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Complex Numbers
(Part 1)
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Basic ConceptsComplex Numbers are an extension to the real
numbers .
The most obvious problem was finding solutions to2 1x =
The are composed of a !eal component" a and an #maginar
component" ib
and are usuall denoted b the letter z
Thus$ for the general complex number we write" z x iy= +
where
2
1i = or 1i= %hile surds can have multiple tpes$ such as 2, 5, 5,
...etc
there is onl one #maginar !oot to consider" 2 1 2 3 1 3
1 2 1 3
2 3i i
= =
= =
= =
&ence the 'coefficient of is alwas real.i
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Basic ConceptsThus the enlarged number field is called the
Complex Numbers $ and it contains the !eal Numbers
(with the !ational Numbers and #ntegers as from previous ears
stud)$ as well as the purel
imaginar numbers M"
M
4
4.5
3
5
3
i
i
4 5
2 3
i
i
+
Purel !eal" Purel #maginar"0a i+ 0 ib+
Notations!eal and #maginar parts for are denoted as andz x iy= +
Re( )z x= Im( )z y=
The con*ugate of is denoted asz x iy= + z x iy=
%e write the term last$ i.e. rather thani 5 4i + 4 5i
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+sing the notation $ we can deduce that addition$ subtraction$
multiplication and divisionwill wor, ver similarl to the processes
we use when doing these operations with s-uare roots.
3 4
2 3
5 7
i
i
i
+ ++
+
3 4
2 3
1 1
i
i
i
+ +
+
3 4 2
2 3 2
5 7 2
+ +
+
+
3 4 2
2 3 2
1 1 2
+
+
+
ddition" /ubtraction"
Basic 0perations
( ) ( )2
3 4 2 3
6 9 8 12
6 12 17
6 17
i i
i i i
i
i
+ +
= + + += += +
( ) ( )
( )2
3 4 2 2 3 2
6 9 2 8 2 12 2
6 12 2 17 2
30 17 2
+ +
= + + +
= + +
= +
ultiplication"
Note how causes anaddition
but that causes asubtractionof the real numbers.
( )2
2 2=
2 1i =
1i=
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Basic 0perations
( )( )
( )( )
( )
2
22
3 4 2 32 3 2 3
6 9 8 12
2 3
1813
i ii i
i i i
i
i
+ +
+ =
=
( ) ( )
( )2
2
3 4 2 2 3 22 3 2 2 3 2
3 4 2 2 3 2
2 3 2
6 9 2 8 2 12 4
4 18
18 2
14
18 2
14
+ +
+ =
+ =
=
=
2ivision"
ultipl b thecon*ugate3
%e 4rationali5e6the denominator
ultipling bthe con*ugatefor complexnumbers3
we 4reali5e6 thedenominator
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Basic #dentities7-ualit of Complex numbers" (where are real)a ib
c id + = + , , ,a b c d
( ) ( ) 0a c i b d + =
( ) ( )0 0a c b d
a c b d
= == =
Thus$ two complex numbers are e-ual iff (if and onl if) both
their real parts are e-ual and boththeir imaginar parts are
e-ual.
+nli,e real numbers$ complex numbers are not ordered (the are
not 'linear).
Nonreal numbers cannot be classified as positive or
negative.
a ib c id + > +The expression has no meaning for complex
numbers8
zThe reciprocal of is $ or1
z
2 2
1 z
z z
z
a b
=+
The multiplications of gives" i.e. the sum of two s-uares8z z (
) ( ) 2 2a ib a ib a b+ = +
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/olutions to 9uadratic 7-uations
2 3 8 0x x+ + =
2 4
2
b b acx
a
=:rom the 9uadratic :ormula we can now define the solution
for
as
7xample" $ hence
Note" due to the smmetr in the roots$ complex solutions to a
-uadratic will occur in con*ugate pairs
0,
