Fig. 15.1 in .NET

Encoder QR Code 2d barcode in .NET Fig. 15.1

Let z1 x1 iy1 and z2 x2 iy2 be two complex numbers. The vector representation of these numbers [Fig. 15.2(a)] suggests the familiar parallelogram law for determining graphically the sum z1 z2 (x1 iy1) (x2 iy2). Since z1 z2 (x1 iy1) (x2 iy2) (x1 iy1) ( x2 iy2), the difference z1 z2 of the two complex numbers may be obtained graphically by applying the parallelogram law to x1 iy1 and x2 iy2. [See Fig. 15.2(b).] In Fig. 15.2(c) both the sum OR z1 z2 and the difference OS z1 z2 are shown. Note that the segments OS and P2P1 (the other diagonal of OP2RP1) are equal. (See Prob. 15.11.)

Let the complex number x yi be represented [Fig. 15.3(a)] by the vector OP. This vector (and hence the complex number) may be described in terms of the length r of the vector and any positive angle which the vector makes with the positive x axis (axis of positive reals). The number r 2x2 y2 is called

the modulus or absolute value of the complex number. The angle , called the amplitude of the complex number, is usually chosen as the smallest positive angle for which tan y/x, but at times it will be found more convenient to choose some other angle coterminal with it. From Fig. 15.3(a), x r cos and y r sin ; then z x yi r cos ir sin r(cos i sin ). We call z r(cos i sin ) the polar or trigonometric form and z x yi the rectangular form of the complex number z. An abbreviated notation is sometimes used and is written z r cis .

i 23 in polar form. [See Fig. 15.3(b).] ( 23)2 2. Since tan y/x 23>1 23, the amplitude is either 120 r(cos i sin ) i sin (300 2[cos (300 n360 )

#(1)2

(See Probs. 15.12 and 15.13.)

The modulus of the product of two complex numbers is the product of their moduli, and the amplitude of the product is the sum of their amplitudes.

The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modulus of the divisor, and the amplitude of the quotient is the amplitude of the dividend minus the amplitude of the divisor. For a proof of these theorems, see Prob. 15.14.

EXAMPLE 15.7 Find (a) the product z1z2, (b) the quotient z1/z2, and (c) the quotient z2/z1 where z1 i sin 210 ). i sin 300 ) and z2 8(cos 210

(a) The modulus of the product is 2(8) 16. The amplitude is 300 we shall use the smallest positive coterminal angle 510 360 (b) The modulus of the quotient z1/z2 is 2/8 i sin 90 ). (c) The modulus of the quotient z2/z1 is 8/2 smallest positive coterminal angle 90 z2/z1

210 510 , but, following the convention, i sin 150 ). 150 . Thus z1z2 16(cos 150 210 300 90 . Thus z1/z2

and the amplitude is 300 4. The amplitude is 210 360 270 . Thus 4(cos 270 i sin 270 )

[NOTE: From Examples 15.5 and 15.6, the numbers are z1 in rectangular form. Then z1z2 as in (a), and z2 z1 423 1 4i ( 4 23 (1 i sin 270 ) 4i) (1 i 23) i 23) 16i 4 4i (1 i 23)( 4 23 4i) 8 23 8i 16(cos 150 i sin 150 ) 1 i 23 and z2 4 23 4i

i 23

i 23) (1

4( cos 270 as in (c).]

(See Probs. 15.15 and 15.16.)

A 23

(See Prob. 15.18.)

We state, without proof, the theorem: A complex number a bi r(cos i sin ) has exactly n distinct nth roots. The procedure for determining these roots is given in Example 15.9.