Complex Numbers

For the complex number z = a + ib, a is called the real part, denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z.

Let us denote √−1 by the symbol i. Then, we have i^{2} = −1. This means that i is a solution of the equation x^{2} + 1 = 0.

Let z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the sum z_{1} + z_{2} is defined as follows: z_{1} + z_{2} = (a + c) + i (b + d), which is again a complex number.

Given any two complex numbers z_{1} and z_{2}, the difference z_{1} – z_{2} is defined as follows: z_{1} – z_{2} = z_{1} + (– z_{2}).

Let z_{1} = a + ib and z_{2} = c + id be any two complex numbers. Then, the product z_{1} z_{2} is defined as follows: z_{1} z_{2} = (ac – bd) + i(ad + bc).

Given any two complex numbers z_{1} and z_{2}, where z_{2} ≠ 0 , the quotient z_{1}/z_{2} is defined by

Let z = a + ib be a complex number. Then, the modulus of z, denoted by  z , is defined to be the nonnegative real number √a^{2}+b^{2}, i.e.,  z  = √a^{2} + b^{2} and the conjugate of z, denoted as z` is the complex number a – ib, i.e., z` = a – ib.
z z`=z^{2}

Polar representation the nonzero complex number z = x + iy  z = r ( cosθ + i sinθ )
Where r=√x^{2}+y^{2}
Quadratic Equations
Roots of equation ax^{2}+bx+c
Or
Examples
Question
Find the multiplicative inverse of 2 – 3i.
Solution
z = 2 – 3i
z` = 2 + 3i
z^{2}= (2^{2}+ (3)^{2}) = 13
z^{1 }= z`/z^{2}=(2+3i)/13 = (2/13)+(3/13)i
Question
Represent the complex number z =1+ i √3 in the polar form.
Solution
1 = r cos θ, √3 = r sin θ
r^{2} ( cos^{2} θ + sin^{2} θ ) = 4
r = 2
cos θ = (1/2), sin θ = (√3/2)
θ=(π/3)
z = 2(cos π/3+i sin π/3)
Question
Solve x^{2} + 2 = 0
Solution
x^{2} + 2 = 0
or
x^{2} = – 2 x
or
x^{2} = – 2 x = ± √−2 = ±√ 2 i