# Remembering that any complex number can be written in the form re" by (9.4),we getSection 9Euler's Formula632 e2 = ri r2 e4(02+@2),e21 22(9.6)In words, to multiply two complex numbers, we multiply their absolute valuesand add their angles. To divide two complex numbers, we divide the absolutevalues and subtract the anglesExample. Evaluate (1 + i)?/(1 - i). From Figure 5.1 we have1 2et/4. We plot 1 - i in Figure 9.5 and findr2, 0=/4 (or +7/4), so 1-i= /2e-i#/4. Then(VEetr/a)2V2e-i7/4=J%-in/s = /2e3i#/42 eir/2(1+i)21-1Figure 9.5From Figure 9.6, we find =-1, y = 1, so(1i)21-1=riy-1+iWe could use degrees in this problem. By (9.6), we findthat the angle of (1 i)2/(1-i) is 2(45°) - (-45°) 135°as in Figure 9.6Figure 9.6PROBLEMS, SECTION 9Express the following complex numbers in the r + iy form. Try to visualize each complexnumber, using sketches as in the examples if necessary. The first twelve problems youshould be able to do in your head (and maybe some of the others-try it!) Doing aproblem quickly in your head saves time over using a computer. Remember that the pointin doing problems like this is to gain skill in manipulating complex expressions, so a goodstudy method is to do the problems by hand and use a computer to check your answers.3. 93ri/2e-2i -4mi -2. ei/21. ei/4(a/3)(344mi4.6.5.7. 3e2(1+i2esri/69.2e-i/2/44e-Sin/310.11.12.(i)1-i(1+ W)15. (1 (1 i)*14.13.((-)(1+v)17.16()19. (1-)2120.

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