If z is a complex number,then which of the fo | vedclass

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(C) For any complex number $z = x + iy$,we have $|z| = \sqrt{x^2 + y^2}$.$(a)$ $|z^2| = |(x + iy)^2| = |x^2 - y^2 + 2ixy| = \sqrt{(x^2 - y^2)^2 + (2xy

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$z = \bar{z}$

Vedclass · Answer

(C) For any complex number $z = x + iy$,we have $|z| = \sqrt{x^2 + y^2}$.$(a)$ $|z^2| = |(x + iy)^2| = |x^2 - y^2 + 2ixy| = \sqrt{(x^2 - y^2)^2 + (2xy)^2} = \sqrt{x^4 + y^4 - 2x^2y^2 + 4x^2y^2} = \sqrt{(x^2 + y^2)^2} = x^2 + y^2 = |z|^2$. This is true.$(b)$ Since $|z| = |\bar{z}|$,it follows that $|z|^2 = |\bar{z}|^2$. Thus,$|z^2| = |\bar{z}|^2$ is true.$(c)$ $z = \bar{z}$ implies $x + iy = x - iy$,which means $2iy = 0$,or $y = 0$. This is only true if $z$ is purely real. It is not true for all complex numbers $z$.$(d)$ $\bar{z^2} = (\bar{z})^2$ is a standard property of complex conjugates. This is true.Therefore,the statement that is not true is $z = \bar{z}$.

If $z$ is a complex number,then which of the following is not true?

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