document.write( "Question 389436: HOW TO DETERMINE i2=−1 FROM THE EULER`S EQUATION USING LOGARITHM/WITHOUT LOGARITHM? PLEASE TELL ME IN DETAILS. \n" ); document.write( "
Algebra.Com's Answer #275965 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"i%5E2+=+-1\" by definition of i\r\n" );
document.write( "\r\n" );
document.write( "By Euler's equation:\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute p\r\n" );
document.write( " for q\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=+cos%28pi%29+%2B+i%2Asin%28pi%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=++-1+%2B+i%2A0\"\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=+-1\"\r\n" );
document.write( "\r\n" );
document.write( "Go back to Euler's equation.\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Atheta%29+=+cos%28theta%29+%2B+i%2Asin%28theta%29\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"pi%2F2\"\r\n" );
document.write( " for q\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Aexpr%28pi%2F2%29%29+=+cos%28pi%2F2%29+%2B+i%2Asin%28pi%2F2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%2F2%29+=++0+%2B+i%2A1\"\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%2F2%29+=+i\"\r\n" );
document.write( "\r\n" );
document.write( "Square both sides:\r\n" );
document.write( "\r\n" );
document.write( "\"%28e%5E%28i%2Api%2F2%29%29%5E2+=+i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28e%5E%28i%2Api%2Fcross%282%29%29%29%5Ecross%282%29+=+i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=+i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( " From the first part we have\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=+-1\"\r\n" );
document.write( "\r\n" );
document.write( "and from the last part we have\r\n" );
document.write( "\r\n" );
document.write( "\"e%5E%28i%2Api%29+=+i%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "So \"i%5E2=-1\" because things equal to the \r\n" );
document.write( "\r\n" );
document.write( "same thing are equal to each other\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );