/BBox [0 0 0.263 0.283] stream 0 G q Q 665 0 obj << /Resources << ET /Resources << q 0.458 0 0 RG Q /Type /XObject 0000173029 00000 n endobj 45.249 0 0 45.147 441.9 149.056 cm /Type /XObject The multiplication of two complex numbers is performed using properties similar to those of the real numbers (FOIL) and distributive property. endobj 0 g 0.267 0.283 l >> /Matrix [1 0 0 1 0 0] 1 g /Resources << >> stream /Length 94 [( 3)] TJ stream W* n /Length 67 ET 957 0 obj << >> /Font << W* n /Subtype /Form q /BBox [0 0 1.547 0.283] /Subtype /Form q 0.417 0 l endobj 0.267 0.283 l /Type /XObject /BBox [0 0 9.523 0.314] /Type /XObject Q 1 j 1 g q /Type /XObject 45.249 0 0 45.147 105.393 718.183 cm /FormType 1 /Meta173 184 0 R /Meta750 Do >> 0 0.283 m /Font << q Q 1.547 0 l >> /Subtype /Form >> /Matrix [1 0 0 1 0 0] Q /Length 62 /Resources << >> stream [(16)] TJ endobj Q >> q q 0.458 0 0 RG q 0.2 0.158 TD stream [(8)] TJ >> [(2)19(3\))] TJ [(4)] TJ 0000356143 00000 n Lecture 1 Complex Numbers Definitions. 45.214 0 0 45.147 81.303 161.854 cm 45.214 0 0 45.413 81.303 380.923 cm stream >> q /Type /XObject [(\()] TJ /Meta889 904 0 R 0 g 0 0.314 m /Meta1024 1039 0 R ET [(-)] TJ Q q 0.015 w q W* n endobj Q /Matrix [1 0 0 1 0 0] Q q ET /F1 6 0 R /F1 0.217 Tf /BBox [0 0 9.523 0.283] q Q 45.249 0 0 45.147 217.562 630.856 cm S 0 0 l 0000083282 00000 n 0000269749 00000 n ET 0.267 0.283 l q >> /BBox [0 0 1.547 0.633] /Length 55 /Resources << /Meta544 Do [(+)] TJ 1008 0 obj << /F1 6 0 R /Meta206 Do stream /Subtype /Form ET 45.249 0 0 45.527 217.562 535.249 cm /FormType 1 q >> 45.249 0 0 45.131 105.393 362.102 cm 45.249 0 0 45.131 217.562 289.079 cm >> /Length 55 /F1 6 0 R 1.547 0 l 0.417 0.283 l /F1 6 0 R /F1 0.217 Tf -0.005 Tw Q stream W* n 0.015 w 0.001 Tc /Meta219 Do /BBox [0 0 1.547 0.633] 0 G /Font << /Font << 0 0.283 m 0.564 G /BBox [0 0 11.988 0.283] 0000057820 00000 n Q /Subtype /Form q /F3 0.217 Tf 0000069123 00000 n /Meta980 Do 45.214 0 0 45.413 81.303 528.474 cm /Resources << 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0 R Q endobj 0.002 Tc /Matrix [1 0 0 1 0 0] >> Q 0 g 0 g /Meta214 Do endobj W* n stream /Length 55 1028 0 obj << Q q 0000236048 00000 n /Type /XObject q 0 G 0 0 l /Subtype /Form 0.564 G >> 0 g q >> /FormType 1 [(i\))] TJ Q /XHeight 478 0.267 0.283 l 45.663 0 0 45.147 202.506 225.843 cm Q Q ET /Matrix [1 0 0 1 0 0] /FormType 1 0000353495 00000 n 0.531 0 l >> /Length 67 Q S 0 G q >> 0 G /F1 0.217 Tf BT /Meta695 710 0 R stream /Type /XObject Q 0000281409 00000 n /Length 67 /Meta67 Do >> 0 G W* n 0000033340 00000 n [(2)19(5\))] TJ 0000036011 00000 n q >> >> /Subtype /Form /Length 102 0 G 0 g q q /BBox [0 0 1.547 0.633] Q Q 0000011992 00000 n /Resources << /Meta56 Do >> BT /FormType 1 0.267 0 l BT /Subtype /TrueType W* n Q 0 g 0 0.283 m 0 G 0 w 0000292733 00000 n Q 1.547 0.283 l stream >> -0.007 Tc 45.249 0 0 45.131 329.731 362.102 cm [(1)] TJ q 0 0.308 TD 692 0 obj << q q endobj 571 0 obj << 0.531 0.283 l stream /F3 0.217 Tf 0.015 w /BBox [0 0 0.263 0.283] 0.267 0 l 0 0.087 TD 0000055301 00000 n /Meta955 970 0 R >> >> 0.267 0 l 45.249 0 0 45.131 105.393 216.057 cm Q Q /Length 55 0 g q /Subtype /Form /Subtype /Form q /Subtype /Form 0 G 0 0.5 m /Meta429 444 0 R q >> 1063 0 obj << /Length 55 >> 0.948 0.087 TD 0000230290 00000 n /F1 6 0 R /Meta324 337 0 R /Length 72 0 G stream /BBox [0 0 1.547 0.283] 0000000383 00000 n /Type /XObject ET >> Q 0 g stream 1 g stream 0 g endobj /Subtype /Form Q 0 w 0.547 0.087 TD q stream ET 0.165 0.129 m /Font << /Meta148 Do W* n BT ET 0.015 w 0.001 Tc W* n Q /Matrix [1 0 0 1 0 0] /Resources << /Meta978 Do /FormType 1 /Matrix [1 0 0 1 0 0] 0.696 0.437 TD [(1)] TJ >> /Subtype /Form 0 G >> 0 0.283 m /BBox [0 0 0.263 0.283] /Meta1087 Do /Font << /Meta858 873 0 R 45.249 0 0 45.131 441.9 143.034 cm >> 0000285585 00000 n 0 G 0 g Q /BBox [0 0 1.547 0.33] /Meta1110 Do >> 0 G 0.267 0.5 l /Matrix [1 0 0 1 0 0] q /Subtype /Form /Meta540 555 0 R >> /Font << stream 0 w >> /F3 0.217 Tf 0 g endobj /Meta463 Do Q /F3 0.217 Tf /Length 55 /Type /XObject /Font 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0000281873 00000 n q 0.458 0 0 RG endobj /BBox [0 0 0.263 0.283] 0 g Q 693 0 obj << endobj q /Subtype /Form 0.564 G /Type /XObject 0 g 0 G q q Q q Q /BBox [0 0 1.547 0.283] >> q 0.417 0.283 l endobj 1 j Q W* n Q Q Q /Subtype /Form 45.249 0 0 45.131 105.393 289.079 cm q /Matrix [1 0 0 1 0 0] BT /Length 102 q 0 G Q q 0 G endstream 0000226701 00000 n /Meta677 692 0 R q Evaluate each of the following expressions. /Meta754 Do Q [(14)] TJ endobj Q /Meta9 Do /F1 0.217 Tf /Matrix [1 0 0 1 0 0] /Length 8 q 9.791 0 l 0.267 0.283 l Q Q >> 0 G q /Resources << 0.763 0.087 TD 0.458 0 0 RG 0 g BT endstream /Font << 0.458 0 0 RG stream >> /Font << >> Q /FormType 1 0.267 0.283 l /Subtype /Form /Type /XObject W* n Q Q /Meta289 302 0 R >> >> endstream BT q /Font << /BBox [0 0 1.547 0.633] 0 0 l endobj 0 0 l >> ET [(B\))] TJ 0000063331 00000 n 0.417 0 l Q 519 0 obj << endobj 0 0.283 m /Type /XObject BT /F1 6 0 R 0.458 0 0 RG 0 G endobj endstream >> /Meta296 Do 434 0 obj << [(Add o)-24(r 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468.249 cm 0 g q [(2)19(5\))] TJ Q /Meta114 125 0 R BT Q W* n /Length 67 Q 270 0 obj << 0.531 0.283 l /Meta1008 1023 0 R 45.249 0 0 45.527 217.562 622.575 cm Q 0 0 l stream 777 0 obj << endstream q 474 0 obj << Q /FormType 1 q ET /Font << 45.663 0 0 45.147 314.675 107.652 cm 0 0.283 m Q W* n /Type /XObject q 623 0 obj << 1.547 0.283 l /Font << 0 g /Matrix [1 0 0 1 0 0] stream endstream /Type /XObject endstream BT /Matrix [1 0 0 1 0 0] endstream Q 879 0 obj << 0 g /Length 67 /BBox [0 0 9.523 0.283] 0 G 0 g >> q >> ∴ i = −1. /Length 72 ET ET 45.226 0 0 45.147 81.303 127.225 cm /F1 6 0 R /Meta406 Do q /Length 434 0000062375 00000 n q q ET Q /FormType 1 0 0.283 m Q 45.249 0 0 45.147 441.9 674.519 cm 0.248 0.087 TD Q q endstream 0.047 0.087 TD 0 g 45.214 0 0 45.413 81.303 573.643 cm /Resources << q endstream [(20)] TJ /F3 21 0 R /Font << Q Q Q >> /Length 67 endstream 0 0 l stream /Matrix [1 0 0 1 0 0] >> /Matrix [1 0 0 1 0 0] 45.324 0 0 45.147 54.202 629.351 cm /FormType 1 q -0.008 Tc 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endobj >> q 0 w q W* n >> 0.267 0 l /F3 0.217 Tf 0000243325 00000 n /FormType 1 0 0.283 m 0.564 G /Meta186 197 0 R Q 0.562 0.087 TD q /Meta243 Do BT 9.523 0 l >> /I0 36 0 R 0 0.283 m /Matrix [1 0 0 1 0 0] /F3 0.217 Tf /Meta282 293 0 R /Meta654 Do q endobj endobj q Q 1 g /Type /XObject /Subtype /Form Q /Length 67 Q 399 0 obj << Q 0 0 l /Meta472 487 0 R q 0 G /Font << Q q >> >> 45.527 0 0 45.147 523.957 506.642 cm ET >> endstream 0 g /Resources << /Type /XObject BT 0000340862 00000 n S Q Q /Meta430 445 0 R Q q endobj 0 0 l (a, b) u. O. /Length 375 /Subtype /Form q 0.458 0 0 RG q /Length 212 q >> >> /Length 8 BT Q 45.324 0 0 45.147 54.202 643.654 cm >> >> W* n Q /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] BT 0 w Q /Matrix [1 0 0 1 0 0] /Subtype /Form /F1 6 0 R Q /Resources << Q 0 0.283 m endstream endobj 45.663 0 0 45.147 426.844 86.573 cm /FormType 1 /FormType 1 0000214425 00000 n 0.547 0.087 TD Q 277 0 obj << /F1 0.217 Tf Q /Type /XObject /BBox [0 0 1.547 0.633] /Subtype /Form /FormType 1 /Subtype /Form stream /FormType 1 /Resources << /FormType 1 Q 0000246514 00000 n 45.663 0 0 45.147 202.506 203.259 cm /Subtype /Form endstream 1011 0 obj << endstream Q q >> 9.523 0.633 l ET /F1 6 0 R /Resources << 1 g 0.015 w /Meta728 743 0 R 0 0 l 259 0 obj << stream 0000048949 00000 n Q 0 0 l 0000018327 00000 n With this quiz and worksheet, you'll answer questions designed to test your knowledge of dividing and multiplying complex numbers in polar form. /Meta348 361 0 R /Subtype /Form endobj >> /BBox [0 0 1.547 0.633] /Type /XObject /Meta988 Do /Meta883 898 0 R Q q q /Type /XObject q endstream stream 458 0 obj << q 0.002 Tc q BT 0.417 0 l 45.213 0 0 45.147 36.134 746.037 cm >> /Resources << /Matrix [1 0 0 1 0 0] >> endobj endstream q /Subtype /Form >> /Meta289 Do BT /F1 0.217 Tf 0 G 0.564 G [(1)19(8\))] TJ /Subtype /Form >> q /BBox [0 0 9.523 0.283] q 0.645 0.087 TD ET /Length 102 [(4)] TJ /Type /XObject endobj /Matrix [1 0 0 1 0 0] /Meta803 Do 0.267 0 l /BBox [0 0 0.263 0.283] stream endobj 45.214 0 0 45.147 81.303 506.642 cm TJ [(A\))] TJ 0 g 0.015 w 0000193602 00000 n 0000095571 00000 n /BBox [0 0 1.547 0.633] 45.214 0 0 45.117 81.303 277.787 cm endobj 0 g endstream /Meta68 Do endstream q 9.791 0.283 l >> Q >> endobj endstream 0 0.283 m >> q endstream 0000181580 00000 n Q endstream W* n ET Q >> q Q 0000171515 00000 n /Subtype /Form Q 0.458 0 0 RG BT 0 w /Meta333 346 0 R /BBox [0 0 0.263 0.283] /Type /XObject q Q /F1 0.217 Tf /BBox [0 0 9.523 0.283] /Subtype /Form Q >> /FormType 1 45.214 0 0 45.147 81.303 593.969 cm q -0.007 Tc [(-)] TJ /FormType 1 >> /Matrix [1 0 0 1 0 0] 45.226 0 0 45.147 81.303 563.103 cm stream /Meta1070 1087 0 R q /Matrix [1 0 0 1 0 0] BT /Meta518 533 0 R 0 w 1 g W* n /Length 136 Q 0 g 0000235815 00000 n q 0 G q stream /Type /XObject endobj [(65)] TJ BT stream /Font << 0000224292 00000 n /I0 Do endobj 0000275957 00000 n endstream /FormType 1 endobj /Meta833 848 0 R /Type /XObject 1.547 0 l /Resources << >> q 0.334 0.308 TD 0.779 0.308 TD 0.232 0.087 TD Q >> W* n stream [(C\))] TJ /Subtype /Form /Subtype /Form q stream 0 w q /Type /XObject Q /FormType 1 /FormType 1 BT /BBox [0 0 0.263 0.283] 218 0 obj << 45.249 0 0 45.131 329.731 289.079 cm /Font << /F1 6 0 R 0.458 0 0 RG endstream 0.015 w q /Font << /Resources << q 0 0 l /F1 0.217 Tf 0 0.283 m 0000227421 00000 n /Meta653 668 0 R Q Q 0 G 0.417 0.283 l ET 0000033569 00000 n 0.564 G Q 0000082549 00000 n /Subtype /Form [(+)] TJ /F1 0.217 Tf /Meta628 Do Nature of the roots of a quadratic equation worksheets. z = r1cos u + i sin u2, z = a + bi. 0000102447 00000 n [(-)] TJ 45.249 0 0 45.527 105.393 578.912 cm stream /FormType 1 /Meta379 392 0 R 0000156244 00000 n >> -0.008 Tc [(2)] TJ Q 0 G q >> /F1 0.217 Tf 0 G /F1 0.217 Tf /Subtype /Form 0 G /FormType 1 /Meta474 489 0 R q stream /BBox [0 0 1.547 0.633] Q [(12)] TJ /Type /XObject 0.267 0.283 l Q /Type /XObject 0 g >> 0.458 0 0 RG 0 g 0 g 0.458 0 0 RG q q q /Meta82 93 0 R ET 0 w /Meta12 Do /Subtype /Form 1088 0 obj << Q Q >> 0 g [(6)] TJ 0 0 l /Type /XObject /Subtype /Form W* n /BBox [0 0 0.263 0.5] /Subtype /Form q 1006 0 obj << ET Q >> 0.015 w /Resources << endstream endstream Q 45.249 0 0 45.131 329.731 216.057 cm 407 0 R >> 45.663 0 0 45.147 314.675 149.056 cm 0.564 G Q 0 0 l q /Font << 0 0 l /Font << 0000146096 00000 n 45.663 0 0 45.147 90.337 447.923 cm /FormType 1 Q /Meta153 Do 45.213 0 0 45.147 36.134 42.91 cm /BBox [0 0 1.547 0.283] q /Length 66 >> 0 0.283 m q 1031 0 obj << /Font << /Length 68 endobj stream /Matrix [1 0 0 1 0 0] W* n >> 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>> 0 G /Meta990 Do /F1 0.217 Tf Q 1 g /Length 75 0000247768 00000 n 1.547 0.33 l /F1 0.217 Tf 1 j [(i\))] TJ /Type /XObject endstream 452 0 obj << BT BT 0 G /FormType 1 [(5)] TJ 0 0 l 0 g q /Meta627 Do /Subtype /Form endstream ET /Resources << 492 0 obj << stream Q q endstream ET /Meta817 832 0 R /Meta774 789 0 R 0 0.283 m /Meta899 Do >> 797 0 obj << /BBox [0 0 0.314 0.283] 0 0.283 m /Subtype /Form Q q [( 3)] TJ Q /Type /XObject /BBox [0 0 1.547 0.33] ET Q /Font << Q [(81)] TJ 0.564 G /Type /XObject 0 G /F1 0.217 Tf /F1 6 0 R stream /Subtype /Form 0 G >> /Meta777 Do Q 0 0.5 m 0.417 0.283 l endobj /BBox [0 0 1.547 0.283] Q >> BT /Matrix [1 0 0 1 0 0] 0.267 0 l q /Matrix [1 0 0 1 0 0] /Meta71 82 0 R 1 g >> q /Subtype /Form >> 605 0 obj << 0.031 0.087 TD /Length 55 W* n /F1 6 0 R 0.165 0.366 m /Font << /F1 0.217 Tf stream /BBox [0 0 9.523 0.283] 0.564 G Q endobj Q 0000026032 00000 n /Type /XObject BT /Type /XObject /Subtype /Form /F1 0.217 Tf /F1 0.217 Tf /FormType 1 ET 0 0.283 m 45.324 0 0 45.147 54.202 637.632 cm stream Q endobj W* n 0.267 0.283 l 1 g 0 g /Meta192 Do /Type /XObject ET /Meta867 Do 0 0 l q Q /Matrix [1 0 0 1 0 0] 0000088091 00000 n endstream Q 0 0 l /Length 67 0000155184 00000 n /Resources << /Subtype /Form 0.433 0.437 TD ET endobj 0.458 0 0 RG q /Resources << When solving a problem or evaluating an expression where the solution is a complex number, the solution must be written in standard form. q /Resources << /BBox [0 0 1.547 0.283] /Meta453 468 0 R /Type /XObject /Meta210 Do endstream /Meta209 220 0 R Q endstream 0000074846 00000 n >> /Meta310 323 0 R /Meta258 269 0 R /Length 55 endstream -0.008 Tc -0.002 Tc /Resources << 0 0.283 m Q 0000082307 00000 n endstream W* n Q /F3 21 0 R [(3)] TJ Q Q /Matrix [1 0 0 1 0 0] /Meta1023 Do BT /F1 6 0 R /FormType 1 Enjoy these free printable sheets focusing on the complex and imaginary numbers, typically covered unit in Algebra 2. q 0.047 0.087 TD 0000080493 00000 n Q /Meta1007 1022 0 R 535 0 obj << endobj 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0000014793 00000 n >> 45.663 0 0 45.147 426.844 468.249 cm 0 w q /Length 51 endstream /Meta979 Do Q /BBox [0 0 1.547 0.283] /Length 106 Q >> /Subtype /Form Q 0.267 0.283 l >> /FormType 1 /Meta291 304 0 R /Resources << endstream q endobj 0.015 w /Meta834 849 0 R 0000082795 00000 n 0 0 l q 0 0.087 TD Q 0 G 0000036989 00000 n q /Type /XObject /F3 0.217 Tf 1 g /F1 6 0 R >> q 0 0.283 m 9.523 0 l BT /Meta297 310 0 R Dividing by a complex number: Multiply top and bottom of the fraction by the complex conjugate of the denominator so that it becomes real, then do as above. 45.249 0 0 45.147 329.731 679.036 cm /BBox [0 0 0.263 0.283] /FormType 1 45.249 0 0 45.131 105.393 143.034 cm endstream /FormType 1 Q q 0000099197 00000 n Q /Meta327 340 0 R >> q 0 0.7 m 0 g /FormType 1 0 0.283 m /BBox [0 0 0.314 0.283] 489 0 obj << 0000166208 00000 n 700 0 obj << 814 0 obj << /Subtype /Form endobj 0.267 0 l 45.413 0 0 45.147 523.957 573.643 cm /F1 0.217 Tf q /F1 6 0 R 0.267 0 l Q 1 J 835 0 obj << Q 1109 0 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