Algebra of Complex Numbers

•  Addition for 2 complex number z1 = a1 + ib1 and z2 = a2 + ib2, their sum is defined as z = z1 + z2 = (a1 + a2) + i(b1 + b2)
•  Subtraction for two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2, the subtraction of z2 from z1 is defined as z1 – z2 = (a1 − a2) + i(b1 − b2)
•  Multiplication of two complex numbers z1 = a + ib and z2 = c + id is defined as z1.z2 = (ac − bd) + i(ad + bc)
•  Division of complex numbers z1 = a + ib and z2 = c + id where z2 ≠ 0 then \frac{z_{1}}{z_{2}} = \frac{a + ib}{c + id}
•  Multiplicative inverse of non-zero complex number z = a + ib is defined as \tt z^{-1} = \frac{1}{z} = \frac{1}{a + ib}.

View the Topic in this video From 19:58 To 52:48

Disclaimer: Compete.etutor.co may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. Compete.etutor.co makes no representations whatsoever concerning the content of these sites and the fact that compete.etutor.co has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by compete.etutor.co with respect to such sites, its services, the products displayed, its owners, or its providers.

1. Addition of Complex Numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex number, then their sum defined as
z1+z2=(x1+iy1)+(x2+iy2=(x1+x2)+i(y1+y2)

Properties of Addition
a) Commutative z1 + z2 = z2 + z1
b) Associative (z1 + z2) + z3 = z1 + (z2 + z3)
c) Additive Identity z + 0 = z = 0 + z
Here, 0 is additive identity.

Subtraction of Complex Numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their difference is defined as
z− z2 = (x1 − x2) + i (y1 − y2)

Multiplication of Complex Numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their multiplication is defined as
z1 z2 = (x1 + iy1) (x2 + iy2) = (x1x2 − y1 y2) + i (x1y2 + x2 y1)

Properties of Multiplication
a) Commutative z1 z2 = z2 z1
b) Associative (z1 z2) z3 = z1 (z2 z3)
c) Multiplicative Identity z.1 = z = 1.z
Here, 1 is multiplicative identity of an element z.
d) Multiplicative Inverse Every non-zero complex number z there exists a complex number z1 such that z.z1 = 1 = z1 . z
e) Distributive Law
i) z1 (z2 + z3) = z1z2 + z1 z3
ii) (z2 + z3) z1 = z2z1 + z3 z1

Division of Complex Numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their division is defined as
\tt \frac{z_{1}}{z_{2}}=\frac{\left(x_{1}x_{2}+y_{1}y_{2}\right)+i\left(x_{2}y_{1}-x_{1}y_{2}\right)}{x_2^2+y_2^2}(where z2 ≠ 0).

2. Integral Power of lota (i)
\tt i=\sqrt{-1},i^{2}=-1,i^{3}=-i,i^{4}=1
So, \tt i^{4n+1}=i,i^{4n+2}=-1,i^{4n+3}=-i,i^{4n+4}=i^{4n}=1
In other words, \tt \begin{cases}i^{n}=\left(-1\right)^{n/2},if \ n \ is \ an \ even \ integer & \\ \tt i^{n}=\left(-1\right)^{\frac{n-1}{2}}.i,if \ n \ is \ an \ odd \ integer & \end{cases}