REVIEW OF COMPLEX NUMBERS 1 i and The Imaginary Number i : i 2 1 Complex numbers are always written in the form a + bi part) (a is the real part and b is the imaginary Examples: Write as complex numbers. a) 25 = 25 1 25 1 25 5i b) 17 = 17 117 1 17 i 17 c) 9 = 9 1 9 1 9 3 i 3 i d) (2 3i ) ( 4 6i) = 2 3i 4 6i 2 4 3i 6i 2 9i e) (6 10i ) (9 3i) = 6 10i 9 3i 6 9 10i 3i 15 13i f) (6 4i ) (2 6i) = (6 4i) (2 6i) 6 4i 2 6i 8 2i g) (2 i) (3 10i) = (2 i) (3 10i) 2 i 3 10i 1 9i h) (8 3i )(1 2i ) = 8 16i 3i 6i 2 8 19i 6(1) 8 19i 6 2 19i i) 3i (5 6i) = 15i 18i 2 15i 18(1) 15i 18 18 15i The conjugate of a + bi is a – bi . Examples: Find the conjugate of the following complex numbers. a) 2 + 6i 2 – 6i b) –3 – 4i –3 + 4i c) 15i –15i To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. Examples: Divide. Be sure to write each answer in standard form. a) 5 5 2i 5(2 i ) 2i 2i 2i (2 i )(2 i ) 10 5i 4 2i 2i i 2 10 5i 4 (1) 10 5i 5 10 5i 5 5 2i b) 1 4i 1 4i 5 3i (1 4i )(5 3i ) 5 3i 5 3i 5 3i (5 3i )(5 3i ) 5 3i 20i 12i 2 25 15i 15i 9i 2 5 23i 12(1) 25 9(1) 7 23 i 34 7 23 i 34 34 Powers of i : i= i i5 = i i2 = –1 i6 = –1 i3 = –i i7 = –i i4 = 1 i8 = 1 So, the pattern for the powers of i is: { i, –1, –i, 1 } Examples: Simplify a) i12 = 1 (count through the pattern until you get to 12) b) i67 = –i (count through the pattern until you get to 67) c) 4 + i3 = 4 + (–i) = 4 – i d) 4i4 – 2i2 + i = 4(1) – 2(–1) + i = 4 + 2 + i = 6 + i When solving the quadratic equation, ax 2 bx c 0 , if b2 – 4ac < 0, then there will be 0 real solutions (because the 2 answers are complex) Examples: Solve a) x2 2 x 5 0 a = 1, b = –2, c = 5 2 (2)2 4 1 5 2 4 20 2 16 2 4i x 2 2i 2(1) 2 2 2 b) 6 x 2 4 x 1 0 a = 6, b = 4, c = 1 x 4 (4)2 4 6 1 4 16 24 4 8 4 2 2 i 4 2 2 i 1 2i 2(6) 12 12 12 12 12 3 6 Notice that complex solutions ALWAYS come in complex conjugate pairs!! Example: If 5i is a solution to a quadratic equation, then so is – 5i If –4 – 6i is a solution to a quadratic equation, then so is –4 + 6i