Wk_2_Review of Complex number ( ERIC )

Wk_2_Review of Complex number ( ERIC ) - BME_311_Network...

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Unformatted text preview: BME_311_Network Theory Review of Complex Numbers YU-YEN HUANG ( ERIC ) 2011.01.21 Complex Numbers Complex Quadratic Equation x +1=0 x =-1 x = -1 2 2 Definition i = -1 Complex Numbers Complex i i i i i i 2 = -1 -1 = = = = = 3 4 5 6 i i i i 2 2 4 4 i = -i x i = (-1) x (-1) = 1 x i = 1xi=i x i = 1 x i = -1 x 2 2 2 Complex Numbers Complex Real numbers and Imaginary numbers are subsets of the set of complex numbers Standard form If b = 0 a + bi a + bi = a is a real number a + bi is called an imaginary number If a = 0 Write the complex number in “ Standard form ” 1+ −8 =1+ ( −1× 8 ) =1+ i 8 =1+ i 4×2 =1+ 2 i 2 Complex Numbers Complex Addition ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i ( 3 + 2 i ) + ( 6 + 3i ) = 9 + 5 i Subtraction ( a + bi ) − ( c + di ) = ( a − c ) + ( b − d ) i ( 3 + 2 i ) − ( 6 + 3i ) = −3 − i Multiplication 2 ( a + bi ) × ( c + di ) = ac + adi + bci + bdi ( 3 − 2 i ) × ( 6 − 3i ) = 18 − 9 i − 12 i + 6 i 2 = 12 − 21i Complex Numbers Complex Can the product of two complex numbers be a real number ? ( a + bi ) × ( a − bi ) = ?? ( a + bi ) × ( a − bi ) = a − abi + abi − b i (3 + 2i ) × (3 − 2i ) = 9 − 6i + 6i − 4i 2 2 22 = a +b 2 2 = 9 + 4 = 13 Complex Conjugates Complex Complex conjugates A pair of complex numbers of the form a + bi and a − bi where a and b are real numbers. a + bi Quotient of two complex numbers : c + di Multiply the numerator and denominator by the conjugate of the denominator a + bi c − di × c + di c − di ac − adi + bci −bdi 2 = c2 +d 2 ac + bd + ( bc − ad ) i = c2 +d 2 Rectangular form Rectangular Any Complex Number can be expressed as : X + Y i Number can be plotted as an ordered pair in a two-dimensional plane Imaginary axis -7 + 6 i 5+4i Real axis -10 - 10 i Polar form Polar Im z= z z θ =r θ r θ where y x Rm r = x +y , or 2 2 2 θ = tan −1 y x x = r cos θ , that is, y = r sin θ z = x + jy = r cos θ + r sin θ Exponential form Exponential Almost the same as the polar form, because we use the same magnitude r and the angle θ z = re , jθ r = x + y , θ = tan 2 2 −1 y x Polar form z=r Rectangular form θ r = x + y , θ = tan 2 2 −1 y x z = x + jy , x = r cos θ , y = r sin θ Euler’s Formula Euler’s e jθ = cos θ + j sin θ cos θ = Re(e ), sin θ = Im(e ) jθ jθ 1 jθ cos θ = (e + e − jθ ) 2 1 jθ − jθ sin θ = (e − e ) 2j Thanks Thanks for your attention ...
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This note was uploaded on 02/01/2011 for the course BME 311 taught by Professor Zhang during the Spring '08 term at University of Texas at Austin.

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