Textbook Notes (270,000)
CA (160,000)
McMaster (10,000)
MATH (400)
Chapter 8.2

# MATH 1C03 Chapter Notes - Chapter 8.2: Local Access And Transport Area, Prope, Uver River

Department
Mathematics
Course Code
MATH 1C03
Professor
Miroslav Lovric
Chapter
8.2

Page:
of 4
8.2 Complex Numbers
Acomplex number will be defined as apair of real numbers (x.gl EIRXR ,like points
on aCartesian plane ,but we will write it as :
XtYI iis asymbol with the
property it -Ior i=F
Definition :Acomplex number zin standard form is an expression of the
form x=yi ,where xand yare real numbers (x,yER ).The set of
all complex numbers is denoted by
E- {xtyilx,YER}
Zcan be broken up into real and imaginary parts :
z= x+Yi
The real number xis called the real part of 7and is denoted
by Retz).
The real number yis called the imaginary part of zand is denoted
by Imla ).
ExÂ¥Ã·
-ti o+Mi 3+02 .
Rebel =5 Re (x):O Relx )=3
Imlyt -4 PImly )=D >In (y)=O
"
purely imaginary
Real number "
8.21 Addition &Multiplication of Complex Numbers
(at bi)+(ctdi )=(at
c) +(bid );
These operations can be treated just as the usual
of algebraic expressions where iis handled as an algebraic Symbol with the
property
that whenever is occurs ,it is replaced by -1
.
Example
Ã·4;)+(3ti )(5+4
;) .(3ti )(2-3 ;)â€¢#.
)(
1+42=(5+3)+1++1
.)i =5.3+4.31+5 .li+4i2 =;-ftp.2 =I+2itj2
=8+5 ;=15+17 ;
-4=1+2;
-1=11+17
;=3zti =2;
Consider these operations ontwocomplexnumberswithzeroimainary parts
(attn ).Â¢t0i )=actoi real multiplication
This means that all real numbers lie on the complex
plane ,and .therefore
thesetof real numbers (Bhisasvbsetotthe complex plane (Â¢)
Wehavenow achieved our objective of
extending the
number system to include the
Square roots of all negative numbers
.
Example
Ã·
-z=4tiandw=-3i2i be complex numbers.Find
Dztw ii )z -wNoteipowersofi
z.,w=H+i)H -3+2 ;) z.wftti )-(-3+21)=7 .ilifn :O (modt)
=1+3 ;=7 -jiifn=1(mod4 )
iiiew nine
in Â¥iHnIYem
:b:p
z2w=( tti.pt -3+2.)w4=( -3+2 ;)t
H6t8itiYC3t2il =-34+41-31312 ;) '- 4Ã—-3512.5+41-3 )fzjH2D"
=t5+8i)f3t2 ;) -
-81-216 ;+216in
-963+14.4=-45-24
;
-138+167=81-216
;i96i+l6
=-61+6 ;=-119-120 ;
Proposition
-
Let Z,w,
tEE and let a=xtgi and w=u +vi
where
,
x,y,v.v ER .Then :
i)
,?
=wif and only if x=o and y=v
Two
complex numbers are equal if and only if their real parts
are equal &their imaginary parts are equal
Proof:The complex numbers z=x+yi and w=u+vi represent
the ordered
pair of real numbers (x,y)and (v.v )
respectively .These two ordered pairs are equal it
and only if their first elements are equal and their
Second elements are equal
ii)(z + w)it =z+(wit )(associatively of addition
).
iii)z+w= wtz (com mutativity of addition )
iv )0=0+0 ithen 0+7=2-1
existence of azero )
v)The number -z =-x-Yi EE is such that Â¥+1 -z)=O (existence of negatives)
vi)(z.w).t =zfw .tl lasso ciativity of multiplication)
vii Z.w=W.z(com mutativity of multiplication
.)
viii)1=1+0 ;is such that I.z=z (existence of aunit )
Dz .(wit )=z.W+zt (distributive law)
Proof:All these
parts of the proposition can be proved by
calculating each expression directly.
ix )If zto ,the element
Z
''
=zt =x-yi=
-y2
is the inverse of zand satisfies zz
''=I (existence of inverses )
Proof :The
complex number z=x+yi is zero if and only if x=o and
y=O.Hence it Zto ,then either xor yis nonzero ,and it follows
that oity '>0
.Therefore
I7Ã·z=xÃ·y -Itta
(
Ã—j+YgÃ·D( xtgi )=x2txg=y =1
x2+y 2
Hence
the complex number ,xjyyÃ· is the inverse of xiyi