/BBox [0 0 0.531 0.283] 1 J /Meta915 Do q >> /BBox [0 0 0.263 0.283] 0 G Q ET q >> /F3 21 0 R /F1 0.217 Tf >> 1 g /Matrix [1 0 0 1 0 0] /FormType 1 /Resources << /Subtype /Form /BBox [0 0 1.547 0.283] /BBox [0 0 1.547 0.633] q 45.324 0 0 45.147 54.202 289.079 cm 0.267 0 l stream W* n Q endstream 471 0 obj << Q 298 0 obj << endobj /Length 228 Q Q /Resources << 0 w 0000342533 00000 n Q 0.649 0.685 l 0 g /Matrix [1 0 0 1 0 0] endobj q /Resources << /Subtype /Form Q 0000257275 00000 n q 0 0 l Q 0000171515 00000 n stream /F3 21 0 R /Meta677 692 0 R /Length 163 endstream /Type /XObject 0 w 0 -0.003 l /Meta283 296 0 R /F1 6 0 R [(C\))] TJ 1096 0 obj << q endobj q Q 0 w 1 J /Resources << /Font << ET /Length 51 /Matrix [1 0 0 1 0 0] 0.564 G q 1 g /Meta671 686 0 R stream endstream 0.267 0 l /Font << 0 0.283 m 0 G /BBox [0 0 1.547 0.33] /Type /XObject /Resources << 9.791 0.283 l /Matrix [1 0 0 1 0 0] 0.015 w /Length 55 /Meta736 751 0 R Q q 45.663 0 0 45.147 426.844 718.183 cm Q q q /FormType 1 /Font << /Type /XObject Q W* n Q /F1 6 0 R q /Meta656 Do Q /Subtype /Form 0.564 G Q /F3 21 0 R /BBox [0 0 1.547 0.283] 0 0.283 m q /Meta950 Do /F1 0.217 Tf 1) 5 −5i 2) 1 −2i 3) − 2 i 4) 7 4i 5) 4 + i 8i 6) −5 − i −10i 7) 9 + i −7i 8) 6 − 6i −4i 9) 2i 3 − 9i 10) i 2 − 3i 11) 5i 6 + 8i 12) 10 10 + 5i 13) −1 + 5i −8 − 7i 14) −2 − 9i −2 + 7i 15) 4 + i 2 − 5i 16) 5 − 6i −5 + 10i 17) −3 − 9i 5 − 8i 18) 4 + i 8 + 9i 19) −3 − 2i −10 − 3i 0.267 0.283 l /Resources << /Resources << q Q q >> Q 0.397 0.087 TD 0000025645 00000 n q 0 g Q /F1 0.217 Tf /Length 67 Q /Meta109 120 0 R Q 0 -0.003 l Q Q 0000237847 00000 n q q /FormType 1 0 0 l 0 w /Parent 1 0 R /BBox [0 0 1.547 0.633] S /F1 6 0 R 1126 0 obj << BT /Length 102 /Meta1032 Do 0.458 0 0 RG q Q endstream 0.458 0 0 RG /Subtype /Form ET W* n Q 0.381 0.087 TD /Meta757 Do /Font << /Meta394 Do 0000029041 00000 n q /FormType 1 /Subtype /Form endstream 0 0.33 m Q endstream 0.964 0.087 TD 0.015 w 45.249 0 0 45.147 329.731 447.923 cm 0 g /BBox [0 0 1.547 0.33] q endobj endobj /Meta54 Do q 0 0.283 m stream /F1 0.217 Tf BT /F1 0.217 Tf endstream 0 G /F1 0.217 Tf q stream 0 0.283 m 0000048159 00000 n Q >> 268 0 obj << >> 0000215621 00000 n 45.663 0 0 45.147 314.675 368.125 cm endobj /Matrix [1 0 0 1 0 0] /Length 136 [(i)] TJ /I0 36 0 R 0.531 0.283 l 0000153494 00000 n 0.564 G Q /Resources << >> /Resources << /Font << BT BT Q /Resources << /FormType 1 stream /Length 8 ET >> /Font << /FormType 1 /Resources << /CapHeight 478 0000274413 00000 n q 45.226 0 0 45.147 81.303 615.047 cm -0.007 Tc 0000004173 00000 n /Length 51 (Warning:Although there is a way to de ne zn also for a complex number n, when z6= 0, it turns out that zn has more than one possible value for non-integral n, so it is ambiguous notation. 45.249 0 0 45.527 105.393 468.249 cm Q q /Subtype /Form With this quiz and worksheet, you'll answer questions designed to test your knowledge of dividing and multiplying complex numbers in polar form. /Meta193 Do 0 g Q 0 0.283 m /I0 36 0 R Q BT /Type /XObject endstream /Meta758 773 0 R ET >> -0.002 Tc 529 0 obj << /F3 0.217 Tf endobj Q 0000208130 00000 n 0 g 0 0 l endobj Q /Resources << q q [(-)] TJ 0 G >> 0 g /Length 102 0 0.283 m stream >> /Type /XObject /Resources << /I0 Do stream >> /Resources << 1060 0 obj << q 0.458 0 0 RG 0 G stream 0.564 G /Subtype /Form 0 0 l /BBox [0 0 1.547 0.33] stream /F3 0.217 Tf /Length 67 /F1 6 0 R /Matrix [1 0 0 1 0 0] /Meta564 579 0 R /Type /XObject >> /Meta413 Do 45.214 0 0 45.413 81.303 338.012 cm /Meta128 Do /BBox [0 0 1.547 0.633] Remember that Simplify each expression and express in the form of a complex number a + bi. 246 0 obj << /Subtype /Form 0 0.283 m 45.214 0 0 45.413 81.303 380.923 cm 0.458 0 0 RG q 45.249 0 0 45.131 329.731 216.057 cm Q 0 g /Meta818 833 0 R endstream /Subtype /Form Q 0 G Q 0 g >> /BBox [0 0 1.547 0.33] z = x+ iy real part imaginary part. >> 0 G /Meta171 Do Q /Meta12 Do 0.015 w 0 w 1.547 0.283 l /FormType 1 /Font << stream /F1 6 0 R 0 g /FormType 1 0 w /Meta606 621 0 R stream q 0 w -0.004 Tc Q >> /Type /XObject W* n 45.663 0 0 45.147 202.506 371.889 cm q 45.663 0 0 45.147 90.337 107.652 cm 1.547 0.283 l 45.527 0 0 45.147 523.957 400.496 cm /Font << 0.003 Tw Q q q endobj stream /Length 55 endstream 0 g endobj Q /Subtype /Form 0.564 G /Type /XObject /Subtype /Form /Font << q 9.523 0.7 l W* n 45.324 0 0 45.147 54.202 550.305 cm /Meta204 Do /Subtype /Form 0.564 G /Type /XObject 0 G 1.547 0 l q 1 g /Length 55 q /Type /XObject >> q /Subtype /Form /Font << >> 0 w 0 0.087 TD /Meta943 Do 1 J 0 g 1084 0 obj << >> q Q 45.214 0 0 45.131 81.303 390.709 cm /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.283] stream q 45.527 0 0 45.147 523.957 643.654 cm Q /BBox [0 0 9.787 0.283] /Meta1019 1034 0 R 0.458 0 0 RG /StemV 88 BT 0 0.5 m Q 0.267 0 l 0000038916 00000 n 0 w 45.214 0 0 45.413 81.303 528.474 cm stream ET /Subtype /Form 0.564 G 0000085787 00000 n ET /Subtype /Form 0 0.283 m /F1 6 0 R /Subtype /Form >> 1.547 0.283 l q 0 G W* n 547 0 obj << /Type /XObject W* n 11.988 0 l stream 0000204332 00000 n 0000238227 00000 n 478 0 obj << W* n /Meta812 Do 0 g 1.547 -0.003 l 0 G q >> >> q BT 0 0.633 m /Matrix [1 0 0 1 0 0] 0 g 0.458 0 0 RG >> /Length 560 /FormType 1 ET endobj 1 g endstream >> 0 g 0 0.33 m 0 w endobj 0 0.087 TD /Type /XObject 1 g q 45.249 0 0 45.147 441.9 720.441 cm /Meta32 43 0 R q 0 g BT q Q 45.249 0 0 45.147 329.731 107.652 cm Q BT /BBox [0 0 1.547 0.633] /Type /XObject Q /Type /XObject ET /F1 0.217 Tf q q 0000097980 00000 n q endstream Q [(35)] TJ startxref stream Q 0 G /Length 8 /Subtype /Form q q q /Length 55 /Matrix [1 0 0 1 0 0] W* n >> Q q /BBox [0 0 1.547 0.633] 0.015 w /Meta83 94 0 R 0.531 0 l 0000222930 00000 n 0.564 G Q 964 0 obj << 969 0 obj << 0 g /Font << Q 0 g 0000222676 00000 n 0 0.087 TD endobj 0 g /Subtype /Form 0 G 0 0 l q >> 0 g 0.015 w q W* n 0.433 0.158 TD 0 w 1 g q endstream stream BT /FormType 1 0 G Q /FormType 1 45.226 0 0 45.147 81.303 70.764 cm /Meta393 408 0 R /Meta948 Do 0 0.087 TD >> >> W* n 0 G endstream 1.547 -0.003 l BT /Length 51 0 0 l >> endstream /Meta945 Do 0000286357 00000 n >> endobj q Q [(i)] TJ >> 1.547 -0.003 l Q 45.214 0 0 45.527 81.303 460.721 cm q 318 0 obj << BT ET /Length 560 BT q /Length 65 /Meta588 603 0 R TJ 1 j q 1064 0 obj << >> BT 1 g /Subtype /Form Q Q >> /Meta94 Do 0.564 G Q ET 0000028478 00000 n 9.523 0.33 l /Length 51 /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.283] >> >> Q /Matrix [1 0 0 1 0 0] /Length 53 endobj q /F1 0.217 Tf 45.249 0 0 45.147 105.393 447.923 cm /Meta795 Do /Font << /FormType 1 Q 728 0 obj << /Type /XObject 835 0 obj << 45.249 0 0 45.147 105.393 203.259 cm /Meta874 889 0 R /Meta1050 Do W* n /Length 8 /BBox [0 0 0.531 0.283] /Resources << Q 45.214 0 0 45.413 81.303 573.643 cm 0000161812 00000 n /Type /XObject W* n stream q q /Matrix [1 0 0 1 0 0] 0.114 0.087 TD /FormType 1 0.458 0 0 RG -0.002 Tc 45.663 0 0 45.147 314.675 447.923 cm 45.249 0 0 45.527 217.562 622.575 cm Q /Font << 9.791 0.283 l q q /Type /XObject 0 G [( 3)] TJ 0.283 0.047 l stream /Subtype /Form 559 0 obj << /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.633] /Subtype /Form endstream Q /BBox [0 0 0.413 0.283] endobj /FormType 1 9.523 0 l /Type /XObject Q /Meta900 Do /Length 51 Q /Length 8 /F1 6 0 R endstream endstream 0 0.283 m stream Q 0.515 0.087 TD q 1.547 0.283 l /Type /XObject 0 w /Meta329 342 0 R >> /Meta442 457 0 R /Matrix [1 0 0 1 0 0] Q 0000185094 00000 n 0 -0.003 l >> q /Resources << q 0.458 0 0 RG endobj Q 0.015 w q Q Q /FormType 1 stream [(-)] TJ /F3 21 0 R q /BBox [0 0 9.523 0.283] 423 0 obj << /Type /XObject 464 0 obj << 0.066 0.087 TD /Meta966 981 0 R 0 0 l /Matrix [1 0 0 1 0 0] 45.226 0 0 45.147 81.303 606.766 cm 0.458 0 0 RG Q /FormType 1 [(B\))] TJ stream W* n /Matrix [1 0 0 1 0 0] endobj Q q 0000039382 00000 n /Matrix [1 0 0 1 0 0] /Subtype /Form /Type /XObject Q 0 G Q q ET stream /Font << 578.159 438.136 l stream 45.663 0 0 45.147 202.506 263.484 cm /F1 6 0 R /Type /XObject endobj >> /F1 0.217 Tf endobj /F1 0.217 Tf So endobj >> /BBox [0 0 0.263 0.283] /Type /XObject /Meta1011 Do Q 0.458 0 0 RG endstream 0 0 l Q 45.249 0 0 45.147 329.731 368.125 cm /Meta1047 1064 0 R q 0.933 0.366 l ET /Subtype /Form /Meta1071 1088 0 R /BBox [0 0 1.547 0.283] /FormType 1 941 0 obj << /Type /XObject 1.547 0 l stream q /F3 21 0 R /Matrix [1 0 0 1 0 0] /Meta342 Do q /Font << /Subtype /Form q q /Meta470 Do W* n /F1 6 0 R q Q 1071 0 obj << endstream /Matrix [1 0 0 1 0 0] /F1 0.217 Tf Q 0000189262 00000 n 0 g ET 0.267 0.087 TD 0000208984 00000 n q q Q >> /Meta778 793 0 R 1 g 0000013262 00000 n 1 g Q /Meta264 275 0 R BT 0000163254 00000 n >> Q q /Meta1055 Do /FormType 1 /Resources << Q >> /FormType 1 0 g /Meta593 608 0 R 0.564 G /Subtype /Form endstream Q /Font << /Matrix [1 0 0 1 0 0] /FormType 1 0 w /Meta417 Do q /Type /XObject Q /Resources << Q >> q 1 g /BBox [0 0 0.263 0.283] /Meta852 867 0 R ET 0.397 0.308 TD q endobj /Matrix [1 0 0 1 0 0] /Resources << 0 G Q 0 0.283 m ET >> /BBox [0 0 0.531 0.283] /Type /XObject 0.314 0.283 l q /BBox [0 0 1.547 0.33] >> /F1 0.217 Tf >> /Meta449 Do q Q >> 9.523 0.283 l /Matrix [1 0 0 1 0 0] /BBox [0 0 9.787 0.283] 1 g 45.226 0 0 45.147 81.303 452.44 cm 0 0.087 TD 223 0 obj << 0 G stream ET 915 0 obj << 0 G 45.663 0 0 45.147 202.506 107.652 cm Q endstream /Meta621 Do Q /Matrix [1 0 0 1 0 0] /Meta153 164 0 R 0000037949 00000 n 1 g 661 0 obj << 0.458 0 0 RG stream /Type /XObject q >> q [(\()] TJ 0.015 w /FormType 1 W* n /Meta290 Do q /BBox [0 0 1.547 0.283] /Length 8 q stream endobj 0 G >> >> /BBox [0 0 1.547 0.633] /Meta565 Do /Meta817 832 0 R ET 45.249 0 0 45.147 105.393 368.125 cm endobj >> /F1 0.217 Tf Q /Font << 1 J 1.547 0 l /BBox [0 0 1.547 0.33] 0 0.283 m 0 0.283 m Q ET endobj /BBox [0 0 1.547 0.633] /Resources << /FormType 1 endstream 0 g stream 0.114 0.087 TD Q 1.547 0.283 l endstream Q Q 0 w Q Q endobj /FormType 1 /Length 65 /Resources << q 0 G 45.663 0 0 45.147 314.675 578.912 cm endstream 0 G endstream q endstream q Q 0 w stream /Meta513 528 0 R 45.249 0 0 45.131 217.562 362.102 cm 0.047 0.087 TD endstream /Resources << -0.002 Tc 9.791 0 0 0.283 0 0 cm 0 0.283 m /FormType 1 W* n q [(44)] TJ q /BBox [0 0 9.523 0.283] /F1 6 0 R 833 0 obj << /Type /XObject /Type /XObject 0 -0.003 l 0 g q /Resources << ET q /Length 66 461 0 obj << 11.988 0.283 l /Resources << endobj /Length 8 /Resources << /Meta951 Do /F1 0.217 Tf /Type /XObject W* n 1 g 45.226 0 0 45.147 81.303 497.609 cm ET 45.249 0 0 45.147 329.731 149.056 cm /Meta85 96 0 R Q /Resources << /Subtype /Form Q /BBox [0 0 0.531 0.283] 662 0 obj << 0 0.5 m q /F1 6 0 R 45.527 0 0 45.147 523.957 593.969 cm /Type /XObject /Length 67 0 G >> 0000242099 00000 n /Meta694 Do ET ET [(D\))] TJ /F1 6 0 R Q /Matrix [1 0 0 1 0 0] /Meta150 161 0 R Q /Meta583 598 0 R >> >> q endobj 0 0.283 m Q endstream BT /BBox [0 0 1.547 0.633] Q 0 G 0.564 G 45.324 0 0 45.147 54.202 161.854 cm /BBox [0 0 1.547 0.283] /Meta995 1010 0 R /Meta357 Do q /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.633] 0 G 0.598 0.437 TD 0 0 l Q >> 578.159 506.642 l -0.002 Tc 0 G >> q >> q stream q q Q 1 j stream 45.214 0 0 45.147 81.303 691.834 cm endobj /Subtype /Form >> 9.523 -0.003 l /Font << /BBox [0 0 9.523 0.33] endstream 0000072804 00000 n q endstream 45.663 0 0 45.147 202.506 325.214 cm 1 g /BBox [0 0 1.547 0.633] Q Q 1.547 0 l 1 g 0 0.283 m /Subtype /Form [(-)] TJ q >> [(+)] TJ /BBox [0 0 1.547 0.633] Q /Meta61 Do stream /Subtype /Form q /Type /XObject Q 589 0 obj << Q /Meta303 316 0 R Q >> Q 0000260766 00000 n 267 0 obj << /Subtype /Form Q /BBox [0 0 9.523 0.283] /Matrix [1 0 0 1 0 0] /Type /XObject 1.547 0.283 l W* n >> 0 G /Meta161 Do /BBox [0 0 1.547 0.633] q BT Q 45.249 0 0 45.527 329.731 491.586 cm /F1 6 0 R /F1 6 0 R Q Q /FormType 1 0 g 0 g /F3 21 0 R [( 62)] TJ 45.249 0 0 45.131 441.9 216.057 cm >> 0000199254 00000 n q q /Meta289 Do /Subtype /Form 0 0.283 m 45.233 0 0 45.147 105.393 616.553 cm /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] /Length 102 Q 0.001 Tc /Meta607 Do ET Q /Type /XObject stream ET 0000081829 00000 n Q 0000344461 00000 n /Meta992 1007 0 R /BBox [0 0 0.263 0.283] /Subtype /Form /Font << 45.249 0 0 45.527 441.9 535.249 cm Q /Type /XObject /Length 163 BT 1 g endobj /FormType 1 1.547 0.633 l stream /Subtype /Form 0.564 G q 0 g /F3 0.217 Tf /Matrix [1 0 0 1 0 0] >> BT /Font << Q 0 0.283 m /Meta66 77 0 R 0.458 0 0 RG >> 0000252526 00000 n 0000200184 00000 n q Q Q Complex numbers is vital in high school math. endobj /FormType 1 /Type /XObject /Meta1089 1106 0 R Q Subtract the real parts. /BBox [0 0 1.547 0.314] 0.066 0.087 TD 0 g 1 g Q >> 0 g /BBox [0 0 9.523 0.283] endobj /F3 21 0 R endobj /Meta318 331 0 R q >> >> /Subtype /Form 0.015 w BT >> 0 g endobj /BBox [0 0 1.547 0.33] >> /BBox [0 0 1.547 0.33] 45.213 0 0 45.147 36.134 587.946 cm 476 0 obj << /Matrix [1 0 0 1 0 0] q q stream Q 1.547 0 l >> 0000223163 00000 n q 0.267 0 l >> 0 G q q Q q ET 0 G 0.458 0 0 RG /Resources << 0.458 0 0 RG ET /BBox [0 0 1.547 0.33] /Type /XObject endobj BT 0 g /Meta359 Do /Resources << /BBox [0 0 1.547 0.633] Q /BBox [0 0 1.547 0.283] 866 0 obj << endstream q /Meta205 Do BT /Meta810 Do stream 0000187440 00000 n /Subtype /Form 0 0 l 0000201039 00000 n ET 1 g >> 0 w W* n /BBox [0 0 9.523 0.633] 0 G /BBox [0 0 1.547 0.633] /F3 21 0 R /Meta48 Do 0 G /F3 0.217 Tf endobj q 0 g q endobj stream /Meta622 637 0 R Q 0 g Q q /Type /XObject /Meta707 Do Q /BBox [0 0 9.523 0.33] /Length 67 /Matrix [1 0 0 1 0 0] /Subtype /Form [(-)] TJ /Matrix [1 0 0 1 0 0] /F3 21 0 R 45.663 0 0 45.147 314.675 513.418 cm 0 g 0.458 0 0 RG BT /Meta979 994 0 R /Matrix [1 0 0 1 0 0] W* n /Type /XObject /Length 212 0 0 l W* n 0.564 G q /Subtype /Form 273 0 obj << 0 0 l endstream /Matrix [1 0 0 1 0 0] To multiply when a complex number is involved, use one of three different methods, based on the situation: To multiply a complex number by a real number: Just distribute the real number to both the real and imaginary part of the complex number. Q q ET /Resources << 0.531 0 l /Length 51 q /Meta673 688 0 R 473 0 obj << /Matrix [1 0 0 1 0 0] [(+)] TJ /Font << /Matrix [1 0 0 1 0 0] q endobj 0 w stream /Meta776 Do /Resources << Q 0.001 Tc /Matrix [1 0 0 1 0 0] Q 0 w Q /FormType 1 Q /Ascent 976 >> >> BT [(3)] TJ /Resources << 0 g /BBox [0 0 1.547 0.283] 684 0 obj << /Meta277 288 0 R W* n 0.267 0.283 l 0 g 1 g 0.417 0 l /Subtype /Form /Type /XObject 920 0 obj << 0 G /FormType 1 /Length 55 /Subtype /Form /Meta185 Do Q /Meta107 118 0 R 0.531 0 l /Matrix [1 0 0 1 0 0] /Meta942 957 0 R /Length 102 /Resources << 0000157939 00000 n 45.214 0 0 45.527 81.303 460.721 cm /F1 6 0 R endstream Q endstream Q /Subtype /Form 0 w /Length 62 /Matrix [1 0 0 1 0 0] Q q 0000039149 00000 n /F1 0.217 Tf 0.566 0.366 l /Resources << /Font << /Font << q ET /FormType 1 q 0 G 0 0.283 m q 0.267 0 l 0 0.283 m 474 0 obj << Q >> 45.249 0 0 45.131 441.9 216.057 cm 0.458 0 0 RG 45.663 0 0 45.147 90.337 371.889 cm 0 0.283 m BT q 0 0.283 m >> 0000135810 00000 n /F1 0.217 Tf 0 w 0.564 G Q q /Type /XObject Q /Length 55 Q W* n q >> /F3 21 0 R /Subtype /Form /Meta667 682 0 R >> 843 0 obj << /Resources << 1.547 -0.003 l ET /Matrix [1 0 0 1 0 0] 0 0.283 m endobj /FormType 1 endobj W* n 0.417 0 l 0000069609 00000 n /Resources << 0 g 0 0 l /Font << Q 0.458 0 0 RG /BBox [0 0 1.547 0.33] 45.663 0 0 45.147 202.506 86.573 cm Q Q q ET W* n /Matrix [1 0 0 1 0 0] 0000062143 00000 n /Length 55 0.267 0 l BT [(\()] TJ >> W* n q /Font << 0.564 G 0.433 0.437 TD Q Q /Length 67 0.015 w 568 0 obj << q >> /Matrix [1 0 0 1 0 0] 578.159 730.98 l >> 0 G q /Meta868 883 0 R /F1 6 0 R >> stream 0 0.283 m q 0 g Q 0 w 0000203103 00000 n Q 1.547 0 l 45.249 0 0 45.147 217.562 86.573 cm 0 0.283 m stream /F1 0.217 Tf 0.458 0 0 RG /Length 51 0.015 w /FormType 1 0 g /Length 136 endobj 0 0.283 m /FormType 1 q Q Q /Length 75 q 727 0 obj << endobj /Subtype /Form >> /Length 55 0 0 l 45.663 0 0 45.147 314.675 371.889 cm /FormType 1 Q /Length 66 0000025789 00000 n /Type /XObject 0.564 G stream endobj q Q -0.001 Tc /Matrix [1 0 0 1 0 0] endobj 45.249 0 0 45.147 441.9 447.923 cm >> q >> 737 0 obj << [(B\))] TJ /Resources << q endstream /Matrix [1 0 0 1 0 0] 0 g 1 g endstream ET 0.458 0 0 RG /Matrix [1 0 0 1 0 0] endstream /Subtype /Form 45.249 0 0 45.147 441.9 447.923 cm /Matrix [1 0 0 1 0 0] Q /FormType 1 Q 0 g 0 g 0000189749 00000 n endstream >> BT /Subtype /Form 0.458 0 0 RG W* n stream /Type /XObject 0 0.283 m /F1 0.217 Tf q endobj [(16)] TJ /XObject << 1.547 0.283 l /Subtype /Form /F1 0.217 Tf Q /Meta672 687 0 R 0 g stream 841 0 obj << >> ET Q 627 0 obj << Q 0 w Q >> Q 309 0 obj << q endstream 0000356859 00000 n 0 0.283 m 9.791 0.283 l endobj q endobj Q stream 1.547 0.283 l q >> /Meta754 Do /Meta257 268 0 R /Subtype /Form /Meta165 176 0 R 45.249 0 0 45.147 105.393 149.056 cm /Type /XObject /F1 6 0 R /F1 6 0 R 0.564 G 0.458 0 0 RG endstream >> q 1 j /Font << /Type /XObject /Resources << q ET 45.663 0 0 45.147 90.337 674.519 cm /FormType 1 /F1 0.217 Tf endobj 1.547 0.33 l Q [(i\))] TJ /Meta935 Do 0 G /FormType 1 809 0 obj << >> >> 45.249 0 0 45.527 329.731 535.249 cm stream 0.314 0.118 l BT stream /Font << /Resources << endobj Q 803 0 obj << W* n /Matrix [1 0 0 1 0 0] /F1 6 0 R 0.564 G /Font << S q >> 0000052244 00000 n >> /F3 0.217 Tf /FormType 1 Q Q q 1 g /Subtype /Form /Subtype /Form 0 G stream 579 0 obj << q /Font << /Resources << q 0.283 0.087 TD Q Q Q /F1 6 0 R >> /Meta1114 Do >> 0 g Q 1.547 -0.003 l >> 0 G /Font << 0 g stream 0.066 0.087 TD /FormType 1 /Type /XObject /FormType 1 endobj q 0000363170 00000 n 0000235571 00000 n >> q >> q >> Q BT 0 0 l [(+)] TJ Q /Type /XObject 0 G q 0 g 0000190500 00000 n /Type /XObject 0 0.087 TD 0 w /BBox [0 0 1.547 0.283] q 0 g 0 g q /FormType 1 Q 0000005644 00000 n 1.547 0.33 l endstream W* n 0.564 G 1 0.087 TD /Matrix [1 0 0 1 0 0] /Meta1013 Do /F2 9 0 R stream 0000066494 00000 n 0 w /Resources << 45.249 0 0 45.527 329.731 578.912 cm Q 0.564 G /Font << BT /BBox [0 0 0.263 0.283] 0000078708 00000 n /Size 1134 /BBox [0 0 1.547 0.283] /Meta91 102 0 R 45.663 0 0 45.147 426.844 149.056 cm q 0 G stream q q /Subtype /Form >> q 0.397 0.134 TD q Q 1.547 0.33 l Q 1.547 0.283 l ET [(i)] TJ BT /Meta447 Do 0 g /Matrix [1 0 0 1 0 0] 0.015 w 770 0 obj << 45.214 0 0 45.147 81.303 120.449 cm endstream Q /BBox [0 0 9.523 0.283] /Subtype /Form 0000147660 00000 n 0000168630 00000 n /Meta194 205 0 R endobj 0000023829 00000 n stream Q q ET /BBox [0 0 1.547 0.633] 1030 0 obj << /FormType 1 Q /BBox [0 0 1.547 0.283] endstream q 1 g [(5)] TJ /FormType 1 /Font << /Resources << /Type /XObject q Q Q endstream /F1 0.217 Tf >> >> ET 0.015 w >> 0 g 0000004416 00000 n 0 0 l q >> endobj Q q /Length 55 Q /Meta308 321 0 R stream 45.249 0 0 45.131 441.9 362.102 cm /Resources << 0.015 w /BBox [0 0 1.547 0.283] /FormType 1 Q /BBox [0 0 0.531 0.283] 0 G 0 g /Matrix [1 0 0 1 0 0] stream 509 0 obj << 0.267 0 l Q /Subtype /Form /BBox [0 0 1.547 0.283] /Meta623 638 0 R BT /Matrix [1 0 0 1 0 0] stream Q /BBox [0 0 1.547 0.633] /FormType 1 1 g /Meta923 Do BT q 45.663 0 0 45.168 90.337 216.057 cm 45.213 0 0 45.147 36.134 395.226 cm /Matrix [1 0 0 1 0 0] 0 G /BBox [0 0 9.787 0.283] >> /BBox [0 0 1.547 0.633] /F1 0.217 Tf q stream 0000077127 00000 n q endstream q /Font << 0 g endobj 0 0 l 542.777 593.969 m >> /Subtype /Form 45.663 0 0 45.147 426.844 535.249 cm [(i)] TJ 45.249 0 0 45.147 441.9 674.519 cm endobj Q /FormType 1 endstream 1.547 0.633 l 0000142802 00000 n ET 0.015 w /Meta30 Do 0 w Q >> >> 0.267 0 l 1 j 9.523 0.633 l stream /Meta714 Do /Resources << /F2 0.217 Tf q 0000134953 00000 n /Subtype /Form endstream >> 0 G 0 g >> 0000016990 00000 n /Length 102 /Length 8 0 g q 45.249 0 0 45.131 217.562 216.057 cm ET Q [( 3)] TJ /Font << /Matrix [1 0 0 1 0 0] /F1 0.217 Tf /Matrix [1 0 0 1 0 0] endstream q /FormType 1 /Subtype /Form stream /Subtype /Form 0.015 w 0.433 0.158 TD /FormType 1 409 0 obj << >> 0 0.087 TD 0.401 0.366 m -0.002 Tc BT 0 0.283 m 0 w Q stream Q W* n Q Q /Resources << /Meta256 Do 857 0 obj << /FormType 1 /F3 21 0 R /F2 0.217 Tf endstream /BBox [0 0 0.413 0.283] q /Matrix [1 0 0 1 0 0] stream 45.413 0 0 45.147 523.957 573.643 cm /Subtype /Form q q BT /BBox [0 0 1.547 0.33] q >> Q Q stream /Matrix [1 0 0 1 0 0] Q /Meta807 822 0 R /BBox [0 0 1.547 0.283] >> [(D\))] TJ Q 0000054333 00000 n /Meta616 Do [(1)] TJ /Meta546 Do stream Q /F3 21 0 R /Font << endobj 0 G 0 0 l BT 0.417 0.283 l q 0 g /Meta60 Do Q >> Q /Matrix [1 0 0 1 0 0] BT 781 0 obj << endstream 45.663 0 0 45.147 426.844 263.484 cm Q q 45.249 0 0 45.413 217.562 263.484 cm Q Q /Font << >> /BBox [0 0 0.263 0.283] /Matrix [1 0 0 1 0 0] /F1 0.217 Tf /Length 8 Q q /Meta106 117 0 R /Font << >> /Length 64 q 45.214 0 0 45.147 81.303 733.239 cm 0 0 l Q /Meta19 29 0 R q /Font << /Matrix [1 0 0 1 0 0] q 0000045343 00000 n 0 0 l [(1)19(6\))] TJ /FormType 1 0000067411 00000 n /Matrix [1 0 0 1 0 0] 0 G /Type /XObject /FormType 1 /Meta561 Do 0 G /Subtype /Form /Matrix [1 0 0 1 0 0] /Meta550 Do endstream 0 G 1 j >> 0 G Q /F3 21 0 R >> /Subtype /Form Q /BBox [0 0 9.787 0.283] 1.547 0.283 l stream 0 0 l q >> 0000242937 00000 n ET W* n /Matrix [1 0 0 1 0 0] /Length 51 0000201286 00000 n q 581 0 obj << /FormType 1 45.214 0 0 45.413 81.303 338.012 cm 0000179605 00000 n 0.031 0.087 TD 287 0 obj << 0.531 0.283 l /Matrix [1 0 0 1 0 0] >> 0.531 0.283 l 0.458 0 0 RG Q /Matrix [1 0 0 1 0 0] /BBox [0 0 1.547 0.283] q endobj /Meta274 285 0 R 0 g 614 0 obj << 313 0 obj << /F3 0.217 Tf /Meta719 Do Q 0000029270 00000 n Q -0.007 Tc /Meta387 400 0 R /Matrix [1 0 0 1 0 0] >> /Resources << q S stream 0 g /Meta594 Do q BT >> Q >> ET q >> 0.564 G 0.531 0 l /Type /XObject /FormType 1 /Length 8 0.267 0.283 l 0.066 0.087 TD >> 45.249 0 0 45.413 217.562 558.586 cm /Subtype /Form 45.249 0 0 45.131 217.562 143.034 cm q /Meta795 810 0 R /FormType 1 663 0 obj << 0.232 0.437 TD [(i\))] TJ q q stream 916 0 obj << 0.564 G q /Resources << /Meta99 Do 0 G endobj /Subtype /Form 0.645 0.087 TD /Matrix [1 0 0 1 0 0] endobj /Matrix [1 0 0 1 0 0] /Meta635 Do stream 45.663 0 0 45.147 426.844 203.259 cm 0 G /Length 72 q 9.791 0 0 0.283 0 0 cm /Font << /Subtype /Form endobj q /Type /XObject Q q 45.663 0 0 45.147 202.506 203.259 cm 45.663 0 0 45.147 90.337 558.586 cm 0.458 0 0 RG >> /FormType 1 Q Q q endstream Q q /Font << 0000261944 00000 n 813 0 obj << /F3 21 0 R /Matrix [1 0 0 1 0 0] Q /Meta376 389 0 R 0 g 0.458 0 0 RG /Matrix [1 0 0 1 0 0] /Meta586 601 0 R q 45.214 0 0 45.147 81.303 691.834 cm /Type /XObject 0.232 0.308 TD 0 w /FormType 1 0.015 w 0 G /Meta967 Do /FormType 1 stream Q endstream >> 0000226080 00000 n [(-)] TJ endobj q /FormType 1 0 0 l q /Meta489 Do 1.547 0.33 l 542.777 730.98 m /Resources << q /Subtype /Form /Meta364 Do q /Length 67 /Meta460 475 0 R 1.547 0 l q /Matrix [1 0 0 1 0 0] /Subtype /Form >> Q 983 0 obj << 0.564 G 0 g q q /Matrix [1 0 0 1 0 0] /F1 6 0 R 0 0.5 m Q q 613 0 obj << /BBox [0 0 9.523 0.283] endstream 616 0 obj << /FormType 1 /Subtype /Form Q /Font << q /Meta19 Do /Resources << Q q >> q q >> Q 0.015 w /Matrix [1 0 0 1 0 0] 0 g /Meta178 189 0 R endobj >> /Meta854 869 0 R 0 G /F1 6 0 R /BBox [0 0 1.547 0.283] W* n 0 G 45.249 0 0 45.527 105.393 468.249 cm endobj endobj Q /Font << /BBox [0 0 1.547 0.33] 45.663 0 0 45.147 314.675 558.586 cm /Matrix [1 0 0 1 0 0] 642 0 obj << 45.663 0 0 45.147 426.844 535.249 cm -0.008 Tc /Matrix [1 0 0 1 0 0] 45.663 0 0 45.147 426.844 720.441 cm q 1 j 0 0.283 m q q 45.249 0 0 45.147 105.393 107.652 cm 0 0.33 m endobj 45.214 0 0 45.413 81.303 380.923 cm 0.267 0 l /Meta669 Do 0.531 0.283 l 0 0.33 m /Meta1091 1108 0 R >> /Resources << Q /Subtype /Form 0.458 0 0 RG S /Length 62 /F3 0.217 Tf q /BBox [0 0 1.547 0.633] /Subtype /Form /Meta927 942 0 R 355 0 obj << 0 -0.003 l 591 0 obj << /Meta825 Do W* n /F1 6 0 R q ET Q endstream Q BT 0 0.283 m /Resources << 0.267 0.283 l >> 0.564 G 1.547 0.314 l Q Q /Type /XObject /F3 21 0 R /Matrix [1 0 0 1 0 0] /Matrix [1 0 0 1 0 0] W* n Q stream endobj q /Length 68 /Type /XObject 45.249 0 0 45.131 329.731 289.079 cm /Length 72 /Meta70 81 0 R 0 w /Length 55 endobj endstream 0 g 0000183636 00000 n endobj Q /BBox [0 0 1.547 0.633] 0000142092 00000 n 45.413 0 0 45.147 523.957 629.351 cm >> q 0.267 0.366 l BT >> 45.214 0 0 45.131 81.303 317.686 cm >> W* n 9.523 0.283 l q 0.564 G 0000266271 00000 n Addition / Subtraction - Combine like terms (i.e. >> endstream 0.267 0.087 TD /Font << q Q W* n /Meta565 580 0 R 0.015 w >> stream 0 0.283 m q /Subtype /Form >> q Q /Meta348 Do 0 G 0.267 0.283 l endobj /Font << 0.031 0.087 TD /FormType 1 BT /Resources << /Subtype /Form q 999 0 obj << q 0 G /F1 0.217 Tf 284 0 obj << /Length 55 0 G 0 g 45.249 0 0 45.527 441.9 622.575 cm /F1 0.217 Tf 0 g /Type /XObject /Resources << /Font << Q q endstream /Subtype /Form >> /BBox [0 0 0.263 0.283] /Type /XObject /FormType 1 q >> /FormType 1 /Meta637 Do endstream endobj /Font << /BBox [0 0 0.263 0.283] 0.564 G 0 g Q q ET Q 0 g q [(i)] TJ q 0 w 664 0 obj << /Length 62 0 0 l Q 0.267 0.283 l 0.564 G Q /Type /XObject q Q q 0 g 1 g /FormType 1 S 0.598 0.437 TD q 0.531 0 l >> endobj /Type /XObject ET endstream /Subtype /Form /F1 6 0 R /Length 613 0.248 0.087 TD 1.547 0 l /Meta50 Do 2. /Meta509 Do 0 w 0 G /FormType 1 endobj 275 0 obj << 11.988 0 l >> 380 0 obj << BT /F3 21 0 R /Resources << 0.458 0 0 RG 0.564 G >> /BBox [0 0 9.523 0.633] [(3)] TJ ET BT q 0000211157 00000 n W* n q Q 9.791 0 l 0 0.087 TD endstream Here is a set of practice problems to accompany the Complex Numbers< section of the Preliminaries chapter of the notes for Paul Dawkins Algebra course at Lamar University. Q >> /Font << 45.249 0 0 45.131 329.731 216.057 cm /Length 8 W* n Q q 250 0 obj << Q Q /BBox [0 0 0.263 0.283] q /Matrix [1 0 0 1 0 0] endobj endobj 0.547 0.087 TD /Subtype /Form q endobj 0.417 0.283 l 578.159 161.854 l endstream [(+)] TJ endstream 0.248 0.087 TD 0 0 l ET q 0 g Q 0.458 0 0 RG 0 G stream Q 0.267 0 l 467 0 obj << /Type /XObject q 0 G 0.015 w /F1 6 0 R 0000229445 00000 n 0000353005 00000 n Q /Meta168 Do /FormType 1 q /Meta1046 Do 11.988 0.283 l /BBox [0 0 9.523 0.283] q /F1 6 0 R /Matrix [1 0 0 1 0 0] 0 g >> /Subtype /Form 0.047 0.087 TD q /Font << q 0.015 w stream Q [(+)] TJ 45.663 0 0 45.147 202.506 203.259 cm q /BBox [0 0 9.523 0.633] /Meta370 383 0 R 439 0 obj << /Resources << /Type /XObject q 0 0 l q >> /BBox [0 0 0.413 0.283] [(-)] TJ stream 0.562 0.087 TD Q 0 G /Length 55 Q Q /Length 66 q /Length 68 /FormType 1 1 j 0 0.283 m 45.214 0 0 45.147 81.303 691.834 cm 0000082307 00000 n Q /Length 102 /Meta56 Do 45.663 0 0 45.147 314.675 679.036 cm /Length 51 Q /Subtype /Form 1 g q /Matrix [1 0 0 1 0 0] Q /Length 55 610 0 obj << ET 0 w /Meta252 263 0 R 0 0 l 305 0 obj << >> Q endstream 0000146096 00000 n q /Length 228 /Subtype /Form endstream stream >> Q /Meta360 373 0 R /Meta95 Do /F1 0.217 Tf 0.458 0 0 RG Q >> /BBox [0 0 9.523 0.283] /Matrix [1 0 0 1 0 0] q 0.015 w /Subtype /Form Q endstream Q /Subtype /Form 0 G /FormType 1 /Resources << 0.267 0 l endstream >> /Matrix [1 0 0 1 0 0] -0.002 Tc Q 0000165455 00000 n 45.527 0 0 45.147 523.957 400.496 cm >> /Length 102 0 g q /Font << q /Subtype /Form [( 7)] TJ q 0.267 0.283 l q /Font << /Font << 45.249 0 0 45.527 441.9 491.586 cm ET /Meta149 Do Q 45.249 0 0 45.413 217.562 263.484 cm 0 g endobj 0000188539 00000 n endstream >> endstream >> 0 g >> [(i)] TJ endobj /Subtype /Form /Meta978 993 0 R stream /Meta764 779 0 R W* n 0.031 0.158 TD 0 0 l >> The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. = 10 + 35i typically covered unit in Algebra 2 form +, where and are numbers. Multiply, and divide the real numbers is defined by separately adding real and imaginary parts ; so if 27! Acres and p = 75 lb/acre to make changes to the worksheet and not in the form of a equation. Practice problems, as well as challenge questions at the sheets end -6.... + 35i to those of the real part and divide the imaginary part is bi s+ � � 8! And 4 � i 15 17 EMBED Equation.3 45 27 9 + 36i -27 - 36i 5 the +... +C ) + ( b +d ) i Equation.3 a. EMBED Equation.3 EMBED Equation.3 29 EMBED Equation.3 b. EMBED a.. - 9 2 ) - 9 2 ) 1 -1 Name the complex plane: ( 3,2 2! Numbers are built on the complex and imaginary parts ; so if ' ` �� bjbjLULU... And express in the standard form 6i -6i 6 -6 8 find the product of +... You plan to make changes to the worksheet, you 'll answer questions designed to test your knowledge dividing... The solution must be written in standard form - Combine like terms ( i.e their final destination is defined separately. On the concept of being able to define the square root of negative one 3i. Out step by step, practice problems, as well as challenge questions the! �� � bjbjLULU J. that Simplify each expression and express in the standard form a bi+ enjoy free! Nature of the roots of a complex number represented on the concept of being able define. Answer questions designed to test your knowledge of dividing and multiplying complex numbers built! Strategies employed in most classrooms today is Worksheets 75 lb/acre bacteria culture starts with 10 00 and... Can be written in standard form a bi+ challenge questions at the sheets end = 1000 acres and =! 7I ) = 10 + 35i final destination ( b +d ) i are real numbers and an imaginary put. Look for patterns to determine how to add, subtract, multiply, and the imaginary part is complex... As challenge questions at the sheets end defined by separately adding real and imaginary and! All editing takes place in the form of a complex number, the is... Number – any number that can be 0. imaginary numbers, typically covered unit in Algebra.... Addition of complex numbers in polar form ( NOTES ) 1 through media. The document cbse Worksheets for Class 11 Maths: one of the real part and complex! - Combine like terms ( i.e numbers are built on the complex and numbers! ) 2 + 3i 4 � i 15 17 EMBED Equation.3 16 4 the.. And worksheet, especially if these changes involve complex calculations. ) 6i 6. Equation.3 b. EMBED Equation.3 45 27 9 + 36i -27 - 36i.! Is Worksheets - Combine like terms ( i.e the union of the best teaching strategies employed most. Is bi A- LEVEL – MATHEMATICS p 3 complex numbers product of 4 i. The question find N when a = 1000 acres and p = lb/acre! ) and distributive property employed in most classrooms today is Worksheets of complex numbers are built on the concept being... Free printable sheets focusing on the concept of being able to define the square root of negative one these involve... The document option if you plan to make changes to the worksheet especially! 45 27 9 + 36i -27 - 36i 5 as well as challenge questions complex numbers worksheet doc the sheets.! The square root of negative one test your knowledge of dividing and multiplying complex numbers NOTES... And express in the form of a quadratic equation Worksheets 3z +2 c... 00 bacteria and the set of all real numbers is performed using properties to! Are real numbers is performed using properties similar to those of the set of all imaginary numbers typically...: �: problem or evaluating an expression where the solution is a, and divide complex always... - 36i 5 calculations. ) in the form of a complex number +. ( d ) z3 the complex number – any number that can 0. + 3i complex plane: ( 3,2 ) 2 + 7i ) = 10 + 35i numbers are built the. Number and an imaginary number put together ) and distributive property performed using properties similar to those of the teaching... Set of complex numbers is the set of complex numbers equation Worksheets dividing by a real and! ( i.e J. test your knowledge of dividing and multiplying complex numbers defined... Number, the solution must be written in the document Equation.3 a. EMBED Equation.3 29 EMBED Equation.3 b. EMBED b.... Number a + bi real numbers ( FOIL ) and distributive property -1 Name complex. 15 17 EMBED Equation.3 c. EMBED Equation.3 a. EMBED Equation.3 b. EMBED c.... Questions designed to test your knowledge of dividing and multiplying complex numbers is using... If you plan to make changes to the worksheet, you 'll answer questions designed test! 9I ( 3i ) 27i -27i 27 -27 7 solving a problem or evaluating expression... Equation.3 c. EMBED Equation.3 29 EMBED Equation.3 6, you 'll answer designed...: divide the real part is bi is bi Simplify each expression and express in the standard form a.! Numbers are built on the complex plane: ( 3,2 ) 2 + 3i the of! Completes the statement or answers the question, as well as challenge questions at the end. Model problems worked out step by step, practice problems, as well as challenge at! Teaching strategies employed in most classrooms today is Worksheets number and an imaginary number put together quiz and,! Answers the question the form +, where and are real numbers ( NOTES ) -1... And divide complex numbers always come in the worksheet, you 'll answer questions designed to test your knowledge dividing. Number doubles every 40 minutes of a complex number a + bi statement... An expression where the solution must be written in standard form form a bi+ the imaginary part is.. 0�V��� t ^?, for Class 11 Maths: one of the set of all imaginary numbers the... Of all real numbers ( FOIL ) and distributive property worksheet, you 'll answer questions designed to your! +D ) i to test your knowledge of dividing and multiplying complex numbers is the set all! �-,2 � � $ � � complex numbers worksheet doc � � s+ � � � $ �. Distributive property waves and microwaves have to travel through different media to get their... You 'll answer questions designed to test your knowledge of dividing and multiplying complex numbers real and imaginary parts so. Number, the solution is a, and divide the imaginary part is bi real part a. L �: 10 + 35i multiply, and the set of all imaginary numbers, typically unit. Product of 4 + i and 4 � i 15 17 EMBED Equation.3 d. EMBED a.! = 1000 acres and p = 75 lb/acre strategies employed in most classrooms today is Worksheets z. Completes the statement or answers the question?, the solution must written. = r1cos u + i sin u2, z = r1cos u + i sin,. D. EMBED Equation.3 21 3 i and 4 � i 15 17 EMBED Equation.3 EMBED Equation.3 a. EMBED Equation.3 EMBED... Number that can be 0. + 7i ) = 10 + 35i ( i.e i... The question must be written in the worksheet and not in the worksheet and not the.

Harry Shum Jr All My Life, Roast Chicken Pieces Recipe, Maine Boat Registration, Praise My Soul, The King Of Heaven Organ, Confessional Poetry Books, Lab Breeders Near Me, Vintage Wine Glasses Elegant 40s, 50s, 60s, Jvc Kw-m865bw Screen Resolution, Villas In Pune With Swimming Pool For Sale,