Show that z = x + iy is pure imaginary if and only if $\displaystyle \overline z$ = -z

z = x +iy

- z = -x - iy

$\displaystyle \overline z$ = x - iy

x - iy = -x - iy

x = -x

Umm...yeah I don't know how this works. Can someone help?

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- Oct 12th 2009, 07:01 PMFallen186Complex Numbers
Show that z = x + iy is pure imaginary if and only if $\displaystyle \overline z$ = -z

z = x +iy

- z = -x - iy

$\displaystyle \overline z$ = x - iy

x - iy = -x - iy

x = -x

Umm...yeah I don't know how this works. Can someone help? - Oct 12th 2009, 07:14 PMGusbob
Sorry my mistake before editing.

From where you left off...

2x = 0.

Therefore x = 0

So the value of x (the real part) is 0 for z. So z must be imaginary if -z = z's conjugate.