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Aus Online Mathematik Brückenkurs 2

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Version vom 10:45, 11. Mär. 2009

If we treat the expression w=z−iz+i as an unknown, we have the equation

w2=−1.

We know already that this equation has roots

w=−ii

so z should satisfy one of the equation's

z−iz+i=−i or z−iz+i=i.

We solve these equations one by one.

• (z+i)(z−i)=−i:

Multiply both sides by z−i,

z+i=−i(z−i).

Move all the z-terms over to the left-hand side and all the constants to the right-hand side,

z+iz=−1−i.

This gives

z=1+i−1−i=1+i−(1+i)=−1.

• (z+i)(z−i)=i:

Multiply both sides by z−i,

z+i=i(z−i).

Move all the z-terms over to the left-hand side and all the constants to the right-hand side,

z−iz=1−i.

This gives

z=1−i1−i=1.

The solutions are therefore z=−1 and z=1.