Melody

Suppose z  is a complex number such that \(z^3 = 100+75i\) . Find |z|

\(|z^3| = \sqrt{100^2+75^2}\\ |z^3| = \sqrt{100^2+75^2}\\ |z^3| = 125\\ z^3=125[\frac{100+75i}{125}]\\ z^3=125[\ 0.8+0.5i]\\ \)

Let

\(z=re^{i\theta}\\ z^3=r^3e^{3i\theta}\\ z^3=r^3(cos(3\theta) +isin(3\theta))\\ \)
\(r^3(cos(3\theta) +isin(3\theta))=125[\  0.8+0.6i]\\ r^3=125\\ r=5\\ cos(3\theta)=0.8\\ sin(3\theta)=0.6\\ 3\theta \approx 0.6435\\ \theta \approx 0.2145\;radians\\ \)

anyway, I digress,

\(|z|=5\)

\(i=\sqrt{-1}\)

I did not give you the minus but without any explanation your answer has absolutely no value.

Someone else was expressing that when they gave you a minus point.

This question has ALREADY been asked today.

So you do not need to ask again!

Looks good to me Tertre

Thea has a key on her calculator marked . If an integer is displayed, pressing the  key chops off the first digit and moves it to the end. For example, if 6138 is on the screen, then pressing the  key changes the display to 138.

Wouldn't   6138  become  1386 ?    You said the first digit gets moved to the end ....

Thea enters a positive integer into her calculator, then squares it, then presses the  key, then squares the result, then presses the  key again. After all these steps, the calculator displays 243. What number did Thea originally enter?

Thanks guest :)

Thanks Chris,

This looks quite interesting. Pity there is not a lot more hours in a day :)

There is so many interesting questions and answer and so much I can learn on this site.

Here is CPhill's graph.

Hi somebody,

If any of my complex number questions prove to be wrong (or you have reason to question my answers) can you please let me know. :)

Find all complex numbers  z such that

|z - 3| = |z+i| = |z -3i|

\(let\;z=a+bi\\ |(a-3)+bi|=|a+(b+1)i|=|a+(b-3)i|\\ (a-3)^2+b^2=a^2+(b+1)^2=a^2+(b-3)^2\\ a^2-6a+9+b^2=a^2+b^2+2b+1=a^2+b^2-6b+9\\ -6a+9=2b+1=-6b+9\\~\\ 2b+1=-6b+9\\\\ 8b=8\\ b=1\\~\\ -6a+9=3\\ -6a=-6\\ a=1 \)

So I get     

\(z=1+i\)