# Positive and Negative Real numbers

I stumbled upon this question on Yahoo answers here :

Prove that for every positive real number x, l/x is also a positive real number.?

Answering to the question is [now] closed, but I was thinking about the answer, when I sketched this argument below, but am not sure if am right, and whether this could be a part of the required proof (or whether this is a mere echo of obvious statements?)

if $p$ is positive real number then this is true:

$p \ge 0 \implies |p| - p = 0$

A similar argument for negative number $n$ would be:

$n \lt 0 \implies |n| - n = 2|n|$

With the above truths, I can then say:

If $x \ge 0$ then ${1 \over x} \ge 0$ since $|{1 \over x} | - {1 \over x} = {0 \over x} = 0$

Is this right?

-
What is $a$???? Do you mean $|n| - n = 2|n|$? And, how do you know $\left|\frac{1}{x}\right| - \frac{1}{x} = 0$ if you don't know that $\frac{1}{x}$ is positive? You're assuming what you want to prove. If you know properties of absolute value, then since you already know $x$ is positive and thus $|x| = x$, we have $\left|\frac{1}{x}\right| = \frac{|1|}{|x|} = \frac{1}{x}$. –  Graphth Oct 16 '12 at 12:43
I guess he meant $|n|-n=2|n|$ for $n<0$. –  coffeemath Oct 16 '12 at 12:47
@Graphth, sorry a was supposed to be n. corrected that (typo) –  nemesisfixx Oct 16 '12 at 13:00

I think perhaps what you are doing could work but there are key details missing. We don't know that $\frac{1}{x}$ is positive, that's what we're trying to prove. So, to say that $\left|\frac{1}{x}\right| - \frac{1}{x} = 0$, which is your definition of $\frac{1}{x}$ being positive, with no other justification, is assuming what you want to prove.
But, if you are allowed to use properties of absolute value, you get $$\left|\frac{1}{x}\right| = \frac{|1|}{|x|} = \frac{1}{x},$$ which proves that little step, since we already know $x$ is positive and thus $|x| = x$.
note that this would hold even for negative $x$. Our task is to prove $1 \over x$ as positive. –  nemesisfixx Oct 16 '12 at 13:21
{"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": 1, "mathjax_asciimath": 1, "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": 0.9511204957962036, "perplexity": 268.9494646784386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2014-41/segments/1410657138980.37/warc/CC-MAIN-20140914011218-00069-ip-10-234-18-248.ec2.internal.warc.gz"}