# How many solutions does $z + i|z|= 0$ have?

I figured out that it can be transformed in $|z| = iz$ and using the trigonometric method I get: $|z| = |z|(\cos(x) + i\sin(x))(\cos(\pi/2) + i\sin (\pi/2))$ which becomes $|z| = |z|(\sin(x) - i\cos(x))$

I delete $|z|$ from both sides and get $1 = \sin(x) - i\cos(x)$

But don't know how to continue... PS: I am open to other types of solutions, if more elegant. Thank you

-

## 5 Answers

$|z|$ is a positive number. So $iz$ must be an imaginary number with positive imaginary part. That implies $z$ is on the imaginary axis, with a sign opposite that of $i$. That's all that is needed. In other words $z=ci$ is a solution for every $c<0$. You get as many solutions as there are negative numbers $c$.

-

Write $z = x + iy$ with $x, y \in \mathbb R$. Then $x + i(y + \sqrt{x^2 + y^2}) = 0$. Since $x, y \in \mathbb R$, we find $x = 0$ and $y = -\sqrt{x^2 + y^2} = |y|$. These conditions are fulfilled for all $y \le 0$, so the set of solutions is $\{ iy \mid y \le 0\}$.

-

Clearly, $z=-i|z|$ is purely imaginary $=iy$ (say) where $y$ is real.

So, $iy+i|y|=0\implies |y|=-y\implies y\le 0$

-

Infinitely many, $z = -i\alpha$ is a solution for every $\alpha \geq 0$ ($\alpha \in \mathbb{R}$, of course).

-

Let $z=r e^{i \theta}$, this gives the equation $r ( e^{i \theta} + i) = 0$, with $r \geq 0$.

Hence either $r=0$ or $\theta = 2 \pi k -\frac{\pi}{2}$, for some integer $k$.

It follows that the solutions are $z = -r i$, with $r \geq0$.

-
{"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": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "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": 0.9809547662734985, "perplexity": 104.92644768813938}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2015-22/segments/1432207929023.5/warc/CC-MAIN-20150521113209-00161-ip-10-180-206-219.ec2.internal.warc.gz"}