Complex Numbers

Express the following in the form a + bi, where a and b are real numbers:      sqrt(i)

$\sqrt i=a+bi\\ i=(a+bi)^2\\ i=a^2+2abi-b^2\\ i=(a^2-b^2)+2abi\\ so\\ a^2-b^2=0\;\; and\;\ 2ab=1\\ (a-b)(a+b)=0\\ a=\pm b\\ but \;\;since\;\; 2ab=1, \;\;a=b\\ 2a^2=1\\ a=\pm\frac{1}{\sqrt{2}}\\ a=\pm\frac{\sqrt 2}{2}\\ so\\ \sqrt{i}=\pm(\frac{\sqrt 2}{2}+\frac{\sqrt 2i}{2})$