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if [tex]e^{\pi\imath}=-1[/tex] then:

[tex]-e^{\pi\imath}=1[/tex] and,

[tex]e^{2\pi\imath}=1[/tex]

then:

[tex]-e^{\pi\imath}=e^{2\pi\imath}[/tex]

[tex]\rightarrow e^{2\pi\imath}+e^{\pi\imath}=0[/tex]

[tex]\rightarrow (e^{\pi\imath})^2+e^{\pi\imath}=0[/tex]

[tex]\rightarrow (e^{\pi\imath}+1)e^{\pi\imath}=0[/tex]

then:

[tex]e^{\pi\imath}=0[/tex]

and

[tex]e^{\pi\imath}+1=0[/tex]

Can somebody explain this contradiction to me?

[tex]-e^{\pi\imath}=1[/tex] and,

[tex]e^{2\pi\imath}=1[/tex]

then:

[tex]-e^{\pi\imath}=e^{2\pi\imath}[/tex]

[tex]\rightarrow e^{2\pi\imath}+e^{\pi\imath}=0[/tex]

[tex]\rightarrow (e^{\pi\imath})^2+e^{\pi\imath}=0[/tex]

[tex]\rightarrow (e^{\pi\imath}+1)e^{\pi\imath}=0[/tex]

then:

[tex]e^{\pi\imath}=0[/tex]

and

[tex]e^{\pi\imath}+1=0[/tex]

Can somebody explain this contradiction to me?

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