left of O in the line OX'.

729. Since a is the same as (+ a) × (− 1), it follows from Art. 728 that the product of a by -1 is represented by turning the line OA, which represents the number + a, through two right angles, in a direction opposite to the motion of the hands of a clock.

-

We may then regard — 1, in the product of any real number by -1, as an operator which turns the line which represents the first factor through two right angles, in a direction opposite to the motion of the hands of a clock.

730. Consider the expression (+a) ×i×i (Art. 327, Note 1).

By Art. 7, this signifies that the number + a is multiplied by i, and the result multiplied by i.

If we could assume the Associative Law for Multiplication (Art. 59) to hold with respect to the product (+a) × i×xi, we should have

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(+ a) xixi = (+ a) × i2 = (+ a) × (−1) (Art. 328).

That is, to multiply a by i, and then multiply the result by i, is the same thing as to multiply

a by-1.

But by Art. 729, (+ a) × (−1) is represented by turning the line which represents the number + a through two right angles, in a direc tion opposite to the motion of the hands of a clock.

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We may then define i, or √1, in the product of any real number by i, as an operator which turns the line which represents the real number through one right angle, in a direction opposite to the motion of the hands of a clock.

B

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+i

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10+a A

-ai
Β'

Hence, if XX' and YY' are fixed straight lines which are perpendicular to each other and intersect at O, and if a is represented! by the line OA, where A is a units to the right of O in the line OX, then + ai may be represented by the line OB, where B is a units above in the line OY.

Again, with the notation of Art. 28, -ai may be represented by the line OB', where B' is a units below O in the line OY'.

The imaginary numbers + and -i are represented by the lines OC and OC', where C and C' are, respectively, one unit above, and one unit below 0, in the line YY'.

Note. It will be understood hereafter that, in any figure where the lines XX' and YY' occur, they are fixed straight lines which are perpendicular to each other and intersect at 0; that all positive or negative real numbers are represented by lines laid off to the right or left of O, respectively, in the line XX'; and that all positive or negative pure imaginary numbers are represented by lines laid off above or below O, respectively, in the line YY'.

731. We will now show how to represent any complex number (Art. 327).

Let the number be a+bi; and let the real number a be represented by the line OA, and the pure imaginary number bi by the • line OB.

Y

B

bi

a

X

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-bi
B'

Draw AC equal and parallel to OB, on the same side of XX' as OB, and join OC.

Then the complex number a + bi is represented by the line OC.

With the notation of Art. 28, the complex number (a + b) may be represented by the line OC', where

OC' is equal in length to OC, and is drawn in the opposite direction from 0.

In like manner, any complex number whatever may be represented by a straight line drawn from 0.

It follows from Arts. 730 and 731 that we may regard -1, in the product of any real, pure imaginary, or complex number by -1, as an operator which turns the line which represents the first factor through two right angles, in a direction opposite to the motion of the hands of a clock. (Compare Art. 729.)

732. In the figure of Art. 731, let C'A' be drawn perpendicular to OX'; then the right triangles OA'C' and OAC are equal, having the hypotenuse and an acute angle of one equal to the hypotenuse and an acute angle of the other, respectively.

Then OA' and A'C' are equal to OA and AC, respectively; that is,
OA' represents the real number -a, and A'C' is equal and parallel to
OB', where OB' represents the imaginary number — bi.
Therefore OC' represents the complex number — a― bi.
But OC' also represents
Whence, (a + bi) = — a — bi.

(a + bi) (Art. 731).

733. The modulus of a real, pure imaginary, or complex number is the length of the line which represents the number.

The amplitude is the angle between the line which represents the number and OX, measured from OX in a direction opposite to the motion of the hands of a clock.

If, for example, in the figure of Art. 731, the angle XOC is 30°, the amplitude of the complex number represented by OC is 30o, and the amplitude of the complex number represented by OC' is 210°.

The modulus is always taken positive, and the amplitude may have any value between 0° and 360°.

The pure imaginary numbers +ai and -ai have the modulus a, and the amplitudes 90° and 270°, respectively; and the real numbers + a and -a have the modulus a, and the amplitudes 0° and 180°,

respectively.

We have, in the figure of Art. 731, OC= √OA2 + AC2 = √ a2 + b2 ; that is, the modulus of the complex number a+bi is √a2 + b2; and this is also the modulus of each of the complex numbers a±bi. Whatever number is represented by a, the amplitude of always equal to the amplitude of a increased by 180°.

Note. We may regard zero as having the modulus zero.

Addition and Subtraction of Imaginary Numbers.

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734. The representation of a complex number, as explained in Art, 731, shows that the result of adding a pure imaginary to a real number may be represented by a straight line drawn from 0.

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We will now show how to represent the result of adding b to a where a and b represent any two real, pure imaginary, or complex numbers.

B

a

a+b

Let a be represented by OA, and b by OB.

Draw AC equal and parallel to OB, in such a way that C shall be in the same direction fron A that B is from 0.

Then the result of adding b to a is represented by the line OC.

That is (Art. 5), a + b is represented by OC.

Note 1. The above construction holds equally when OA and OB lie in the same direction, or in opposite directions, from 0.

Note 2. The form of addition exemplified in the above construction is known as Geometric Addition.

In like manner, the result of adding any number of real, pure imaginary, or complex numbers may be represented by a straight line drawn from 0.

735. In the figure of Art. 734, draw BC.

By Geometry, OACB is a parallelogram, and therefore BC is equal and parallel to OA.

Then OC represents the result of adding a to b.

But OC also represents the result of adding b to a.
Whence, a+b=b+a. (Compare Art. 36.)

The above result holds if either of the letters a and b represents the sum of any number of real, pure imaginary, or complex numbers.

B

736. We shall define the subtraction of b from a, where a and b

b

a

B'

represent any two real, pure imaginary, or complex numbers, as the process of finding a number such that, when b is added to it, the sum shall be equal to a. (Compare Art. 41.) Let a be represented by OA, and b by OB; and complete the parallelogram OBAC.

By Art. 734, OA represents the result of adding the number represented by OB to the number represented by OC; that is, if b is added to the number represented by OC, the sum is equal to a.

Therefore, ab is represented by the line OC.

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