Complex Numbers
, 4A: Cartesian form
i = F
Worked example :
a .
find
?
b .
Hence ,
solve the equation x -6 + 13 = 0
a .
F = N x F
: 42
b .
x =
-
(b) = 5 -
4(1)(3) s use the quadratic
2 formula
=> 6= # = 4: from part a.
2
=> 6 = 4;
2
=
3 = Li
Conjugate roots
Key Point :
A
complex number
, E ,
can be written in Cartesian form as :
z =
x +
iy
where , y
ER
The real part of the complex number E = 2+
Ly is 2 , denoted
by Re <2) .
The imaginary this is denoted
part Of E isy by Im(z)
:
t .
gifz = 3 -
2i Re(z) =
3 [m(z) = -
2
if then its complex Z, is
= x+
My conjugate ,
zx =
x -
iy
Sums, products and quotients in Cartesian form:
worked example
= = 2+ i and w = 5 -
3;
find :
a .
z + W b .
z -
W C .
zw
a z + w =
2 + i + 5 -3 ; and
group real
imaginary parts
.
>
= 7 -
2i
b .
z -
w = 2 + i -
(5 3i) -
=
2+ i -
5 + 3i
= 3 + 4;
c . zw =
(2 + i) (5 -
3i) ~
expand brackets as usual
2
= 10-Gi + Si-3 ;
= 10 -
Gi + 5i + 3
= 13 -
i
, 4A: Cartesian form
i = F
Worked example :
a .
find
?
b .
Hence ,
solve the equation x -6 + 13 = 0
a .
F = N x F
: 42
b .
x =
-
(b) = 5 -
4(1)(3) s use the quadratic
2 formula
=> 6= # = 4: from part a.
2
=> 6 = 4;
2
=
3 = Li
Conjugate roots
Key Point :
A
complex number
, E ,
can be written in Cartesian form as :
z =
x +
iy
where , y
ER
The real part of the complex number E = 2+
Ly is 2 , denoted
by Re <2) .
The imaginary this is denoted
part Of E isy by Im(z)
:
t .
gifz = 3 -
2i Re(z) =
3 [m(z) = -
2
if then its complex Z, is
= x+
My conjugate ,
zx =
x -
iy
Sums, products and quotients in Cartesian form:
worked example
= = 2+ i and w = 5 -
3;
find :
a .
z + W b .
z -
W C .
zw
a z + w =
2 + i + 5 -3 ; and
group real
imaginary parts
.
>
= 7 -
2i
b .
z -
w = 2 + i -
(5 3i) -
=
2+ i -
5 + 3i
= 3 + 4;
c . zw =
(2 + i) (5 -
3i) ~
expand brackets as usual
2
= 10-Gi + Si-3 ;
= 10 -
Gi + 5i + 3
= 13 -
i
