MHB Effie's question via email about Complex Numbers

AI Thread Summary
The discussion revolves around calculating the argument of the complex number \( z^5 \) where \( z = -2 + 2i \). The polar form of \( z \) is determined to be \( 2\sqrt{2} e^{\frac{3\pi}{4} i} \), leading to \( z^5 = 128\sqrt{2} e^{\frac{15\pi}{4} i} \). The angle \( \frac{15\pi}{4} \) is adjusted to fit within the defined range of \( (-\pi, \pi] \), resulting in \( \text{arg}(z^5) = -\frac{\pi}{4} \). The solution confirms the correct calculation of the argument based on the properties of complex numbers.
Prove It
Gold Member
MHB
Messages
1,434
Reaction score
20
If $\displaystyle \begin{align*} z = -2 + 2\,\mathbf{i} \end{align*}$ what is $\displaystyle \begin{align*} \textrm{arg}\,\left( z^5 \right) \end{align*}$?

First let's write this number in its polar form.

$\displaystyle \begin{align*} \left| z \right| &= \sqrt{\left( -2 \right) ^2 + 2^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\,\sqrt{2} \end{align*}$

and as the number is in Quadrant 2

$\displaystyle \begin{align*} \textrm{arg}\,\left( z \right) &= \pi - \arctan{ \left| \frac{2}{-2} \right| } \\ &= \pi - \arctan{ \left( 1 \right) } \\ &= \pi - \frac{\pi}{4} \\ &= \frac{3\,\pi}{4} \end{align*}$

thus we can say

$\displaystyle \begin{align*} z &= -2 + 2\,\mathrm{i} \\ &= 2\,\sqrt{2}\,\mathrm{e}^{ \frac{3\,\pi}{4}\,\mathrm{i} } \\ z^5 &= \left( 2\,\sqrt{2}\,\mathrm{e}^{\frac{3\,\pi}{4}\,\mathrm{i}} \right) ^5 \\ &= 128\,\sqrt{2}\,\mathrm{e}^{ \frac{15\,\pi}{4}\,\mathrm{i} } \end{align*}$

so that means $\displaystyle \begin{align*} z^5 \end{align*}$ makes an angle of $\displaystyle \begin{align*} \frac{15\,\pi}{4} \end{align*}$ with the positive real axis, but as we define $\displaystyle \begin{align*} \textrm{arg}\,\left( Z \right) \in \left( -\pi , \pi \right] \end{align*}$, that means we keep adding or subtracting integer multiples of $\displaystyle \begin{align*} 2\,\pi \end{align*}$ until we have an angle in this region.

Thus $\displaystyle \begin{align*} \textrm{arg}\,\left( z^5 \right) = -\frac{\pi}{4} \end{align*}$.
 
Mathematics news on Phys.org
Prove It said:
First let's write this number in its polar form.

$\displaystyle \begin{align*} \left| z \right| &= \sqrt{\left( -2 \right) ^2 + 2^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\,\sqrt{2} \end{align*}$

and as the number is in Quadrant 2

$\displaystyle \begin{align*} \textrm{arg}\,\left( z \right) &= \pi - \arctan{ \left| \frac{2}{-2} \right| } \\ &= \pi - \arctan{ \left( 1 \right) } \\ &= \pi - \frac{\pi}{4} \\ &= \frac{3\,\pi}{4} \end{align*}$

thus we can say

$\displaystyle \begin{align*} z &= -2 + 2\,\mathrm{i} \\ &= 2\,\sqrt{2}\,\mathrm{e}^{ \frac{3\,\pi}{4}\,\mathrm{i} } \\ z^5 &= \left( 2\,\sqrt{2}\,\mathrm{e}^{\frac{3\,\pi}{4}\,\mathrm{i}} \right) ^5 \\ &= 128\,\sqrt{2}\,\mathrm{e}^{ \frac{15\,\pi}{4}\,\mathrm{i} } \end{align*}$

so that means $\displaystyle \begin{align*} z^5 \end{align*}$ makes an angle of $\displaystyle \begin{align*} \frac{15\,\pi}{4} \end{align*}$ with the positive real axis, but as we define $\displaystyle \begin{align*} \textrm{arg}\,\left( Z \right) \in \left( -\pi , \pi \right] \end{align*}$, that means we keep adding or subtracting integer multiples of $\displaystyle \begin{align*} 2\,\pi \end{align*}$ until we have an angle in this region.

Thus $\displaystyle \begin{align*} \textrm{arg}\,\left( z^5 \right) = -\frac{\pi}{4} \end{align*}$.
This problem is solved correctly.
 
Well, we can see that ##\arg (z) = \frac{3\pi}{4}## by considering its position in the plane. Then ##arg (z^5) = \frac{15\pi}{4} = -\frac {\pi}{4}##.
 
  • Like
Likes Greg Bernhardt
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
1
Views
6K
Replies
1
Views
10K
Replies
2
Views
10K
Replies
4
Views
11K
Replies
1
Views
11K
Back
Top