complex number question

• October 15th 2009, 03:45 PM
oblixps
complex number question
if z = a + bi and|z| = 1, let w = (z+1)/(z-1). prove that w is a pure imaginary number of the form bi.

so from the given i have a^2 + b^2 = 1 since sqrt(a^2 + b^2) = 1 = |z|.

so i plugged in a + bi for z into w = ((a + bi) + 1) / ((a + bi) - 1) i multiplied the top and bottom by the conjugate of the bottom witch is ((a + bi) + 1) and i got something complicated: (2a^2 + 2abi + 2a + 2bi) / (2abi - 2b^2) when i split this up into separate fractions i don't get a pure imaginary number. how do i do this?
• October 15th 2009, 04:11 PM
Plato
Quote:

Originally Posted by oblixps
if z = a + bi and|z| = 1, let w = (z+1)/(z-1). prove that w is a pure imaginary number of the form bi.?

You should know that $\left| z \right| = 1\; \Rightarrow \;z\overline z = 1$ and that $\left| {z - 1} \right|^2$ is a real number.
Also that $\overline z - z = - 2\text{Im} (z)$.

$\frac{{z + 1}}{{z - 1}} = \frac{{\left( {z + 1} \right)\left( {\overline z - 1} \right)}}{{\left| {z - 1} \right|^2 }} = \frac{{z\overline z + \left( {\overline z - z}\right) - 1}}{{\left| {z - 1} \right|^2 }}$

Now put it all together.
• October 15th 2009, 10:10 PM
oblixps
Quote:

Originally Posted by Plato
You should know that $\left| z \right| = 1\; \Rightarrow \;z\overline z = 1$ and that $\left| {z - 1} \right|^2$ is a real number.
Also that $\overline z - z = - 2\text{Im} (z)$.

$\frac{{z + 1}}{{z - 1}} = \frac{{\left( {z + 1} \right)\left( {\overline z - 1} \right)}}{{\left| {z - 1} \right|^2 }} = \frac{{z\overline z + \left( {\overline z - z}\right) - 1}}{{\left| {z - 1} \right|^2 }}$

Now put it all together.

what does the notation $\overline z$ and $\ - 2\text{Im} (z)$ mean? i haven't encountered those before in my class.
• October 16th 2009, 02:08 AM
red_dog
We have, as Plato said, $z\cdot\overline{z}=1\Rightarrow \overline{z}=\frac{1}{z}$

A complex number w is pure imaginary if and only if $\overline{w}=-w$.

$\overline{w}=\overline{\left(\frac{z+1}{z-1}\right)}=\frac{\overline{z+1}}{\overline{z-1}}=\frac{\overline{z}+1}{\overline{z}-1}=$

$=\frac{\displaystyle\frac{1}{z}+1}{\displaystyle\f rac{1}{z}-1}=\frac{1+z}{1-z}=-\frac{z+1}{z-1}=-w$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8808525800704956, "perplexity": 765.0552245508004}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765002.8/warc/CC-MAIN-20141217075245-00097-ip-10-231-17-201.ec2.internal.warc.gz"}