By Dr. Shahid Ali Yousafzai

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As the Euler's formula is $$ { e }^{ ix }=\cos { x } +i\sin { x } $$ Now if I replace \(x\) here by \(i\pi \) $$ { e }^{ i\pi }=-1 $$ $${ e }^{ i\pi }+1=0$$ This is the Euler's identity. It can be also written as $$ { e }^{ i\pi }=\cos { \left( 1 \right) +i\sin { \left( 1 \right) } } =0.54+0.84i $$ Here the 1 is in radians, not degrees. We can plot the complex numbers as well but you have to keep in mind that in a unit circle we have \(r = 1\), then $$ x=\cos { \theta } $$ $$ y=\sin { \theta } $$ By squaring we get $$ \cos ^{ 2 }{ \theta } +\sin ^{ 2 }{ \theta } =1 $$ Example: \(3+4i\) $$ { r }=\sqrt { { 3 }^{ 2 }+{ 4 }^{ 2 } } =5 $$ $$ \theta =\tan ^{ -1 }{ \frac { 4 }{ 3 } } =0.927 $$

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