Square root of the complex number z = a + bi, is called the complex number w = x + yi for which it is valid that or .
Find the square root of z=5-12i.
The square root of
z is the complex number x + yi for which
So, or 2xy = 12 (1) or xy = 6 (2)
which is impossible or
If x=3 the equation (2) gives y = 2
If x=-3 the equation (2) gives y = -2
Therefore the number z = 5 - 12i has two
square roots: 3 + 2i, -3 - 2i
If z = a + bi a complex number and M(a,b)
the corresponding element on the Complex plane.
If and then and .
Trigonometric form of z
The angle is called argument of z and is denoted arg(z).
Now, if , then easily and not using much trigonometry, it can be proved that:
Also the following can be proved:
De Moivre's Theorem
If n integer and a complex number, then