Summary Complex numbers

Complex numbers

i :




i=F Pattern : -
l ; -
i ; I ;i

( multiple of 4)
i. 2=-1 Ru , @ : e.
=
,


13=112 I .
= -
t.FI = -

I

it =
ii. it =
1




Complex numbers :




2- =
atbi
← →
Real Imaginary
RECZ) Im (E)

e. 2- =L Zi
g.
-




Recz> =
4


Im (2) =
-2




Complex numbers with operators :




'
Zi ( 3 Zi ) ( 3i ) ( hi )( 3 Zi ) Remember e- 2=-1
:
I .
-
2 .
I -
3. Z - -




=
bi -
↳ it
=
I -
bit quiz
=
6+8 i -
8W

=
=
bit 4 =
-8 -6L 14+8 i




Complex conjugate :




of the
Inverse operator
imaginary part
symbol = *


e. 2- =3 hi
g
-




*
2-
=
3. + ↳ I

, Division of complex numbers :


Don't want
imaginary number in demon .
-
>
.
'


. times by
2É¥
e. g. p=
I -
3i ; q= 2. + hi ; find E-
E- III. -
✗
E¥i
wi )
=L ' 3i)( 2
- -




(2 + hi )(2 -
w - i)
2- 1 Oi +121-2
=

4- 16in
-
I0-1OI
=

20
-10 ( I + i)
=
20
-
Citi )
=
2




Quadratic equation :




I . 3<2+2×+4--0
-2 ± 6
=
✗
-2 I -
12 C- can be split up
:



FT
=
✗ 2 12×-1 = JT2 ✗


-2 ± 12 i '
i
✗
=
z
=
JT2 I

✗
=
_2±{Fsi_
✗
= -
I ± F3J Answers of
any quadratic equation with
numbers will be complex
imaginary
I Fsi Fsi conjugates of
= + OR
✗ 2=-1 eachother
-
-




2 . ✗ (2+2)+5=0
22+2×+5=0

✗
=
-Z±yf
=
-2T¥
2=-1+-2 i