How to calculate the square root of a complex number?

To calculate the square root of a complex number, use the formula √(a+bi) = ±(√[(a+√(a^2+b^2))/2] + i(√[(√(a^2+b^2)-a)/2])).

To understand how to calculate the square root of a complex number, it is important to first understand what a complex number is. A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1.

To find the square root of a complex number, we can use the formula √(a+bi) = ±(√[(a+√(a^2+b^2))/2] + i(√[(√(a^2+b^2)-a)/2])). This formula involves finding the square root of the real part of the complex number plus the square root of the magnitude of the complex number, divided by 2. We then add to this the square root of the magnitude of the complex number minus the real part of the complex number, divided by 2, multiplied by i.

Let's take an example to illustrate this formula. Suppose we want to find the square root of the complex number 3+4i. Using the formula, we have:

√(3+4i) = ±(√[(3+√(3^2+4^2))/2] + i(√[(√(3^2+4^2)-3)/2]))
= ±(√[(3+5)/2] + i(√[(5-3)/2]))
= ±(√4 + i√1)
= ±(2+i)

Therefore, the square roots of 3+4i are 2+i and -2-i.