Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
complex_operations_brief evolution to 09/18/02 8:32 AM (last update) The real number system and the imaginary numbers, bi, were combined to form the complex number system, a + bi. No real numbers, a, are imaginary and no imaginary numbers, bi, are real; the sets are disjoint. However, both the real numbers and the imaginary numbers are complex numbers. Real numbers are also complex numbers: a = a + 0i ; ex) 6 = 6 + 0i. Pure Imaginary are also complex numbers: bi = 0 + bi ; ex) 3i = 0 + 3i. The Conjugate of z = x + yi is z = x − yi. Definition of i i = −1 i 2 = −1 Add/Subtract - add/subtract real parts and add/subtract i parts (a ± bi ) ± (c + di ) = (a ± c) ± (b ± d )i Example (2 + 3i ) − (1 − 2i ) = 2 − 1 + 3i + 2i = 1 + 5i Multiply - FOIL and use i = −1 2 (a + bi )(c + di ) = ac + (ad + bc)i − bd Example (2 + 3i )(1 − 2i ) = 2 − 4i + 3i − 6i 2 = 2 − i − 6( −1) = 2 + 6 − i = 8 − i Divide - use conjugate of denominator to eliminate the i in the denominator a + bi (a + bi )(c − di) ac + (−ad + bc)i − bd ac − bd (− ad + bc)i = = = 2 + c + di (c + di)(c − bi) c2 + d 2 c +d2 c2 + d 2 Example 2 − i (2 − i)(3 − i) = 3 + i (3 + i)(3 − i) 6 + ( −2 − 3)i + i 2 9 − (−1) 6 + ( −2 − 3)i − 1 = 9 +1 5 − 5i 5 − 5i 5 5 = = = − i 10 10 10 10 1 1 = − i 2 2 =