What is the square root of the imaginary number i?

Let's suppose that (0+1i)^0.5 = x+iy, where x and y are both real numbers. This is the most general form of a complex number. Now let's square both sides. This gives 0+1i = x²+2i xy - y². We will equate real and imaginary parts of this equation to solve for the real numbers x and y. Equating real parts gives x² = y² (Equation 1). Equating imaginary parts gives xy=0.5 (Equation 2). Now we will substitute x = 0.5/y into equation 1. This gives us (0.5/y)² = y². Multiplying through by y² results in the following solution for y: y⁴ = 0.5², so that y = 0.5^1/2. Then x = 0.5/0.5^1/2 which gives us that x = y = 0.5^1/2. So the square root of i is the complex number 0.5^1/2+0.5^1/2i.