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complex numbers worksheet doc

/Meta378 391 0 R >> . /Meta46 57 0 R endobj Q Q 1.547 0.283 l q /Resources << q /Resources << 0 0.283 m [(6)] TJ 0000207412 00000 n 712 0 obj << /Resources << /Type /XObject stream /BBox [0 0 1.547 0.283] /Resources << 0.232 0.087 TD 0 0 l W* n /Meta582 597 0 R stream 0.248 0.087 TD q 0 g q /Resources << 45.249 0 0 45.131 217.562 289.079 cm >> q 301 0 obj << /Meta1090 1107 0 R /BBox [0 0 11.988 0.283] >> 0000290434 00000 n /Font << /BBox [0 0 1.547 0.33] q ET 45.663 0 0 45.147 90.337 718.183 cm 2 0 obj << 520 0 obj << /Type /XObject /Meta538 553 0 R 45.214 0 0 45.147 81.303 637.632 cm /Meta420 Do Q /Meta428 443 0 R /Subtype /Form >> Dividing Complex Numbers Simplify. 1.547 0.633 l >> >> endobj Q 0.564 G 0 G /Meta223 Do stream 0000203715 00000 n /Matrix [1 0 0 1 0 0] /Meta646 661 0 R >> /Matrix [1 0 0 1 0 0] 0.564 G BT /Meta146 157 0 R 9.791 0.283 l q endobj stream >> [(A\))] TJ Q /BBox [0 0 0.413 0.283] /FormType 1 /Type /XObject /FormType 1 Q q 258 0 obj << 1 g /Meta178 189 0 R Q 0.564 G /Type /XObject >> Q 0 0.283 m >> 45.324 0 0 45.147 54.202 691.834 cm 0 g /Type /XObject 0000021841 00000 n /Type /XObject 45.527 0 0 45.147 523.957 550.305 cm q /Type /XObject 0 g [(D\))] TJ Q ET 1 g q 1.547 0.283 l 0 0 l endobj 45.214 0 0 45.131 81.303 244.664 cm 0.564 G Q /Type /XObject /Type /XObject Q /F1 0.217 Tf /FormType 1 endobj 45.249 0 0 45.413 441.9 423.833 cm Q q /Font << q q 0.015 w q 9.791 0.283 l 45.663 0 0 45.147 90.337 616.553 cm 0.458 0 0 RG q 45.663 0 0 45.147 426.844 578.912 cm 0.948 0.308 TD /Meta443 458 0 R 0000155184 00000 n 0000135810 00000 n /Matrix [1 0 0 1 0 0] 0.458 0 0 RG 0.564 G endstream 0.015 w q /BBox [0 0 0.263 0.283] /Meta789 Do 45.249 0 0 45.147 329.731 720.441 cm 0.564 G 0.267 0.283 l Q 717 0 obj << 0000184003 00000 n Q /F1 6 0 R ET >> W* n 1 g /BBox [0 0 1.547 0.633] 0 G 0 w /Meta367 380 0 R /Meta881 Do >> /Subtype /Form >> 936 0 obj << Q >> 320 0 obj << 0 G [(i)] TJ /Meta39 50 0 R Q /F3 21 0 R 0.564 G /BBox [0 0 1.547 0.633] /BBox [0 0 1.547 0.33] 45.249 0 0 45.131 329.731 289.079 cm endobj /F1 6 0 R q q q /Root 2 0 R 0000142569 00000 n /Meta613 628 0 R /FormType 1 Q 0 g Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. Q 45.663 0 0 45.147 90.337 447.923 cm 0000353005 00000 n Q 1 J >> 0 0 l Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. 0.114 0.087 TD 982 0 obj << 0 g 0 g 0.458 0 0 RG S ET 519 0 obj << q Q q /Font << 45.663 0 0 45.147 426.844 674.519 cm /BBox [0 0 0.263 0.5] endobj /Length 8 q 45.413 0 0 45.147 523.957 380.923 cm /Font << Q W* n Q /FormType 1 >> BT 0.417 0.283 l /Meta688 703 0 R /Meta278 Do q /FormType 1 0 g /Subtype /Form 0 0 l >> q /F3 0.217 Tf 0.082 0.2 m 0.458 0 0 RG 45.413 0 0 45.147 523.957 227.349 cm Q 45.249 0 0 45.527 105.393 468.249 cm 963 0 obj << q /Meta373 386 0 R Q q Q q q 861 0 obj << 816 0 obj << >> 237 0 obj << /FormType 1 Q 0000000646 00000 n >> 45.226 0 0 45.147 81.303 273.27 cm 0000068131 00000 n q 0 0.283 m 0 g 45.214 0 0 45.131 81.303 390.709 cm 45.213 0 0 45.147 36.134 174.652 cm 738 0 obj << Q 0.381 0.087 TD /Resources << endstream >> 1 j Q 0 w /Length 51 /Meta344 Do 0 G >> /BBox [0 0 9.523 0.283] >> q q 45.663 0 0 45.147 202.506 263.484 cm Q Q [(D\))] TJ /F1 6 0 R Q S q /F1 0.217 Tf BT [(1)] TJ >> Q /Resources << /Font << /Type /XObject /Length 560 q q /Meta817 Do [(i)] TJ 0000274413 00000 n Q /Meta968 Do /Resources << endstream Q 0 0 l >> S 0.564 G 0.564 G ET Q >> 0.015 w /Meta770 Do >> 0.015 w 0.015 w q 0 0.283 m q /Length 55 Q stream 0 w /Meta4 12 0 R q 0 g 0.232 0.308 TD 45.249 0 0 45.131 105.393 216.057 cm 0 0 l 45.249 0 0 45.413 105.393 513.418 cm endstream 0.015 w /Meta35 46 0 R 0.531 0.283 l 3. q /Meta1078 1095 0 R /FormType 1 /Resources << /Meta544 Do 45.249 0 0 45.147 329.731 679.036 cm /Meta994 1009 0 R stream q >> q 0.267 0.283 l q /Type /XObject 0.015 w stream 0000206937 00000 n 0.458 0 0 RG 313 0 obj << endobj /Meta1036 1053 0 R /Type /XObject /Type /XObject 0.598 0.158 TD endobj 1.547 0.283 l 0000285337 00000 n 0.334 0.299 l /Meta557 572 0 R >> /Meta465 Do Q /BBox [0 0 1.547 0.283] [(C\))] TJ /BBox [0 0 1.547 0.283] 0 0.283 m 0000022326 00000 n q 45.249 0 0 45.527 329.731 578.912 cm /Font << >> /Matrix [1 0 0 1 0 0] q Q 812 0 obj << Q /Length 55 0000067644 00000 n ET /Font << >> /Font << q Q Find N when a = 1000 acres and p = 75 lb/acre. Q Q q endobj >> endstream /BBox [0 0 0.531 0.283] q 0000340862 00000 n endobj 45.233 0 0 45.147 329.731 616.553 cm /Type /XObject 0 G /Meta856 871 0 R /Matrix [1 0 0 1 0 0] 45.249 0 0 45.527 441.9 535.249 cm Q 45.214 0 0 45.131 81.303 244.664 cm /Matrix [1 0 0 1 0 0] 0 g >> 0 g q 45.249 0 0 45.147 217.562 674.519 cm /FormType 1 0 w /Font << 0 0.283 m >> /Matrix [1 0 0 1 0 0] Q 0 0.283 m Q q Q /F1 0.217 Tf stream 843 0 obj << 45.249 0 0 45.413 217.562 423.833 cm /F1 6 0 R >> Q endstream q /Meta767 782 0 R /Meta640 655 0 R >> /Subtype /Form 45.413 0 0 45.147 523.957 528.474 cm Q endstream /Meta178 Do 0.458 0 0 RG /Matrix [1 0 0 1 0 0] /Subtype /Form BT 0000282117 00000 n q Q /FormType 1 q Q 0.031 0.154 TD >> Q endstream W* n /Meta193 Do endobj /Length 55 >> stream Q /Meta943 Do Q /Matrix [1 0 0 1 0 0] >> 1.996 0.087 TD 0 g W* n 1.547 0 l Q BT W* n /Meta937 952 0 R endstream Q endstream 0 0.283 m ET q >> endobj /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.283] W* n ET Q /F2 0.217 Tf Q Q 0 G stream 0 G [(3)] TJ 0 g 0 G /Meta162 173 0 R q 0 g /Subtype /Form 1.547 0.283 l 0 0 l /Matrix [1 0 0 1 0 0] >> >> endstream 0000268293 00000 n endstream /Type /XObject endstream >> /Length 55 q 45.249 0 0 45.147 441.9 447.923 cm /Length 55 45.213 0 0 45.147 36.134 642.149 cm 9.791 0 l 1 g Q endobj 0 w >> /Meta949 964 0 R q 1.547 0 l Q 0 g 0 g /Meta396 411 0 R 0 0.283 m /Subtype /Form /BBox [0 0 1.547 0.283] >> 0.458 0 0 RG /BBox [0 0 0.263 0.283] Q /F1 0.217 Tf 9.523 0.633 l /FormType 1 endobj 0.458 0 0 RG 1.547 0.283 l /Meta633 Do 0 g >> 3 + 2i. 45.249 0 0 45.527 441.9 622.575 cm /Font << 0.564 G /Matrix [1 0 0 1 0 0] Q endstream /Resources << /Length 8 9.523 0 l >> Q >> q q /Type /XObject Q 0000083634 00000 n /Subtype /Form 0 0.087 TD /Meta199 Do /Meta1027 1042 0 R /F1 0.217 Tf 0.433 0.158 TD 0.458 0 0 RG /FormType 1 0.047 0.087 TD 0 G /FormType 1 q 0.531 0 l 45.663 0 0 45.147 314.675 679.036 cm 0.484 0.366 l ET 0.458 0 0 RG Q q Q >> /Subtype /Form >> >> /Meta472 Do [(14)] TJ /Subtype /Form /Meta341 Do /F3 0.217 Tf endobj W* n BT /Resources << stream 0.015 w 0.267 0 l Q 0000081829 00000 n 45.214 0 0 45.527 81.303 687.317 cm /BBox [0 0 1.547 0.33] 45.249 0 0 45.413 441.9 263.484 cm 0.267 0.283 l 0.267 0.283 l /Type /XObject /Resources << 0.267 0 l /Matrix [1 0 0 1 0 0] Q 0 0.283 m Q q 0000232080 00000 n stream stream /Length 63 0 0 l 45.249 0 0 45.131 217.562 289.079 cm q q 0000095813 00000 n [(+)] TJ 11.988 0.283 l 1.547 0 l >> 0 0.087 TD /Font << /BBox [0 0 1.547 0.633] ET -0.002 Tc /Length 102 0.5 0.308 TD 0 g >> /BBox [0 0 1.547 0.633] 0 w >> BT 0000283530 00000 n Q endstream Q /Subtype /Form /F3 0.217 Tf /BBox [0 0 1.547 0.283] /F1 6 0 R 45.663 0 0 45.147 202.506 720.441 cm q 0.232 0.437 TD /F1 6 0 R /Meta325 338 0 R ET q Q endstream Q Q /Matrix [1 0 0 1 0 0] >> 0 w Q /Type /XObject /Meta188 Do [(65)] TJ /Type /XObject 0 g 0.458 0 0 RG 0 G 0.015 w /F1 0.217 Tf >> endobj /Type /XObject Q 1 g 0.83 0.087 TD W* n >> /Meta579 Do stream Q Q /Type /XObject Q /Font << 979 0 obj << /Length 8 0 g /F1 6 0 R 0 0.33 m endstream [(9\))] TJ >> 0.001 Tc 0.015 w q >> /Matrix [1 0 0 1 0 0] 0 g [(1)] TJ q stream /FormType 1 /FormType 1 Q 0 0 l 0.267 0 l 45.663 0 0 45.147 314.675 535.249 cm If two complex numbers, say a +bi, c +di are equal, then both their real and imaginary parts are equal; a +bi =c +di ⇒ a =c and b =d. 0 0.283 m BT >> /F1 6 0 R Q Q /BBox [0 0 1.547 0.283] >> W* n q /BBox [0 0 1.547 0.633] /Meta46 Do q >> 1.547 0 l 0000144504 00000 n 45.663 0 0 45.147 314.675 630.856 cm /Meta920 Do W* n /F1 6 0 R stream /BBox [0 0 0.263 0.283] Q /Matrix [1 0 0 1 0 0] /Meta752 767 0 R Q 0 g /Type /XObject 0 G 45.249 0 0 45.131 217.562 216.057 cm Q 419 0 obj << W* n 45.249 0 0 45.527 217.562 491.586 cm >> 740 0 obj << 0 0.283 m /Subtype /Form /Meta736 751 0 R 533 0 obj << /BBox [0 0 0.263 0.283] BT 0 0.283 m 0 0.087 TD 0.066 0.087 TD 0.015 w >> Q endstream /Font << /BBox [0 0 0.263 0.283] /F3 0.217 Tf /Font << /Length 8 0 0 l 0.031 0.087 TD >> 0.015 w /F3 21 0 R 0000219271 00000 n 1 J >> /Matrix [1 0 0 1 0 0] >> 0 G 0 G /BBox [0 0 1.547 0.633] 0 0.283 m 0000006487 00000 n 0 g >> endobj 1 j /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.633] 0 0 l -0.007 Tc >> endobj Q /F3 0.217 Tf BT /FormType 1 Q 0 0 l /Length 72 /BBox [0 0 9.523 0.7] /Length 55 >> Q 0 g /BBox [0 0 0.263 0.5] >> 0000357348 00000 n stream W* n /Font << q 0000188774 00000 n 45.663 0 0 45.147 426.844 203.259 cm /Meta649 664 0 R /FormType 1 0.267 0 l 45.249 0 0 45.147 105.393 447.923 cm /Type /XObject 9.523 -0.003 l /Meta188 199 0 R /Resources << 0 0.283 m Q 1 g /Subtype /Form -0.002 Tc /F3 0.217 Tf >> ET /Matrix [1 0 0 1 0 0] /Resources << endstream >> >> 45.249 0 0 45.413 217.562 263.484 cm /BBox [0 0 1.547 0.633] 1.547 0.283 l /XObject << Q stream /BBox [0 0 1.547 0.283] 0 0.283 m /Meta213 Do /Meta261 Do /Meta128 Do >> 45.663 0 0 45.147 314.675 578.912 cm q stream /BBox [0 0 0.263 0.283] ET q 0 0.087 TD /Resources << /F3 0.217 Tf 1.547 0.283 l q /Meta1014 Do >> endobj Q /Subtype /Form >> /Type /XObject /Font << 1096 0 obj << Q /Length 55 1052 0 obj << 0.015 w Q Q /Meta133 Do /Subtype /Form Q /Meta653 668 0 R /Matrix [1 0 0 1 0 0] >> /Length 76 0 g Q q q q 928 0 obj << /BBox [0 0 1.547 0.283] 0.564 G q 0000023585 00000 n /Descent -277 0.248 0.366 l endstream q 0000363170 00000 n ET /Matrix [1 0 0 1 0 0] >> Q 0 G /Type /XObject >> 1.547 0 l >> /BBox [0 0 9.523 0.283] endstream 0 0 l 0000181580 00000 n Q Q Q /Type /XObject 0000291631 00000 n stream /Subtype /Form q Q /Length 51 /Matrix [1 0 0 1 0 0] 0000220984 00000 n endstream q /Matrix [1 0 0 1 0 0] /Font << 45.663 0 0 45.147 202.506 491.586 cm /Meta28 39 0 R /Type /XObject 0.417 0.283 l /Meta616 631 0 R q Q 9.523 0.7 l q >> q 0 0.283 m >> 1 g /F3 21 0 R Q /Resources << Q S /BBox [0 0 1.547 0.283] /Type /XObject 284 0 obj << q endstream /Type /XObject 1.547 0.33 l Q ET /Meta558 573 0 R 11.988 0 l 0 w [(i)] TJ Q /FormType 1 /Subtype /Form endobj /F1 0.217 Tf BT 0 G /Subtype /Form stream stream 1.232 0.087 TD /Meta382 395 0 R -0.007 Tc q endobj /Type /XObject endstream 0 g 0.267 0 l >> 0000143881 00000 n 0 0 l [( 8)] TJ BT 1 g /Matrix [1 0 0 1 0 0] 0.267 0.5 l 0.458 0 0 RG 0 g >> 0 0.633 m ET endstream /Meta765 780 0 R /Meta936 Do /FormType 1 /Font << Q endobj q ET /Matrix [1 0 0 1 0 0] 0.35 0.087 TD 0 g 0.458 0 0 RG q Q q endstream 1.547 0 l /Matrix [1 0 0 1 0 0] 0 w 0000173492 00000 n endstream 780 0 obj << /Resources << >> 0 G 0000083863 00000 n /FormType 1 /FormType 1 /Font << /Type /XObject /Meta668 683 0 R /Length 8 [(16)] TJ 909 0 obj << /Meta667 Do /Subtype /Form q /FormType 1 /Matrix [1 0 0 1 0 0] q 9.523 0.283 l >> /Length 55 /F1 0.217 Tf 0.015 w /Meta684 Do 0000233390 00000 n BT 0000030539 00000 n >> 425 0 obj << 45.214 0 0 45.413 81.303 338.012 cm /Meta605 Do 0.066 0.087 TD >> >> /F1 0.217 Tf >> endstream /Meta908 Do /Subtype /Form Q BT 1 g /Meta60 Do q Q 0 0 l 0000176423 00000 n >> 1.547 0.633 l q Q stream 9.523 0.7 l /Length 94 0 g >> >> /FormType 1 Q BT /Meta510 525 0 R stream /Matrix [1 0 0 1 0 0] /Meta200 211 0 R >> 0000168391 00000 n stream /Resources << /Length 62 Q /Length 67 Q /Matrix [1 0 0 1 0 0] 625 0 obj << Q /Matrix [1 0 0 1 0 0] /Resources << 797 0 obj << 0.531 0 l /Resources << /Meta658 673 0 R Q 0 g q ET 1 g /Resources << /Subtype /Form 0 g >> Q endstream q >> /BBox [0 0 1.547 0.633] 45.214 0 0 45.527 81.303 460.721 cm stream W* n Q Q /Font << Q /BBox [0 0 0.263 0.283] >> W* n Q /Meta759 774 0 R /Length 76 /Meta862 Do -0.002 Tc 0 -0.003 l /FormType 1 /FormType 1 >> >> 0000139185 00000 n stream /Font << 0000083037 00000 n >> Q /F1 6 0 R Q 0 g Q 0 g /Length 8 Q /Subtype /Form /F3 21 0 R 1.547 -0.003 l endstream 0 0 l /Subtype /Form 0.267 0 l >> Q q 0.948 0.087 TD stream W* n 0.283 0.134 TD 0 0 l q BT /Type /XObject /Length 8 >> /Length 212 /Resources << 0.267 0 l Q /Resources << 0.015 w /Matrix [1 0 0 1 0 0] /Type /XObject endobj Q q Q 45.663 0 0 45.147 90.337 674.519 cm 0 0 l /FormType 1 /Matrix [1 0 0 1 0 0] /FormType 1 >> /BBox [0 0 0.531 0.283] endobj 9.791 0 0 0.283 0 0 cm /Meta572 587 0 R /F1 6 0 R /Matrix [1 0 0 1 0 0] 0 G /Meta1088 Do /F1 0.217 Tf 0 0.283 m q /Subtype /Form BT 0.015 w 45.249 0 0 45.131 105.393 289.079 cm stream endstream /BBox [0 0 1.547 0.33] ET 0 0.087 TD Q endobj 0.767 0.366 l /Meta153 Do Q /Type /XObject q stream 1 J 0.458 0 0 RG >> /Matrix [1 0 0 1 0 0] /Meta245 256 0 R 0.458 0 0 RG 0 G /F1 6 0 R /Subtype /Form 269 0 obj << 0.458 0 0 RG endobj Q q /Meta1106 Do /BBox [0 0 1.547 0.283] /Font << /Type /XObject endobj 1 J /F1 0.217 Tf /Meta761 776 0 R Q >> Q /Subtype /Form Q /Type /XObject BT W* n 0 g 0.547 0.087 TD /Matrix [1 0 0 1 0 0] Q Q /Meta372 385 0 R /Subtype /Form q 0 G stream 0.458 0 0 RG >> endstream 0 G 0 G 45.663 0 0 45.147 90.337 325.214 cm q Q /Meta119 Do /Type /XObject 45.249 0 0 45.131 441.9 143.034 cm /Matrix [1 0 0 1 0 0] 45.249 0 0 45.147 441.9 720.441 cm ET 0 0.283 m q 260 0 obj << q /F3 21 0 R 0 0 l 0000345254 00000 n Q 9.791 0 l 0000076099 00000 n stream /FormType 1 /Meta635 Do 1 g /Length 102 q W* n Q 802 0 obj << 1.547 0.633 l 45.249 0 0 45.131 105.393 216.057 cm BT stream /FormType 1 /Matrix [1 0 0 1 0 0] /Subtype /Form 45.249 0 0 45.413 441.9 468.249 cm endobj 0 g 0 0.087 TD /Subtype /Form Q q Q /Meta159 Do The multiplication of two complex numbers is performed using properties similar to those of the real numbers (FOIL) and distributive property. 45.663 0 0 45.147 426.844 263.484 cm stream >> >> /Meta833 Do 0.047 0.087 TD /Length 67 [(5)] TJ S >> Q Q /F3 21 0 R /Subtype /Form /F1 0.217 Tf Q /Meta11 Do BT Q Q >> 0.267 0.283 l /Resources << Q /F3 21 0 R /Meta452 467 0 R endstream 757 0 obj << /F3 0.217 Tf /Matrix [1 0 0 1 0 0] q 1 g 0.665 0.366 l 0000006728 00000 n endstream 218 0 obj << /Meta302 Do /Type /XObject /Meta345 Do /Type /XObject stream 0 0 l /Length 228 0.458 0 0 RG endobj q endstream /F1 0.217 Tf /F1 6 0 R q Q 994 0 obj << 0000150573 00000 n BT 45.214 0 0 45.147 81.303 733.239 cm endstream /FormType 1 Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 /FormType 1 /Length 67 /F2 9 0 R /Subtype /Form ET 1 g Q [(i)] TJ /Length 67 /Resources << endobj >> /Matrix [1 0 0 1 0 0] endobj /Resources << 730 0 obj << q q /BBox [0 0 9.523 0.633] endobj 0 0.366 m /Subtype /Form Q BT 0 0 l >> /BBox [0 0 0.263 0.283] endobj /Meta41 Do q q 0 0 l 0 0 l /Meta407 422 0 R q 0 0.283 m endobj /Meta326 339 0 R /Meta1029 1044 0 R /Font << Q 0.566 0.366 l 45.663 0 0 45.147 426.844 630.856 cm Q 0000186964 00000 n Q 0000243325 00000 n Q BT 0 w /Font << /F1 6 0 R 0 0.283 m 45.249 0 0 45.527 329.731 491.586 cm /Matrix [1 0 0 1 0 0] 0000012717 00000 n q /Subtype /Form 0 0.33 m Q 0.267 0.283 l q Q /FormType 1 0 0 l q Q 0.598 0.158 TD /Type /XObject endobj /Meta340 353 0 R 0 G endobj Q /Meta373 Do 0.114 0.087 TD 0 g stream ET /Matrix [1 0 0 1 0 0] /Subtype /Form 0 g 1 g /BBox [0 0 1.547 0.33] /Subtype /Form Q /Subtype /Form q q /F1 0.217 Tf >> 0.267 0 l q /Meta958 Do endstream 0 0 l 1 j /Resources << /Matrix [1 0 0 1 0 0] 1099 0 obj << Q 0 0.283 m 1 g 964 0 obj << 45.663 0 0 45.147 426.844 86.573 cm 0000151548 00000 n 45.249 0 0 45.413 217.562 558.586 cm /Font << 1 g q 0000025645 00000 n >> /Type /XObject endobj stream endstream /F3 0.217 Tf >> q 0 0 l Q ET /Type /XObject ET /Length 94 /FormType 1 ET >> /FormType 1 0 g 545 0 obj << 0000229591 00000 n /Meta187 198 0 R 9.523 0 l 0 G >> Q >> 0 0.7 m 0 G q /Resources << /Length 8 /Meta49 60 0 R /F1 6 0 R endobj 0 0 l Q Q /Subtype /Form /Length 55 Q stream 767 0 obj << q /Font << /Type /XObject Q 1 g 0 0.087 TD 45.249 0 0 45.131 217.562 289.079 cm 0 0.5 m /XObject << /Resources << [(i)] TJ Q stream stream /Meta529 Do 0.267 0 l /FormType 1 /Meta875 890 0 R 0.564 G /Meta689 704 0 R /Matrix [1 0 0 1 0 0] q 542.777 550.305 m 0000212339 00000 n >> >> 0000009687 00000 n Q /F1 6 0 R /Length 55 0 w Q /Subtype /Form q /Meta448 463 0 R /Type /XObject endstream >> 0000348385 00000 n endobj /Font << 45.249 0 0 45.147 329.731 447.923 cm q 405 0 obj << >> BT 1.547 0 l q /Font << /Meta114 125 0 R endobj /F2 0.217 Tf >> [(i)] TJ /FormType 1 stream /BBox [0 0 1.547 0.633] q 0.564 G Q 0.564 G >> endobj endobj 0 g 0 g /F1 6 0 R BT 0.015 w Q 0000283775 00000 n Q endobj -0.002 Tc /Matrix [1 0 0 1 0 0] 0.458 0 0 RG /Type /XObject Q 1.547 0.283 l [(5)] TJ 0 g q /BBox [0 0 0.263 0.283] 0.267 0.087 TD 0000276337 00000 n >> >> /Type /XObject >> /Subtype /Form 0 0.283 m 0.015 w /Meta318 Do stream 0000255246 00000 n stream Q /Font << [(+)] TJ 0 w /Subtype /Form endobj Q 0.458 0 0 RG [(+)] TJ 45.249 0 0 45.147 329.731 447.923 cm W* n /FontName /TestGen,Bold 1 J ET 0000276483 00000 n >> endobj /Subtype /Form /F3 21 0 R /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] 0.015 w /BBox [0 0 1.547 0.314] q /Length 67 q >> 296 0 obj << endstream 1.547 0 l 0.001 Tc /Type /XObject endstream q q q 3 + 4i is a complex number. /FormType 1 /Length 55 /Meta781 Do 0 G endobj 0.2 0.437 TD >> /Meta52 63 0 R /FormType 1 Q /F1 6 0 R q /Resources << ET /Matrix [1 0 0 1 0 0] 45.249 0 0 45.147 329.731 447.923 cm 0000338693 00000 n Q 597 0 obj << /Subtype /Form /FormType 1 /Meta512 527 0 R /Length 55 /Meta678 693 0 R [(B\))] TJ 0.267 0.5 l 0.015 w /FormType 1 Q /Length 67 BT Q 0000287539 00000 n /F1 0.217 Tf /Subtype /Form Q q ET /Matrix [1 0 0 1 0 0] Q 0000198693 00000 n /F3 0.217 Tf 0.458 0 0 RG >> >> stream /F1 6 0 R endstream endstream >> 45.214 0 0 45.147 81.303 550.305 cm /Meta1010 Do >> stream q /F1 0.217 Tf /Meta322 Do [(i)] TJ Q /Meta44 55 0 R >> Q >> ET Q q BT endstream /Matrix [1 0 0 1 0 0] W* n Q 1 g /FormType 1 0.248 0.087 TD 0.267 0.283 l >> /Meta645 660 0 R 0000102677 00000 n q q 0 G 0 0.283 m /FormType 1 [(1)19(7\))] TJ 9.523 0 l /BBox [0 0 0.263 0.283] /Type /XObject /BBox [0 0 1.547 0.633] /Matrix [1 0 0 1 0 0] /Meta116 127 0 R q 0 G /Font << /F1 0.217 Tf 0 0 l >> endstream >> >> /BBox [0 0 1.547 0.314] >> /Length 8 Q /F1 0.217 Tf /Meta455 470 0 R q q Q /Font << 0 g 0000228832 00000 n /Subtype /Form 0 g /F1 0.217 Tf q 0 G /Meta349 Do q /BBox [0 0 1.547 0.633] 0 G >> /Meta333 346 0 R [(i)] TJ /Meta183 Do 0 G 45.249 0 0 45.147 329.731 720.441 cm -0.007 Tc 0 g /BBox [0 0 1.547 0.283] /Matrix [1 0 0 1 0 0] /Subtype /Form 0 G 45.249 0 0 45.527 105.393 558.586 cm endstream endobj 0 G W* n [(i)] TJ W* n 0.015 w 0000093990 00000 n /Subtype /Form 11.988 0.283 l 690 0 obj << 454 0 obj << 0.031 0.437 TD W* n The real part is a, and the imaginary part is bi. endstream stream q /Length 102 q Q /Meta846 Do 0000165969 00000 n 9.791 0.283 l endobj [(B\))] TJ 0.267 0 l /BBox [0 0 1.547 0.633] W* n q /Font << q endobj BT 0 g stream q /Resources << stream q 0.564 G q endstream endstream q q 1 j /Meta164 Do 1 g [(-)] TJ /Length 66 /Subtype /Form q /Meta830 845 0 R /Meta981 Do endobj Q /Subtype /Form /F1 6 0 R 0.458 0 0 RG /F1 0.217 Tf /Length 8 Q ET /Resources << /Resources << /FormType 1 >> 1 g 45.249 0 0 45.147 441.9 203.259 cm >> (1) log 5 25 = y (2) log 3 1 = y (3) log 16 4 = y (4) log 2 1 8 = y (5) log 5 1 = y (6) log 2 8 = y (7) 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0] 0 G /FormType 1 0 G /Type /XObject /Subtype /Form q Q endstream Note: and both can be written in the worksheet and not in the worksheet, you answer... 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