Complex Numbers
1. 𝑖 = √−1
2. = −𝑖
3. 𝑖 = 1 ⇒ 𝑖 = 𝑖
4. 𝑧 = 𝑎 + 𝑖𝑏 ⇒ 𝑧̅ = 𝑎 − 𝑖𝑏
5. |𝑧| = √𝑎 + 𝑏 =
6. |𝑧| = |𝑧̅|
7. 𝐴𝑟𝑔(𝑧) = tan
𝑅𝑒(𝑧)
+ 𝐼𝑚(𝑧)
→ Distance of 𝑧 from origin
| | ∈ [− , ]
8. 𝐼𝑓 𝑧 = 𝑘, arg(𝑧) = 0 𝑖𝑓 𝑘 > 0, arg(𝑧) = 𝜋 𝑖𝑓 𝑘 < 0.
9. 𝐼𝑓 𝑧 = 𝑖𝑘, arg(𝑧) 𝑖𝑠 𝑛𝑜𝑡 𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
10. Forms of Complex Numbers:
(i)
Cartesian Form: 𝑧 = 𝑥 + 𝑖𝑦, 𝑥, 𝑦 ∈ 𝑅
(ii)
Polar Form
: 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃)
(iii)
Euler’s Form : 𝑧 = 𝑟𝑒
11. (𝑎 + 𝑖𝑏) ± (𝑐 + 𝑖𝑑) = (𝑎 + 𝑐) ± 𝑖(𝑏 + 𝑑)
)
12. 𝑧 𝑧 = 𝑟 𝑒 . 𝑟 𝑒
=𝑟𝑟𝑒(
13. 𝐼𝑓 𝑧 = 𝑥 + 𝑖𝑦,
14.
=|
|𝑒
=
(
=
̅
→ Multiplicative Inverse
)
15. Square Root of a Complex Number:
(i)
𝐹𝑜𝑟 𝑧 = (𝑎 + 𝑖𝑏): √𝑧 = ±
(ii)
𝐹𝑜𝑟 𝑧 = (𝑎 − 𝑖𝑏): √𝑧 = ±
| |
| |
+𝑖
−𝑖
| |
| |
16. Properties of Conjugate of a Complex Number:
(i)
𝑧𝑧̅ = |𝑧|
̅
(ii)
𝑅𝑒(𝑧) =
(iii)
𝐼𝑚(𝑧) =
(iv)
(v)
(𝑧 ± 𝑧 ) = 𝑧 ± 𝑧 = 𝑧 ∓ 𝑧
𝑧 𝑧 𝑧 …𝑧 = 𝑧 .𝑧 .𝑧 …𝑧
(vi)
̅
=
(𝑖𝑓 𝑧 ≠ 0)
(vii) 𝑧 = 𝑧̅
(viii) arg(𝑧) + arg(𝑧̅) = 2𝑘𝜋
17. Properties of Modulus of a Complex Number:
|𝑧| = 0 𝑖𝑓 𝑧 = 0; |𝑧| > 0 𝑖𝑓 𝑧 ≠ 0
(i)
(ii)
−|𝑧| ≤ 𝑅𝑒(𝑧) ≤ |𝑧|, 𝑎𝑛𝑑 − |𝑧| ≤ 𝐼𝑚(𝑧) ≤ |𝑧|
|𝑧 𝑧 𝑧 … 𝑧 | = |𝑧 ||𝑧 ||𝑧 | … |𝑧 |
(iii)
(iv)
|
=|
|
|
(v)
(vi)
(vii)
(viii)
(ix)
(x)
|𝑧
|𝑧
|𝑧
|𝑧
|𝑧
|𝑧
(xi)
|𝑧 + 𝑧 | = |𝑧 | + |𝑧 | ⟺
± 𝑧 | = |𝑧 | + |𝑧 | ± 𝑅𝑒(𝑧 𝑧 ) = |𝑧 | + |𝑧 | ± 𝑅𝑒(𝑧 𝑧 )
+ 𝑧 | + |𝑧 − 𝑧 | = 2(|𝑧 | + |𝑧 | )
+ 𝑧 | ≤ |𝑧 | + |𝑧 |
− 𝑧 | ≥ |𝑧 | − |𝑧 |
| − |𝑧 | ≤ |𝑧 | + |𝑧 |
+ 𝑧 | = |𝑧 − 𝑧 | ⇔ arg(𝑧 ) − arg(𝑧 ) =
𝑖𝑠 𝑝𝑢𝑟𝑒𝑙𝑦 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦
(xii) |𝑧 + 𝑧 + 𝑧 … + 𝑧 | = |𝑧 + 𝑧 + 𝑧 … + 𝑧 |
(xiii) |𝑧 − 𝑧 | + |𝑧 − 𝑧 | ≥ |𝑧 − 𝑧 |
(xiv) |𝑧| = 𝑧𝑧̅
18. Properties of Argument of a Complex Number:
(i)
arg(𝑧) = − arg(𝑧̅) , 𝑧 ∉ 𝑅
(ii)
arg(𝑧 𝑧 ) = arg(𝑧 ) + arg(𝑧 ) + 2𝑘𝜋, 𝑘 ∈ 𝑍
(iii)
arg
(iv)
arg
= arg(𝑧 ) − arg(𝑧 ) + 2𝑘𝜋, 𝑘 ∈ 𝑍
̅
= 2 arg(𝑧) + 2𝑘𝜋, 𝑘 ∈ 𝑍
(v)
arg(𝑧 ) = 𝑛 arg(𝑧) + 2𝑘𝜋, 𝑘 ∈ 𝑍, 𝑘 𝑑𝑒𝑝𝑒𝑛𝑑𝑠 𝑜𝑛 𝑛
(vi)
arg(𝑧) ∈ (−𝜋, 𝜋]
19. (cos 𝜃 + 𝑖 sin 𝜃) = cos 𝑛𝜃 + 𝑖 sin 𝑛𝜃 *
*Only one of the solutions
20. (cos 𝜃 + 𝑖 sin 𝜃) = cos 𝜃 + 𝑖 sin 𝜃 *
21. log 𝑧 = log |𝑧| + 𝑖(2𝑘𝜋 + 𝜃) =
log(𝑥 + 𝑦 ) + 𝑖 2𝑘𝜋 + tan
22. Properties of nth Roots of 1:
(i)
𝛼
= cos
+ 𝑖 sin
= 𝑒
(ii)
1 + 𝛼 + 𝛼 + 𝛼 …+𝛼
=0
(iii)
1. 𝛼. 𝛼 . 𝛼 … 𝛼
= (−1)
23. Properties of Cube Roots of 1:
(i)
𝜔 = cos
(ii)
𝜔 = cos
(iii)
(iv)
(v)
1+𝜔+𝜔 = 0
𝜔 =1
𝜔=
+ 𝑖 sin
+ 𝑖 sin
=
=
[𝛼 = 1]
√
√
(vi)
𝜔= 𝜔
(vii) If 𝑝 is a cube root of a number, then the other cube roots are 𝑝𝜔 and 𝑝𝜔
24. Geometry of Complex Numbers:
(i)
Section Formula: 𝑃 ≡
(ii)
(iii)
(iv)
Condition for Parallelogram: 𝑧 + 𝑧 = 𝑧 + 𝑧
Condition for Rhombus
: (a) 𝑧 + 𝑧 = 𝑧 + 𝑧
(b) |𝑧 − 𝑧 | = |𝑧 − 𝑧 |
Condition for Square
: (a) 𝑧 + 𝑧 = 𝑧 + 𝑧
(b) |𝑧 − 𝑧 | = |𝑧 − 𝑧 |
(c) |𝑧 − 𝑧 | = |𝑧 − 𝑧 |
(v)
Condition for Rectangle
: (a) 𝑧 + 𝑧 = 𝑧 + 𝑧
(b) |𝑧 − 𝑧 | = |𝑧 − 𝑧 |
25. Rotation of Complex Number 𝑧 by an Angle 𝜃:
(i)
Rotation about Origin: 𝑧 = 𝑧 𝑒 = |𝑧 |𝑒 ( ( ) )
(ii)
Rotation about 𝑧 : 𝑧 − 𝑧 = |
(iii)
Rotation about 𝑧 : 𝑧 − 𝑧 = |
|
|
|
|
|
|
(𝑧 − 𝑧 )𝑒
(𝜃 > 0)
(𝑧 − 𝑧 )𝑒
26. Straight Lines in the Argand Plane 𝛼 =
(𝜃 < 0)
:
(i)
Equation of Straight Line: 𝛼 𝑧 + 𝛼𝑧̅ + 𝑐 = 0
(ii)
Real Slope = −𝑖
(iii)
Complex Slope = −
(iv)
(v)
(vi)
Condition for Parallel Lines (Slopes 𝜇 , 𝜇 ): 𝜇 = 𝜇
Condition for Perpendicular Lines (Slopes 𝜇 , 𝜇 ): 𝜇 + 𝜇 = 0
For two straight lines 𝛼 𝑧 + 𝛼𝑧̅ + 𝑐 = 0 and 𝛽̅ 𝑧 + 𝛽𝑧̅ + 𝑘 = 0:
(a) They are parallel if 𝛼 𝛽 − 𝛼𝛽̅ = 0
(b) They are perpendicular if 𝛼 𝛽 + 𝛼𝛽̅ = 0
Equation of line joining 𝑧 and 𝑧 : 𝑧 = 𝑡𝑧 + (1 − 𝑡)𝑧 , 𝑡 ∈ 𝑅
𝑧 𝑧 1
Three points 𝑧 , 𝑧 , 𝑧 are collinear if 𝑧 𝑧 1 = 0
𝑧 𝑧 1
(vii)
(viii)
(ix)
Perpendicular distance of 𝑧 from a line is 𝑝 =
(x)
Image 𝑧 of 𝑧 wrt to a line: 𝑧 = −
(xi)
Centroid of a triangle with vertices 𝑧 , 𝑧 , 𝑧 is 𝐺 ≡
(xii)
Incentre of a triangle with vertices 𝑧 , 𝑧 , 𝑧 is 𝐼 ≡
(xiii)
Circumcentre of Triangle: 𝑧 =
∑
∑
(
(
)
|
|
|
|
|
| |
|
| |
|
|
or by solving |𝑧 − 𝑧 | =
)
|𝑧 − 𝑧 | = |𝑧 − 𝑧 |
(xiv)
Orthocentre of Triangle: 𝑧 =
∑
(
) ∑|
| (
∑(
)
)
27. Circles in Argand Plane:
(i)
Radius with centre 𝑧 : |𝑧 − 𝑧 | = 𝑟
(ii)
Equation with centre 𝑧 : (𝑧 − 𝑧 )(𝑧̅ − 𝑧 ) = 𝑟
(iii)
General Equation of Circle: 𝑧𝑧̅ + 𝛼 𝑧 + 𝛼𝑧̅ + 𝑐 = 0 (centre ≡ 𝛼)
(iv)
Diametric Equation (𝑧 , 𝑧 ): (𝑧 − 𝑧 )(𝑧̅ − 𝑧 ) + (𝑧̅ − 𝑧 )(𝑧 − 𝑧 ) = 0
(v)
(vi)
(vii)
Circle Passing through 3 points:
(
)(
)
(
)(
(
)(
)
)(
)
Four points 𝑧 , 𝑧 , 𝑧 , 𝑧 are concyclic if
(
( ̅
=(
)
̅
)(
)
)(
)
∈𝑅
The inequality 𝑟 < |𝑧 − 𝑧 | < 𝑅 represents the area between the
concentric circles of radii r and R, centred at 𝑧 .
28. Important loci in the Argand Plane:
|𝑧 − 𝑧 | = |𝑧 − 𝑧 | is the perpendicular bisector of the line joining 𝑧 and 𝑧
(i)
|𝑧 − 𝑧 | + |𝑧 − 𝑧 | = |𝑧 − 𝑧 | is the line segment joining 𝑧 and 𝑧 .
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
|𝑧 − 𝑧 | + |𝑧 − 𝑧 | > |𝑧 − 𝑧 | is an ellipse with foci 𝑧 and 𝑧 .
|𝑧 − 𝑧 | + |𝑧 − 𝑧 | < |𝑧 − 𝑧 | does not exist.
|𝑧 − 𝑧 | − |𝑧 − 𝑧 | = |𝑧 − 𝑧 | is the extension of the line joining 𝑧 and
𝑧 , but does not lie between 𝑧 and 𝑧 .
|𝑧 − 𝑧 | − |𝑧 − 𝑧 | > |𝑧 − 𝑧 | represents a hyperbola with foci 𝑧 and 𝑧
|𝑧 − 𝑧 | − |𝑧 − 𝑧 | < |𝑧 − 𝑧 | does not exist
(viii)
If arg
(ix)
= 0 𝑜𝑟 𝜋, 𝑧 is a straight line passing through 𝑧 and 𝑧 .
= 𝑘 (𝑘 > 0, 𝑘 ≠ 1) represents a circle
(x)
|𝑧 − 𝑧 | + |𝑧 − 𝑧 | = 𝑘 is a circle iff 𝑘 > |𝑧 − 𝑧 |
(xi)
If arg(𝑧) = 𝜃, it represents a ray from origin passing through 𝑧 (origin not
included).
If arg(𝑧 − 𝑧 ) = 𝜃, it represents a ray from 𝑧, which passes through 𝑧 .
If |𝑧 − 𝑧 | + |𝑧 − 𝑧 | = |𝑧 − 𝑧 | , 𝑧 represents a circle with 𝑧 𝑧 as
(xii)
(xiii)
diameter. Then centre≡
and radius≡
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