for a while I have been interested in the details of the construction  of the integers from the natural numbers. credit to the software, for as soon as I began writing this, for it drew my attention to a very apposite discussion from  on MSE about a year ago. i hope to revisit that discussion, as i am still not familiar with some of the references there, but perhaps this exercise will help me to make progress with more abstract ideas. so my question is - is the following method a reasonable way of deriving the integers from the natural numbers, explaining, by the way, a very good reason why a negative times a negative comes out positive. you can actually calculate it, with the $\star$operation introduced here.  the additive monoid $\mathbb{N}$ is also furnished with a multiplicative structure on $\mathbb{N}^+ = \mathbb{N} \setminus\{0\}$ which it pulls back (via the natural bijection) from its own multiplicative monoid of injective endomorphisms.  $\mathbb{N} \times \mathbb{N}$ is an additive monoid with the obvious addition performed elementwise. we define  $$(a,b) \star (c,d) = (ac+bd, ad+bc) $$ the $\star$-multiplication is commutative, associative, distributive over addition, and has an identity element $(1,0)$.  $\mathbb{N} \times \mathbb{N}$ is the union of three sets $\mathbb{M}^+,\mathbb{M}^0$ and $\mathbb{M}^-$ which intersect only in $\{0\}$. each is a submonoid. an element $(a,b)$ is in $\mathbb{M}^+,\mathbb{M}^0$ or $\mathbb{M}^-$ as $a \gt b, a=b$ or $a&lt;b$.  with respect to the $\star$-multiplication $\mathbb{M}^+$ and $\mathbb{M}^0$ are closed under this operation. $\mathbb{M}^0$, however has an ideal-like property -it is absorbing. if $m_0 \in \mathbb{M}^0$ then for any other $m \in \mathbb{N} \times \mathbb{N}$ we have $m_0m \in \mathbb{M}^0$. the remaining relations may be summarized by  $\mathbb{M}^+\star\mathbb{M}^- \subset \mathbb{M}^-$ and $\mathbb{M}^-\star\mathbb{M}^- \subset \mathbb{M}^+$.  now take two copies of $\mathbb{N}^+$, the second of which we shall call $\mathbb{N}^-$, and define $\mathbb{N}^0=\{0\}$. we define the set  $$\mathbb{Z} = \mathbb{N}^+ \cup \mathbb{N}^0 \cup \mathbb{N}^- $$ define  $$\psi: \mathbb{N} \times \mathbb{N} \rightarrow (\mathbb{N},0) \cup (0,\mathbb{N}) $$in terms of three auxiliary functions, $\chi^+$ and $\chi^-$ the indicator functions for $\mathbb{M}^+$ and $\mathbb{M}^-$, and $\tau: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by $\tau(a,b)=max(a,b)-min(a,b)$. then we may write $$\psi(m)=(\chi^+(m)\tau(m),\chi^-(m)\tau(m)) $$ although $\psi$ is not quite a morphism, its kernel is $\mathbb{M}^0$, and restricted to either $\mathbb{M}^+$ or $\mathbb{M}^-$ is a morphism of monoids. moreover it preserves the multiplicative structure, subject to the sign permutation already mentioned. so it gives the usual multiplicative structure on $\mathbb{Z}$.
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.  I went ahead and gave them a proof by contradiction like so:  Assume $(-x) \cdot (-y) = -xy$  Then divide both sides by $(-x)$ and you get $(-y) = y$  Since we have a contradiction, then our first assumption must be incorrect.  I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote.  Is there a better way to explain this?  Is my proof incorrect?  Also, what would be an intuitive way to explain the negation concept, if there is one?
Do potions use concentration for their effects? how to pass objects between intents Why does one get an illusion as moon following him? When you run or ride bike at night if you observe the moon you feel like he moves along with you as the same speed you are going. Why? biblatex: per-entry language switching According to biblatex manual (p.24), the hyphenation field in a bib-entry switches language on per-entry basis:     hyphenation field    The language of the bibliography entry. The identifier must be a language name   known to the babel package. This information may be used to switch hyphenation   patterns and localize strings in the bibliography.   This functionality seemed to not work. I just used an example of bib base that comes with biblatex documentation to print the entire bibliography.   \documentclass[a4paper]{article} \usepackage[T1]{fontenc} \usepackage[german,american]{babel} \usepackage{csquotes} \usepackage[style=numeric,backend=biber]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} \nocite{*} \printbibliography \end{document}   The bibliography contains few entries in German. However in the printed list of references they are formatted with english strings.     [8] Averroes. Drei Abhandlungen über die Conjunction des separaten Intellects   mit dem Menschen. Von Averroes (Vater und Sohn), aus dem   Arabischen übersetzt von Samuel Ibn Tibbon. Ed. and trans. by J. Hercz.   Berlin: S. Hermann, 1869.   Per-entry language switching is provided by the  bundle of bst style for BibTeX. Looking through biblatex internal code, I've got an impression that biblatex needs major re-implementation in order to provide similar functionality. Or am I wrong? Why are there humans in the Star Wars Universe? Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$. I was just chatting on the Ask Different chat - and the reputation numbers for me and the others chatting don't seem to match any other reputation count I can see on the sites here.  Is there some calculation being made? These numbers are the highest of any other combined reputation I have seen. For me - I'm listed over 10k - but no other aggregate reputation counter I can see is above 8k.  Perhaps I'm unaware how reputation is calculated, but the people there suggested a bug report in case this isn't intended.  (after three lines of text are drawn - a number shows beneath our icons in the left hand side) Ignore Pulses Less than a Given Width in a Pulse Train Let's say I have a 5V pulse train that varies between 0 and 75000 pulses per second. Pulse width, let's say, is 5 microseconds. How could I calculate or determine the best way to filter out all pulses less than 4 microseconds in width? Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? How do the review limits work? I got the brand new Custodian badge today for reviewing posts. Inspired by that, I reviewed some more suggested edits. Now I ran into a limit:     You have no more suggested edit votes today; come back in x hours.   How is that limit calculated? I made 61 reviews today which seems to be some calculated limit.  I searched for an answer on meta but found nothing helpful. Maximal order of an element in a symmetric group If  $$\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$$ then prove that  $$\sin 2A \quad\text{ is a root of }\quad x^2 - 44x - 36 = 0$$  I have no idea how to solve it. Plz help. How can you explain that this double negation is not a double negation? Is there a rule in English about this kind of sentence?  PS / Do I have to mention Pink Floyd Copyright ? :-)  Edit : Since there are a lot of Pink Floyd related explanation, I'll bring a Freddy Mercury one : "I don't have time for no monkey business", which I also understand as "I don't have time for monkey business". Am I right ? Convert NTFS to FAT32 without losing data I understand that you can  without formatting or backing up the data and later restoring - is the same thing possible in the other direction?  I want to convert my NTFS portable hard drive to FAT32 so that my PS3 will recognise it.  If it's not possible from within Windows, are there any free programs which will allow me to do it? In "Tasksel" how do I select a package? How to burn a Windows .iso to a USB device? I want to burn a Windows ISO to a USB device in Ubuntu. How do I do this?   I know how to burn a Ubuntu ISO into a USB device, but with a Windows ISO it's not the same. How to display javascript console on Chrome (or a different browser)? I want to quickly try some Javascript code on Android phone.   I cannot find how to open console or something similar on either Chrome or default browser. about:debug doesn't do anything, and I don't want "remote debugging".   I can download another app if necessary (but I don't want to root my device). Can a textbox be enabled after a certain value from a picklist is selected i want to write a device driver but not able to find the header file can someone please help me find them?Also if someone can point out some important site links that would be really appreciated Dynamic SELECT TOP @var In SQL Server How can I have a dynamic variable setting the amount of rows to return in SQL Server? Below is not valid syntax in SQL Server 2005+:  DECLARE @count int SET @count = 20  SELECT TOP @count * FROM SomeTable