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Complex Numbers and Their Geometry

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7/9/2013 Complex Numbers 2 Complex Numbers and the Imaginary i Definition: The number x such that x 2 = –1 is defined to be i Applying the square root property, The Complex Plane –1 = ± x so that –1 i = and –i–i – –1 =

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7/9/2013 Complex Numbers 3 Complex Numbers and the Imaginary i Standard Form The Complex Plane a + bi Complex Numbers a + bi, b = 0 Real Numbers a + bi, b 0 Imaginary Numbers x iyiy a + bi Note: a + bi is also written a + ib for real a and b … especially if b is a radical or function Sometimes b 0 and a = 0 The Complex Plane

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7/9/2013 Complex Numbers 4 Complex Numbers in the Plane Numbers as points in a plane … instead of points on a line Complex numbers as ordered pairs of real numbers Complex number z Point in complex plane The Complex Plane x iyiy The Complex Plane Imaginary Axis Real Axis z = a + ib z z = ( a, b ) a + ib = a ib ( a, b ) =

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7/9/2013 Complex Numbers 5 Radical Expressions and Arithmetic The expression Sum and difference of complex numbers ( a + bi ) ± ( c + di ) = ( a ± c ) ± ( b ± d ) i Examples: (3 + 4 i ) – (2 – 5 i ) (7 – 3 i ) + (2 – 5 i ) The Complex Plane = (3 – 2) + (4 + 5) i = i = (7 + 2) – (3 + 5) i = 9 – 8 i – a –1 a = – a i = a can be written

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7/9/2013 Complex Numbers 6 Radical Expressions and Arithmetic The expression Product of Complex Numbers The Complex Plane – a –1 a = – a i = a can be written ( a + bi )( c + di ) Example: = 6 – 20 i 2 + (8 i – 15 i ) = 26 – 7 i (3 + 4 i )(2 – 5 i ) ac + bdi 2 + ( ad + bc ) i = ( ac – bd ) + ( ad + bc ) i =

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7/9/2013 Complex Numbers 7 Complex Conjugates Definition: a + bi and a – bi are a complex conjugate pair Example: i and 7 – 3 i are complex conjugates The Complex Plane

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7/9/2013 Complex Numbers 8 Complex Conjugates Definition: a + bi and a – bi are a complex conjugate pair Fact: The product of complex conjugates is always real ( a + bi ) ( a – bi ) = a 2 + b 2 Example: (7 + 3 i ) (7 – 3 i ) = The Complex Plane = = 58

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7/9/2013 Complex Numbers 9 Complex-Number Quotients The Complex Plane a + bi c + di = ( bc – ad ) i + ( ac + bd ) c2 + d 2c2 + d 2 = ac + bd c2 + d 2c2 + d 2 + ( bc – ad ) i c2 + d 2c2 + d 2 = + i ac + bd c2 + d2c2 + d2 ( ) bc – ad c2 + d 2c2 + d 2 () c – di = a + bi c + di Complex conjugate of c + di Real denominator Note: We can always multiply by 1 in clever disguise to change form NOT value

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7/9/2013 Complex Numbers 10 Complex-Number Quotients Quotient Examples 1. The Complex Plane i i ( i ) (1 + 2 i ) (1 – 2 i ) = (65 – 30) i = = i i a + bi c + di = + ac + bd c2 + d2c2 + d2 ( ) bc – ad c2 + d 2c2 + d 2 ()

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7/9/2013 Complex Numbers 11 Quotient Examples 2. The Complex Plane 3 i = ( i )( – i ) 3 ( – i ) = –3 i = –i 2–i 2 3 ( – i ) = –(–1) 3 ( – i )

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7/9/2013 Complex Numbers 12 Quotient Examples 3. The Complex Plane (1 + i ) 2 –2 + i = (1 – i ) 2 (1 + i ) 2 (1 – i ) 2 (–2 + i ) = (1 – i ) ) 2 ( (1 + i ) (–2 i ) (–2 + i ) = ( 2 ) i = 1 2 i +

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7/9/2013 Complex Numbers 13 Think about it !

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