Complex number proof

[TsAmE]

Let z be a non-zero complex number. Prove that \(\displaystyle z + \frac{1}{z}\) is real if and only if either z is real or \(\displaystyle |z| = 1\)

Attempt:

\(\displaystyle z = 1\) (since modulus = 1 and z is real)

\(\displaystyle z + \frac{1}{z}\)

\(\displaystyle = 1 + \frac{1}{1}\)

\(\displaystyle = 2\)

but for some reason this is wrong (as compared to the answer).

 

[Deleted member 4993]

Let z be a non-zero complex number. Prove that \(\displaystyle z + \frac{1}{z}\) is real if and only if either z is real or \(\displaystyle |z| = 1\)

Attempt:

\(\displaystyle z = 1\) (since modulus = 1 and z is real)

\(\displaystyle z + \frac{1}{z}\)

\(\displaystyle = 1 + \frac{1}{1}\)

\(\displaystyle = 2\)

but for some reason this is wrong (as compared to the answer).

assume

z = r*e[sup:1ckgcoun]i?[/sup:1ckgcoun]

then

1/z = 1/r * e[sup:1ckgcoun]-i?[/sup:1ckgcoun]

z + 1/z = r*e[sup:1ckgcoun]i?[/sup:1ckgcoun] + 1/r * e[sup:1ckgcoun]-i?[/sup:1ckgcoun] = (r + 1/r)* cos(?) + i (r - 1/r) * sin(?)

What does the above tell you?

 

[TsAmE]

I cant see how you got from:

z + 1/z
= r*ei? + 1/r * e-i?

to

= (r + 1/r)* cos(?) + i (r - 1/r) * sin(?)

but from z + 1/z = r*ei? + 1/r * e-i? I tried doing something else:

\(\displaystyle z + \frac{1}{z} = re^{i\theta} + \frac{1}{re^{-i\theta}}\frac{re^{i\theta}}{re^{i\theta}}\)

\(\displaystyle =re^{i\theta} + re^{i\theta}\)

\(\displaystyle =2re^{i\theta}\)

\(\displaystyle =2r(cos\theta + isin\theta)\)

not sure what to do next

 

[Deleted member 4993]

I cant see how you got from:

z + 1/z

= r*e[sup:38qrkwu2]i?[/sup:38qrkwu2] + 1/r * e[sup:38qrkwu2]-i?[/sup:38qrkwu2]

to

= (r + 1/r)* cos(?) + i (r - 1/r) * sin(?)

from

e[sup:38qrkwu2]i*?[/sup:38qrkwu2] = cos(?) + i*sin(?)

e[sup:38qrkwu2]-i*?[/sup:38qrkwu2] = cos(?) - i*sin(?)



but from z + 1/z = r*ei? + 1/r * e-i? I tried doing something else:

\(\displaystyle z + \frac{1}{z} = re^{i\theta} + \frac{1}{re^{-i\theta}}\frac{re^{i\theta}}{re^{i\theta}}\) <<< Incorrect - check your algebra properly

\(\displaystyle =re^{i\theta} + re^{i\theta}\)<<< Incorrect - does not follow from step above - check your algebra properly



\(\displaystyle =2re^{i\theta}\)

\(\displaystyle =2r(cos\theta + isin\theta)\)

not sure what to do next