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© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR Introduction to Complex Numbers, Standard Form Approximate Running Time - 20 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: 1.Select Slide Show with the menu: Slide Show|View Show (F5 key), and hit Enter 2.You will hear CHIMES at the completion of the audio portion of each slide; hit the Enter key, or the Page Down key, or Left Click 3.You may exit the slide show at any time with the Esc key; and you may select and replay any slide, by navigating with the Page Up/Down keys, and then hitting Shift+F5.

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© 2005 Baylor University Slide 2 T out T Sinusoidal Response Complex Numbers Definitions and Formats Complex Numbers mathematically represent actual physical systems T in SYSTEM T out Feedback Exponential Decay

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© 2005 Baylor University Slide 3 The General Quadratic Equation Take the Square Root Complete the Square The Solution to the General Quadratic Equation

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© 2005 Baylor University Slide 4 Solutions of the Quadratic Equation By solution, we mean roots, or where x=0 2 nd Order 3 rd Order If there is one real root If there are no real roots, as shown If there are two real roots, as above

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© 2005 Baylor University Slide 5 The Imaginary Number Consider: Complete the square: Take the square root: The solution: Because does not exist, we call this an imaginary number, and we give it the symbol or. j i becomes

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© 2005 Baylor University Slide 6 Complex Numbers Substitute into Checks! A general solution for is

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© 2005 Baylor University Slide 7 z=x+iy Complex Numbers Definitions The Standard Form Im(z)=y Re(z)=x z=x+iy and if y=0, then z=x, a real number i 3 =-i i 4 =1 i 2 =-1 i 5 =i i 6 =-1 z 1 =x 1 +iy 1 Given z 2 =x 2 +iy 2 and Then if z 1 =z 2 y 1 = y 2 x 1 = x 2

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© 2005 Baylor University Slide 8 Algebra of Complex Numbers z 1 =x 1 +iy 1 Definitions: Given z 2 =x 2 +iy 2 z 1 +z 2 =z 3 z 3 = (x 1 + x 2 ) + i(y 1 + y 2 ) z 1 -z 2 =z 3 z 3 = (x 1 - x 2 ) + i(y 1 - y 2 ) z 1 * z 2 =z 3 z 3 = (x 1 + iy 1 )(x 2 + iy 2 ) = x 1 x 2 +ix 1 y 2 +ix 2 y 1 +i 2 y 1 y 2 z 3 = x 1 x 2 +i 2 y 1 y 2 +i(x 1 y 2 +x 2 y 1 ) Im(z 3 )= x 2 y 1 -x 1 y 2 Re(z 3 )= x 1 x 2 -y 1 y 2

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© 2005 Baylor University Slide 9 Dividing Complex Numbers To divide, must eliminate the i from the denominator We do this with the Complex Conjugate - by CHANGING THE SIGN OF i

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© 2005 Baylor University Slide 10 Reciprocals of Complex Numbers Multiply by the Complex Conjugate to put in Standard Form

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© 2005 Baylor University Slide 11 This concludes the Lecture

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