Complex Numbers and Quadratic Equations

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

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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
   z1 − 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}