z-x-yi-C-z-x-y-C-proof-z-2-z-2-zz-so-z-0-1-z-z-z-2-

Question Number 33095 by 7991 last updated on 10/Apr/18

z = x + y i \in C

\bar{z} = x - y \in C

p r o o f ∣ z ∣^{2} =∣ z^{2} ∣= z \bar{z}, s o z \neq 0 \to \frac{1}{z} = \frac{\bar{z}}{∣ z ∣^{2}}

Answered by Rasheed.Sindhi last updated on 10/Apr/18

z = x + y i \in C, \overset{―}{z} = x - y i

∣ z ∣=∣ x + y i ∣= \sqrt{x^{2} + y^{2}}

∴ ∣ z ∣^{2} = x^{2} + y^{2} \dots \dots \dots \dots . A

∣ z^{2} ∣=∣ {(x + y i)}^{2} ∣=∣ x^{2} - y^{2} + 2 xyi ∣

= \sqrt{{(x^{2} - y^{2})}^{2} + {(2 xy)}^{2}}

= \sqrt{x^{4} + y^{4} - {2 x}^{2} y^{2} + {4 x}^{2} y^{2}}

= \sqrt{x^{4} + y^{4} + {2 x}^{2} y^{2}}

= \sqrt{{(x^{2} + y^{2})}^{2}}

= x^{2} + y^{2} \dots \dots \dots \dots \dots B

z . \overset{―}{z} = (x + y i) (x - y i) = {(x)}^{2} - {(yi)}^{2}

= x^{2} + y^{2} \dots \dots \dots \dots \dots . . C

From A, B & C

∣ z ∣^{2} =∣ z^{2} ∣= z . \overset{―}{z}

z . \overset{―}{z} =∣ z^{2} ∣\Rightarrow z = \frac{∣ z^{2} ∣}{\bar{z}} \Rightarrow \frac{1}{z} = \frac{\bar{z}}{∣ z^{2} ∣}