COMPLEX NUMBERS
Complex number : A number of the form z = x + iy where x, y are the set of real numbers. Here i = Ö-1 represents an imaginary number or i2 = -1. In other words i0 = 1, i1 = i2 = -1, i3 = -i, i4 = + 1-
The conjugat of z = x + iy is z = x – iy.
Operations of Complex numbers
- Addition of complex numbers : When two complex numbers are added, real part is added with real part and imaginary part with imaginary part as shown below:
(x + iy) + (c + id) = (x + c) + i (y + d)
- Subtraction of complex numbers : When two complex numbers are sub- tracted, real part is subtracted from real part and imaginary part is subtracted from imaginary part as shown below :
(x + y) – (c + id) = (x – c) + i (y – d)
- Multiplication of complex numbers : Multiplication is carried out as follows: (x + iy) (c + id) = (xc – yd) + i(xd + yc)
- Division of complex nubers : Division of two complex numbers is carried out as follows :
x + iy xc + yd yc – xd
——– = ———- + i———-
c + id c2 + d2 c2 + d2
Properties of complex addition
- Commutative law : According to this law two complex numbers on adding follow z1 + z2 = z2 + z1.
- Associative law : According to this law, z1, z2 and z3, three complex numbers
follow (z1 + z2) + z3 = z1 + (z2 + z3).
- Additive identity : A complex number (0, 0) when added to any complex
number, it results in the same complex number i.e., (x, y) + (0, 0) = (x, y) = (0, 0) + (x, y)
Additive inverse : For z = x + iy there is a unique complex number – z = – x – iy which is called additive inverse such that z + (- z) = 0
Properties of complex multiplication
- Commutative Law : If there are two complex numbers z1 and z2 then z1z2 =
z2z1
- Associative Law : If there are three comlex numbers z1, z2 and z3 then
(z1z2)z3 = z1 (z2z3)
- Multiplicative identity : Multiplicative identity or unity is a complex number
which when multiplied to any complex number gives the same complex number i.e.,(x + iy) (l + io) = (x + iy) = (1 + io) (x + iy)
- Multiplicative inverse : For any complex number z = x + iy there exists a
1 x – iy
complex number z-1 = ——- =———- which is known as
x + iy x2 + y2 multiplicative inverse. z.z-1 = (1, 0)
- Distributive law : If there are there complex number z1, z2 and z3 then z1 (z2
+ z3) = z1z2 + z1z3.
Geometrical Representative of a complex number
Argand’s diagram : In Rectangular axes OX and OY a complex numbers a + ib can be represented as follows :
————————–
——————————-
———————————–
POM = q
b then q = tan-1 —
a and r = Öa2 + b2
Plane OX and OY is called Argand plane.
Vector Representation of a complex number
A complex number z = a + ib is represented by a vector OP. The length of the vector OP is added to the modulus of z.
A complex number x + iy is put as g = cosq + i sinq where g is the modulus and — is the argument of the complex number. z = x + iy = g (cos q + i sin q) |z| = Öx2 + y2
The principal value of amplitude is that value of q which satisfies 0 £ x £ P
Properties of the moduli and conjugate of ocmplex numbers :
If suppose complex numbers are A1, A2, A3 etc. then (1) (A1 + A2) = A1 ± A2
A1 A1 (2) —- = —-
A2 A2
(3) (A1A2) = A1A2
(4) (A) = A
- A = A « A is purely real
- A = – A « is purely imaginary
- A + A = 2Re (a) = A – A = 2i im (A)
1 A
(8) A A = |A|2; — = —–
A |A|2
(9) |A| £ 0 and |A| = 0 Û A = 0
- |-A| = |A| =
- |A +A | ³ |A | + |A | (triangle inequality)
1 2 1 2
(12) |A + A | ³ |A | – |A |
1 2 1 2
(13) |A1 – A2| ³ |A1| + |A2|
|
(14) |A – A | £ |A | |A |
(15) |A1.A2| = |A1|.|A2|
A1 A1
(16) —- =——- ; A2 ¹ 0
A2 A2
(17) |A + A |2 = |A |2 + |A |2 ± 2Re (A A )
1 2 1 2 1 2
(18) |A
+ A |2 + |A – A |2 = 2|A |2 + |A |2
1 2 1 2 1 2
z = a + ib is a complex number then z = a – ib is called its conjugate.
De Moivre’s Theorem :
(1) If n is any integer (positive or negative) or rational number then (cos q + i sin
q)n = cos nq + i sin nq
Euler’s Formula :
ei0 = cos q + i sin q and e-10 = cos q – i sin q
Cube Roots of Unity : Let x be a cube root of 1 then x = {cos (2rp) + i sin (2rp)}1/3
2rp 2rp
= cos —— + i sin—— , Putting r = 0, 1, 2, we get
3 3
2p 2p 1 Ö3
= 1, w, w2 where w = cos —-, + i sin —- = – — + i —–
3 3 2 2
Properties of cube root of unity :
- l + w + w2 = 0
- w3 = l
- i.e., (w)2 = w2 and (w2)2 = w.
1 1
- i.e., — = w2 and————————- = w.
w w2
- Cube roots of unity are in P.
- Cube roots of unity lie on a circle |z| = 1 and divide it in three equal
Logarithm of a Complex Number : Consider x + iy and a + ib as two complex
numbers such that a + ib = ex+iy, then x + iy is known as the logarithm of a + ib to the base e and we write x + iy = loge (a + ib)
x + i (2np + y) is also a logarithm of a + ib for all integral values of n. This is also
called the general value of log (a + ib). log (a + ib) = 2npi + log (a + ib)
Suppose a + ib = r (cos q + i sin q) = rei0 where r = |z| and q = arg (z)
b
= tan-1 —
a
Then log (a + ib) = log (reiq) = log r + iq
b
= log Öa2 + b2 + tan-1—- or log(z) = log |z| + i amp (z), where z = a + ib
a
Rule : Öa + ib = x + iy
\ a + ib = x2 – y2 + i 2xy
|
|
Rotation : Let A = rei(q+j) = reiq eij [\ A = B eiq] (z If z = r (cosq + i sin q), then
iz = r (cos q + i sin q)
p p
cos — + i sin — 2 2
p p
= r cos q + — + i sin q + —
2 2
– z0) = (z1
– z ) eij
Therefore multiplication of z with i rotates the vector for z through 900 angle.
Some Important Results :
- If z1 and z2 are two complex numbers, then the distance between z1 and z2 is
|z1 – z2|.
- The line joining z1 and z2 its equation is given by z = tz1 + (1 – t) z2 (Parametric form) where t is a real parameter.
- The equation of the line joining z1 and z2 in non-parametric form is given
z – z1 z – z1 z z 1
by ———- = ——— or z1 z1 1 = 0
z2 – z1 z2 – z1 z2 z2 1
(Non-parametric form)
- The condition of collinearity for three points z1,z2 and z3 is given by z1 z1 1
z2 z2 1 = 0
z3 z3 1
|
- The equation of a circle having centre z1 and radius r is |z – z0| = r or zz – z0z + z z – r2 = 0.
- The general equation of a circle is zz – az + az + b = 0, where b is a real The centre of this circle is – a and its radius is Öaa – b.
- The equation of the circle described on the line segment joining z1 and z2 as diameter is (z – z1) (z -z2) + (z – z2) (z – z1) = 0.
- The equation of a circle is represented by |z – z |2 + |z – z |2 = k (where k is a
real number). It has a centre at 1/2 (z1
1
+ z2) and radius 1/2 Ö2k-|z1
2
|
– z |2 provided k ³ 1/2
|z1 – z | .
|
2
- If the complex number z1 and z2 are such that the sum z1 + z2 is a real number, then they are not necessarily conjugate
- If z1 and z2 are two complex numbers such that the product z1z2 is a real number, then they are not necessarily conjugate
- The centroid of a triangle on the Argand’s plane whose vertices are z1, z2, z3 are the affixed of the point P, Q, R
z1 + z2 + z3 respectively is given by —————
3
- If z1, z2, z3 are the affixes of the points P, Q, R respectively on Argand’s Then angle between PQ and PR.
z3 – z2 i.e.,<QPR is given by ———
z2 – z1
z3 – z1 z3 – z3
\ ——— =———- (cos q + i sinq),
z2 – z1 z2 – z1
|
|
Where q = <QPR = q – q (see figure given below)
———————————
———————————
———————————
- z1, z2, z3 are collinear if arg (z3 – z1) = arg (z2 – z1).
- If z1, z2, z3 are such that PR ^ PQ,
p z3 – z1
then q = —- and so———- is purely imaginary.
2 z2 – z1
- SR and QP are at 90 degrees
z3 – z1 p
arg ——— = ± —
z2 – z1 2
i.e., Provided z3 – z1 = ± ik (z1 – z1), where k is a zon-zero number.
- If z1, z2, z3 are the vertices of a triangle ABC described in anticlockwise
z3 – z1 CA
direction then ——— =—— (cos a + i sin a)
z1 – z1 BA
————————————
—————————————
————————————–
z3 – z1 z3 – z1
or ——— =————- ekz (where angle CAB = a)
z2 – z1 z2 – z2
- Equation of perpendicular bisector of the join of the two points having z1 and
z respectively as affixes, is given by z(z – z ) + z(z – z ) = |z |2 – |z |2
2 1 2 1 2 1 2
- The order relation is not expressed on the set C of all complex numbers as it is not a complete ordered field. Thus the statements z1 > z2 have no meaning unless z1 and z2 both are purely
- Since |z|2 = [Re (z)]2 + [Im (z)]2, therefore Re(z) £ |z|, Im (z) £ |z|.
- For any x, y Î R
(1) Öx + iy + Öx – iy = Ö2{x2 + y2 + x}
(2) Öx + iy – Öx – iy = Ö2{x2 + y2 – x}
- The one and only one case in which |z1|+|z2|+…+|zn| = |z1+z2+…+zn| is that the numbers z1. z2,. zn have the same
- The sum and product of two complex numbers are real if they are conjugate
to each other.
- When xz1 + yz2 + zz3 = 0 where x + y = 0, then the three points are collinear.
- Complex numbers z1z2z3 are the vertices of a triangle A,B,C which is an
isosceles right angled triangle having c as right angle then (z – z )2 = 2 (z – z ) (z – z ).
1 2 1 3 3 2
- The complex numbers z1 z2 z3 be three vertices of an equilateral If z0
is the circumcentre of the triangle then z 2 + z 2 + z 3 = 3z 2
1 2 3 0
- If three complex numbers z1, z2, z3 are in Arithmatical progression then they lie on a straight line in the complex
