Question

Complex Number help

Answer 1

now suppose we had a polynomial of degree n
\[P(z)=a _{n}Z ^{n}+a _{n-1}Z ^{n-1}+...+a _{2}Z ^{2}+a _{1} Z ^{1}+a _{0}\] whose coefficients are an, an-1,... a2, a1 and a0 are all real numbers, then how is P(Z*) is the sum of n+1 terms of the form \[(a _{r}Z ^{r})^*\].

Answer 2

\[(z_1+z_2)^* = z_1^* +z_2^* \\~\\~\\(z^{r})^* = (z^*)^r\]

Answer 3

actually you just need the second property
either use it or prove it

Answer 4

@ganeshie8 why is it the sum of n+1 terms shouldn't the degree of the polynomial degree decrease if its in descending order.

Answer 5

degree of terms will decrease and it will be similar to P(Z)

Answer 6

r is just an index variable, r = 0,1,2,...

Answer 7

\[\begin{align}P(Z) &=a _{n}Z ^{n}+a _{n-1}Z ^{n-1}+...+a _{2}Z ^{2}+a _{1} Z ^{1}+a _{0}\\~\\
P(Z^*) &= ?

\end{align}\]

Answer 8

\[a _{n}(Z ^{*})^{n}+a _{n-1}(Z^*)^{n-1}\]

Answer 9

and so on right?

Answer 10

Yep! you good wid SUM notation ?

Answer 11

\[\begin{align}P(Z) &=a _{n}Z ^{n}+a _{n-1}Z ^{n-1}+...+a _{2}Z ^{2}+a _{1} Z ^{1}+a _{0}\\~\\
P(Z^*) &= a _{n}(Z^*) ^{n}+a _{n-1}(Z^*) ^{n-1}+...+a _{2}(Z^*) ^{2}+a _{1} (Z^*) ^{1}+a _{0}\\~\\

\end{align}\]

Answer 12

using sum notation we can express them as :
\[\begin{align}P(Z) &=\sum\limits_{\color{red}{r}=0}^na _{\color{red}{r}}Z^{\color{red}{r}}\\~\\
P(Z^*) &=\sum\limits_{\color{red}{r}=0}^na _{\color{red}{r}}(Z^*)^{\color{red}{r}}\\~\\

\end{align}\]


next use the second property that i listed earlier

Answer 13

Thanks @ganeshie8 :)

Answer 14

In an argand diagram, points S and T represent 4 and 2i respectively see fig 16.12 identify the points P such that PS<PT.

Answer 15

let P represent a complex number z=x+yi. the length PS and PT are given by\[PS=\left| z-4 \right|\]\[PT=\left| z-2i \right|\]

Answer 16

\[\left| z-4 \right|<\left| z-2i \right|\]\[\left| z-4 \right|^2<\left| z-2i \right|^2\]\[(z-4)(z^*-4*)<(z-2i)(z^*-(2i)^*)\]\[(z-4)(z^*-4)<(z-2i)(z^*+2i)\]\[zz^*-4z-4z^*+16<zz^*+2iz-2iz^*+4\]\[4(z+z^*)+2i(z-z^*)>12\]

Answer 17

\[z+z^*=x+x+yi-yi=2x\]
\[z-z^*=2yi\]
\[8x-4y>12\rightarrow 2x-y>3\]