← Topic 1 · Algebra
AMC 10/12 · Cheatsheet
Topic 1 · Algebra
Chapter 4 · Complex numbers
🖨 Print / PDF
📋 Reference · always available
Powers of i
i
,
−
1
,
−
i
,
1
(
cycle
4
)
i,\,-1,\,-i,\,1\;(\text{cycle }4)
i
,
−
1
,
−
i
,
1
(
cycle
4
)
Modulus
∣
a
+
b
i
∣
=
a
2
+
b
2
|a+bi| = \sqrt{a^2+b^2}
∣
a
+
bi
∣
=
a
2
+
b
2
Product
(
a
+
b
i
)
(
c
+
d
i
)
=
(
a
c
−
b
d
)
+
(
a
d
+
b
c
)
i
(a+bi)(c+di) = (ac-bd)+(ad+bc)i
(
a
+
bi
)
(
c
+
d
i
)
=
(
a
c
−
b
d
)
+
(
a
d
+
b
c
)
i
Polar form
z
=
r
e
i
θ
=
r
(
cos
θ
+
i
sin
θ
)
z = re^{i\theta} = r(\cos\theta + i\sin\theta)
z
=
r
e
i
θ
=
r
(
cos
θ
+
i
sin
θ
)
De Moivre
(
cos
θ
+
i
sin
θ
)
n
=
cos
(
n
θ
)
+
i
sin
(
n
θ
)
(\cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)
(
cos
θ
+
i
sin
θ
)
n
=
cos
(
n
θ
)
+
i
sin
(
n
θ
)
$n^{\text{th}}$ roots of unity
ω
k
=
e
2
π
i
k
/
n
,
k
=
0
,
1
,
…
,
n
−
1
\omega_k = e^{2\pi i k/n},\;\; k=0,1,\dots,n-1
ω
k
=
e
2
π
ik
/
n
,
k
=
0
,
1
,
…
,
n
−
1
Sum of roots of unity
∑
k
=
0
n
−
1
ω
k
=
0
(
n
≥
2
)
\sum_{k=0}^{n-1}\omega_k = 0\;\;(n\ge 2)
k
=
0
∑
n
−
1
ω
k
=
0
(
n
≥
2
)