document.write( "Question 857211: Please help me solve this problem:
\n" ); document.write( "Find all complex square roots of i. in other words find all complex solutions of x^2 = i \n" ); document.write( "

Algebra.Com's Answer #516468 by josgarithmetic(39618) $\"\"$ $\"About$

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This may be only part of the answer, but $\"x%5E2=i\"$ if $\"x=cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "How?
\n" ); document.write( "Draw a unit circle. The horizontal axis is Real and the vertical axis is Imaginary. Horizontal intercepts are 1 and -1. Vertical intercepts are i and -i. Think of multiplications by whole number powers of i to be rotations starting at (1,0). If 1*1, get 1. If 1*-1, get -1. If 1*i, get i. If 1*(i)(i), get same as 1*(-1) which is -1. Thinking this way, the way to go from negative 1 to HALF WAY from negative 1 to positive 1 is the take square root of -1. \r
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\n" ); document.write( "\n" ); document.write( "In that same way, if you start with $\"x%5E2=i\"$ , and you want what is x, this is like starting at (0,i) on this complex unit circle, and rotating half-way from (0,i) to (1,0). This puts your point on an angle of positive $\"pi%2F4\"$ . The coordinates on this point are as (cos(pi/4),i*sin(pi/4)), which you can represent as $\"highlight%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29\"$ . \n" ); document.write( "

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