Question

help complex numbers.

Answer 1

prove that all the values of \[\left( 1-i \right)^{\sqrt{2}i}\] lie on a straight line

Answer 2

i don't know how to find the values ?@Zarkon help

Answer 3

you can write 1-i as
\[ \sqrt{2} e^{-i \frac{\pi}{4} + i n 2\pi} \]

Answer 4

yes i can express it ,so how do i know that all the values lie on a straight line

Answer 5

when you raise it the the sqrt(2) i power, you will multiply the exponents.
the i*i becomes -1, and the expression will be pure real.
so all values will lie on the x-axis (the real axis)

Answer 6

you will get different values for integer n= ....-2,-1,0,1,2,....

Answer 7

but \[\left( \sqrt{2} \right)^{\sqrt{2}i} \] is not real

Answer 8

yes, good point. I forgot about the sqrt(2) !

but the other part is pure real, so you will get a scaled version of
\[ \left( \sqrt{2} \right)^{\sqrt{2}i} \]
and all the points will lie on a line (But not the real-axis)

Answer 9

ok thanx