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------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/master08.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADeAQAAAUAHgAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwCeQChAAMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAOAAAAJAMFAP///////3gA oAB4AKAA////////CAAAAPoCAAAAAAAAAAAAAAQAAAAtAQIABAAAAC0BAAAEAAAAJwH//xwAAAD7 Av7/AAAAAAAAkAEAAAACAEAAElN5bWJvbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABAAAAC0B AwAEAAAALgEYAAQAAAACAQEABQAAAAkC/wAAAgkAAAAyClIAngABAAAA0ycCAAQAAAAuAQAAHAAA APsCEAAHAAAAAAC8AgAAAAABAgIiU3lzdGVtAAAAAAAAAAAAABgAAAABAAAAGAE/BCoAAAAEAAAA LQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKWACeAAEA AABGJwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJp YWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAA CQL/AAACCQAAADIKWgCeAAEAAAByJwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8A AAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKXQCeAAEAAABhJwEABAAAAC4BAAAEAAAALQEE AAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKYACeAAEAAABu JwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/ AAACCQAAADIKYgCeAAEAAABrJwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAA AACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAu ARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKaACeAAEAAABNJwIABAAAAC4BAAAEAAAALQEEAAQA AADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKagCeAAEAAABhJwEA BAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAAC CQAAADIKbQCeAAEAAAAyJwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQ AQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgA BAAAAAIBAQAFAAAACQL/AAACCQAAADIKcACeAAEAAAAwJwEABAAAAC4BAAAEAAAALQEEAAQAAADw AQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAAADIKcgCeAAEAAAAwJwEABAAA AC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/v8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQL/AAACCQAA ADIKdQCeAAEAAAA2JwEABAAAAC4BAAAEAAAALQEEAAQAAADwAQMAHAAAAPsC/f8AAAAAAACQAQAA AAIAQAASU3ltYm9sAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQAAAAuARgABAAA AAIBAQAFAAAACQIAAAACCQAAADIKcwCBAAEAAADTJwIABAAAAC4BAAAEAAAALQEEAAQAAADwAQMA HAAAAPsC/f8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAE AAAALQEDAAQAAAAuARgABAAAAAIBAQAFAAAACQIAAAACHAAAADIKcwCEAA4AAAAyMDA2LCBGcmFu ayBNYQIAAgACAAIAAQABAAIAAQABAAIAAgAAAAMAAQAEAAAALgEAAAQAAAAtAQQABAAAAPABAwAc AAAA+wL3/wAAAAAAAJABAAAAAABAACJBcmlhbAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAQA AAAtAQMABAAAAC4BGAAEAAAAAgEBAAUAAAAJAgAAAAIeAAAAMgoJACsADwAAAENvbXBsZXggTnVt YmVycwAHAAUACAAFAAIABAAFAAIABwAEAAgABQAFAAMABAAEAAAALgEAAAQAAAAtAQQABAAAAPAB AwADAAAAAAA= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0060.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Ó= ; 2006, Frank = Ma
Complex Numbers<= /span>
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0059.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0078.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0078_image001.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0079_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0079_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Complex Numbers<= /span>
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0080_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0080_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the soluti= on of x2 =3D -r using i.<= /div>
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0081_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0081_image007.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0082.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0082_image008.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0082_image009.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0083.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
x2 + 49 =3D 0       Use= the square-root method:
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0083_image010.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0083_image011.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0084.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
x2 + 49 =3D 0       Use= the square-root method:
x2 =3D -49
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0084_image012.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0084_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
x2 + 49 =3D 0       Use= the square-root method:
x2 =3D -49
x =3D = ±Ö-49
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0085_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0085_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0085_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhIQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAf AAEAgAAAAAAAAAIFjI+pewUAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0086.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
x2 + 49 =3D 0       Use= the square-root method:
x2 =3D -49
x =3D = ±Ö-49 =3D ±Ö49Ö-1
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0086_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0086_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0086_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhiAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACG AAEAgAAAAAAAAAIPjI+pe+APo5y0spsqxNwUADs= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0087.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F= = r= a= n= k= =  M= a = 2= 0= 0= 6
Complex Numbers<= /span>
We may write the solution of x2 =3D -r using i.
Example A: Find the solutions of x2 + 49 =3D 0
x2 + 49 =3D 0       Use= the square-root method:
x2 =3D -49
x =3D = ±Ö-49 =3D ±Ö49Ö-1 =3D ±7i
The equation x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D -1 does no have any real number solution since the squ= are of any real number is always non-negative.
So we mak= e up a new imaginary number :
        = ;     Ö-1
to be a s= olution of this equation.
We write = i for Ö-1 for simplicity. Hence, i= 2 =3D -1.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0087_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0087_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAW AAEAgAAAAAAAAAIEjI+pUQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0087_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhiAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACG AAEAgAAAAAAAAAIPjI+pe+APo5y0spsqxNwUADs= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0087_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLgAYAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAu ABcAgAAAAAAAAAJDjI+py20Ao5y0Wvuu3hzkDoZfSGpjiU5nyq4s6r5kLIN0zd24Geyp7qsAg6oe UWQ8doZKT7LJg+aeUiG1qnI6tlxHAQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0071.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers,
Complex Numbers<= /span>
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0088.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0089.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,  6i,  -17 are complex numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0090.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,=   6i,  -17 are complex numbers
for 6i =3D 0 + 6i, -17 =3D -17 + 0i.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0091.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,=   6i,  -17 are complex numbers
for 6i =3D 0 + 6i, -17 =3D -17 + 0i.
Every = real number x is a complex number because
x =3D = x + 0i. <= /div>
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0092.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,=   6i,  -17 are complex numbers
for 6i =3D 0 + 6i, -17 =3D -17 + 0i.
Every = real number x is a complex number because
x =3D = x + 0i. <= /div>
The absol= ute value of a ± bi, writ= ten as | a &pl= usmn; bi|, is defined to be Öa + b2.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0092_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0093.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,=   6i,  -17 are complex numbers
for 6i =3D 0 + 6i, -17 =3D -17 + 0i.
Every = real number x is a complex number because
x =3D = x + 0i. <= /div>
The absol= ute value of a ± bi, writ= ten as | a &pl= usmn; bi|, is defined to be Öa + b2.
Example C:  |5 – 3i|
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0093_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0094.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Ó F<= /span>= rank M<= /span>a 2006
A complex number is a number of the form
        = ;            &n= bsp;            = ;  a ± bi
where a and b are real numbers, a= is called the real part and ±bi is called the imaginary part of the complex number=
Complex Numbers<= /span>
Example B:    5 – 3i,=   6i,  -17 are complex numbers
for 6i =3D 0 + 6i, -17 =3D -17 + 0i.
Every = real number x is a complex number because
x =3D = x + 0i. <= /div>
The absol= ute value of a ± bi, writ= ten as | a &pl= usmn; bi|, is defined to be Öa + b2.
Example C:  |5 – 3i| =3D Ö52 + 32 =3D Ö34
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0094_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQA/ AAEAgAAAAAAAAAIGjI+py+1cADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0094_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhfwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0095.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0096.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i)
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0097.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0098.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0098_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA3 ABwAgAAAAAAAAAJPRI6py+0Popy02ovvybx7u33iCAbkeYbo2qnsW8Jy5c5ybb94vu58avrBfEIR seg5IjnKJabpjEVJ0Omkao1gs1trd/qNhp3jpemBTqsbBQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0099.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i)
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0099_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhNwAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA3 ABwAgAAAAAAAAAJPRI6py+0Popy02ovvybx7u33iCAbkeYbo2qnsW8Jy5c5ybb94vu58avrBfEIR seg5IjnKJabpjEVJ0Omkao1gs1trd/qNhp3jpemBTqsbBQA7 ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0100.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Arithmetic of Complex Numbers
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0100_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0101.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplication of complex numbers:
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0101_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0102.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0102_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0103.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
Example E:        = ;      
(4 – 3i)(2 + 7i)
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0103_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0104.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
Example E:        = ;      
(4 – 3i)(2 + 7i) =3D 8 – 6i + 28i – 21i2  
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0104_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0105.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
Example E:        = ;      
(4 – 3i)(2 + 7i) =3D 8 – 6i + 28i – 21i= 2   set i= 2 =3D (-1)
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0105_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0106.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
Example E:        = ;      
(4 – 3i)(2 + 7i) =3D 8 – 6i + 28i – 21i= 2   set i= 2 =3D (-1)
        = ;            &n= bsp;   =3D 8 – 6i + 28i + 21
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0106_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0107.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Addition and subtraction of complex numbers:
To add or subtract complex numbers, treat the i as a = variable. =
Example D:
(7 + 4i)= + (5 – 3i) =3D 7 + 4i + 5 – 3i =3D 12 + i
(7 + 4i)= – (5 – 3i) =3D 7 + 4i – 5 + 3i =3D 2 + 7i
Ó F= r= a= n= k= =  M= a = 2= 0= 0= 6
Arithmetic of Complex Numbers
Multiplic= ation of complex numbers:
To multip= ly complex numbers, use FOIL, then set i2 to be (-1) and simplify the result= .
Example E:        = ;      
(4 – 3i)(2 + 7i) =3D 8 – 6i + 28i – 21i= 2   set i2 =3D (-1)
        = ;            &n= bsp;   =3D 8 – 6i + 28i + 21 =
        = ;            &n= bsp;   =3D 29 + 22i
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0107_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 ADYAgAAAAAAAAAJ9jI+py90Ao5y02msD3rxX7YXiN5YlaKYcqrakC09sDM90a99prp99/XPxgp4h cXXcJU3GZcY5akJl05C0CrlitVXu1At1iMfkBxZ53oCd62U7+T7GifNg/Xfv5XX7W5/2FxMIlIYx KFRomHhxiLP49EjRqDKpFClTlqnZUAAAOw== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0043.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
The complex numbers a + bi and a – bi are the conjugate of each other. 
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Arithmetic of Complex Numbers
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0109.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
The conjug= ate of a real number is itself.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0110.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
The conjug= ate of a real number is itself.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0111.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(a + bi)(a= – bi) =3D a2 + b2 
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
The conjug= ate of a real number is itself.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0112.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(a + bi)(a – bi) = =3D a2 + b2 
Note that = the product of a conjugate pair is always a <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:89.88%;height:6.0%'= >positive real number.
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
The conjug= ate of a real number is itself.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0117.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(a + bi)(a – bi) = =3D a2 + b2 
Note that = the product of a conjugate pair is always a <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:89.88%;height:6.0%'= >positive real number.
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
Example F:        = ;      
a. (4 – 3i)(4 + 3i)
b. (Ö7 – 2i)(Ö7 + 2i)
The conjug= ate of a real number is itself.
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(a + bi)(a – bi) = =3D a2 + b2 
Note that = the product of a conjugate pair is always a <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:89.88%;height:6.0%'= >positive real number.
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
Example F:        = ;      
a. (4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
b. (Ö7 – 2i)(Ö7 + 2i)
The conjug= ate of a real number is itself.
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(a + bi)(a – bi) = =3D a2 + b2 
Note that = the product of a conjugate pair is always a <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:89.88%;height:6.0%'= >positive real number.
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
Example F:        = ;      
a. (4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
b. (Ö7 – 2i)(Ö7 + 2i) =3D (Ö7)2 + 22 =3D 11
The conjug= ate of a real number is itself.
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0143.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
(a + bi)(a – bi) = =3D a2 + b2 
Note that = the product of a conjugate pair is always a <= span style=3D'position:absolute;top:65.75%;left:5.24%;width:89.88%;height:6.0%'= >positive real number.
Arithmetic of Complex Numbers
The most important complex number multiplication = is the conjugate-multiplication:
<= span lang=3DEN-US style=3D'font-size:88%;mso-fareast-language:ZH-CN'>The complex numbers a + bi and a – bi are the conjug= ate of each other.  For example 3 + 4i and
3 – 4i are conjug= ate of each other.
Example F:        = ;      
a. (4 – 3i)(4 + 3i) =3D 42 = + 32 =3D 25
b. (Ö7 – 2i)(Ö7 + 2i) =3D (Ö7)2 + 22 =3D 11
The conjug= ate of a real number is itself.
If z is a = complex number, we write z for it's conjugate.
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Arithmetic of Complex Numbers
Division of complex numbers:
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Arithmetic of Complex Numbers
Division = of complex numbers:
Division = of complex numbers are carried out
by multip= lying the conjugate of the denominator to the top and bottom.=
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Arithmetic of Complex Numbers
Example G= :  Simplify
4 – 3i
3 + 2i
Division = of complex numbers:
Division = of complex numbers are carried out
by multip= lying the conjugate of the denominator to the top and bottom.=
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0119_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0120.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
Example G= :  Simplify
4 – 3i
3 + 2i
4 – 3i
3 + 2i
=3D
Division = of complex numbers:
Division = of complex numbers are carried out
by multip= lying the conjugate of the denominator to the top and bottom.=
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0120_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0120_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
Example G= :  Simplify
4 – 3i
3 + 2i
4 – 3i
3 + 2i
=3D
4 – 3i
3 + 2i
4 += 3i
4 += 3i
conjugates
Division = of complex numbers:
Division = of complex numbers are carried out
by multip= lying the conjugate of the denominator to the top and bottom.=
------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA6 AAEAgAAAAAAAAAIGjI+py+1XADt= ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0121_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRgAfAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAgA/ ABsAgAAAAAAAAAJlDI6py+2/goS0Wifz3Rzn34XbRxriCZUaynqr2cbJBE+ybCvwzebzzgv5dEHU 8FcUAiPJzlG3bHoqUSnjGalaEdjr9tHNfq/aRlkaJo+RzjX3bIbf0mA3ffq91/NycP+kl7JFVwAA Ow== ------=_NextPart_01C82E61.7DCB73D0 Content-Location: file:///C:/7102A2B3/6ComplexNumbers_files/slide0122.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Complex Numbers
Arithmetic of Complex Numbers
Example G= :  Simplify
4 – 3i
3 + 2i
4 – 3i
3 + 2i
=3D