Complex numbers notes

COMPLEX NUMBERS
CONCEPTS AND RESULTS
* A number of the form (a + ib) where a b  R, the set of real numbers, and i =  1 (iota) is called a
complex number. It is denoted by z, z = a + ib. “a” is called the real part of complex number z and
“b”
is the imaginary part i.e. Re(z) = a and Im(z) = b.
* Two complex numbers are said to be equal i.e. z1 = z2.
 (a + ib) = (c + id)
 a = c and b = d
 Re (z1) = Re (z2) & Im(z1) = Im(z2).
* A complex number z is said to be purely real if Im(z) = 0
and is said to be purely imaginary if Re(z) = 0.
* The set R of real numbers is a proper subset of the set of complex number C, because every real
number
can be considered as a complex number with imaginary part zero.
* i4n = (i4)n = (1)n = 1 i4n+1 = i4n.i = (1).i = i
i4n+2 = i4n. i2 = (1) (–1) = –1 i4n+3 = i4n.i3 = (1)( –i) = – i.
Algebra of Complex Numbers
** Addition of two complex numbers : Let z1 = a + ib and z2 = c + id be any two complex numbers.
Then, the sum z1 + z2 is defined as follows: z1 + z2 = (a + c) + i (b + d), which is again a complex
number.
The addition of complex numbers satisfy the following properties:
(i) The closure law The sum of two complex numbers is a complex number, i.e., z1 + z2 is a complex
number for all complex numbers z1 and z2 .
(ii) The commutative law For any two complex numbers z1 and z2, z1 + z2 = z2 + z1
(iii) The associative law For any three complex numbers z1, z2, z3, (z1 + z2) + z3 = z1 + (z2 + z3).
(iv) The existence of additive identity There exists the complex number 0 + i 0 (denoted as 0), called
the additive identity or the zero complex number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse To every complex number z = a + ib, we have the complex
number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. Thus z + (–z) = 0
(the additive identity).
** Difference of two complex numbers : Given any two complex numbers z1 and z2, the difference
z1 – z2 is defined as follows: z1 – z2 = z1 + (– z2).
** Multiplication of two complex numbers : Let z1 = a + ib and z2 = c + id be any two complex
numbers.
Then, the product z1 z2 is defined as follows: z1 z2 = (ac – bd) + i(ad + bc)
**The multiplication of complex numbers possesses the following properties :
(i) The closure law The product of two complex numbers is a complex number, the product z1 z2 is a
complex number for all complex numbers z1 and z2.
(ii) The commutative law For any two complex numbers z1 and z2, z1 z2 = z2 z1
(iii) The associative law For any three complex numbers z1, z2, z3, (z1 z2) z3 = z1 (z2 z3).
(iv) The existence of multiplicative identity There exists the complex number 1 + i 0 (denoted as 1),
called the multiplicative identity such that z.1 = z, for every complex number z.
(v) The existence of multiplicative inverse For every non-zero complex number z = a + ib or a + bi
a b 1
(a ≠ 0, b ≠ 0), we have the complex number 2 i 2 (denoted by or z 1 ), called the
a b 2
a b 2
z
1
multiplicative inverse of z such that z.  1 (the multiplicative identity).
z
(vi) The distributive law For any three complex numbers z1, z2, z3,
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, (a) z1 (z2 + z3) = z1 z2 + z1 z3 (b) (z1 + z2) z3 = z1 z3 + z2 z3
**Division of two complex numbers : Given any two complex numbers z1 and z2, where z2 ≠ 0 , the
z1 z 1
quotient is defined by 1  z1. .
z2 z2 z2
**Modulus a Complex Number : Let z = a + ib be a complex number. Then, the modulus of z,
denoted by | z |, is defined to be the non-negative real number a 2  b2 , i.e., | z | = a 2  b2
** Properties of Modulus :
If z, z1, z2 are three complex numbers then
(i) |z| = 0  z = 0 i.e., real part and imaginary part are zeroes.
(ii) |z| = | z | = |– z |
(iii) z. z = | z |2
(iv) |z1.z2| = |z1|.|z2|
z1 | z1 |
(v)  , z 0
z2 | z2 | 2
(vi) |z1+z2|2 = |z1|2 + |z2|2+ 2Re(z1 z2 )
(vii) |z1-z2|2 = |z1|2 + |z2|2 – 2Re(z1 z2 )
(viii) |z1+z2|2 + |z1-z2|2 = 2(|z1|2 + |z2|2)
**Conjugate of a Complex Number : Let z = a + ib then its conjugate is denoted by z  (a  ib) .

**Properties of conjugates :
(i) (z) = z
(ii) z+ z = 2Re(z)
(iii) z– z = 2iIm(z)
(iv) z+ z = 0  z is purely real.
(v) z. z = [Re(z)]2 + [Im(z)]2.
(vi) z1  z2  z1  z2
(vii) z1.z2  z1.z2
 z1  z1
(viii)    , z2  0
 z2  z2

**Argand Plane and Polar Representation
Some complex numbers such as 2 + 4i, – 2 + 3i, 0 + 1i, 2 + 0i, – 5 –2i
and 1 – 2i which correspond to the ordered pairs (2, 4), ( – 2, 3), (0, 1),
(2, 0), ( –5, –2), and (1, – 2), respectively, have been represented
geometrically by the points A, B, C, D, E, and F, respectively.
The plane having a complex number assigned to each of its
point is called the complex plane or the Argand plane.

In the Argand plane, the modulus of the complex number
x + iy = x 2  y 2 is the distance between the point P(x, y) to
the origin O (0, 0). The points on the x-axis corresponds to the
complex numbers of the form a + i 0 and the points on the y-axis
corresponds to the complex numbers of the form 0 + i b.
The x-axis and y-axis in the Argand plane are called, respectively,
the real axis and the imaginary axis.
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