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 i2 = −1. This means that i is a solution of the equation x2 + 1 = 0.
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.
Given any two complex numbers z1 and z2, the difference z1 – z2 is defined as follows: z1 – z2 = z1 + (– z2).
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).
Given any two complex numbers z1 and z2, where z2 ≠ 0 , the quotient z1/z2 is defined by
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 √a2+b2, i.e., | z | = √a2 + b2 and the conjugate of z, denoted as z` is the complex number a – ib, i.e., z` = a – ib.
Polar representation the nonzero complex number z = x + iy -- z = r ( cosθ + i sinθ )
Roots of equation ax2+bx+c
Find the multiplicative inverse of 2 – 3i.
z = 2 – 3i
z` = 2 + 3i
|z|2= (22+ (-3)2) = 13
z-1 = z`/|z|2=(2+3i)/13 = (2/13)+(3/13)i
Represent the complex number z =1+ i √3 in the polar form.
1 = r cos θ, √3 = r sin θ
r2 ( cos2 θ + sin2 θ ) = 4
r = 2
cos θ = (1/2), sin θ = (√3/2)
z = 2(cos π/3+i sin π/3)
Solve x2 + 2 = 0
x2 + 2 = 0
x2 = – 2 x
x2 = – 2 x = ± √−2 = ±√ 2 i