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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
=