Consider the following numpy vector of number: a = np.array([.1, .2, .2, .2, .2, .1])  Obviously, the sum of these numbers gives 1. However, when computing b = np.sum(a)  I get print (b) 0.9999999999999999  Could you anyone explain why and how to solve this approximation issue?
Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen?
Is it correct to omit preposition before date? My father wrote some papers on cosmology that remained unpublished at the time of his death.  I have been looking for a publisher for them. I contacted IOP publishing but they said that, while Dad’s writings were interesting, they were not ready for publishing in any of their journals. For example, they need a reference list citing other current published research on the same topic.  I am not a scientist and have no idea how to do this. Where can I get help with this? Would something like  be a good idea? Show that $d(z,a)=|\frac{z-a}{1-\overline{a}z}|$ is a metric on the unit disk For any $z, a$ in the open unit disk, we define$$d(z,a)=\left \vert \frac{z-a}{1-\overline{a}z} \right \vert$$  Show that $d$ is a metric on the open unit disk.     It is easy to show $d(z,a)=d(a,z)$ and $d(z,a)=0$ $\rm iff$ $z=a$. However, I am having trouble showing that it satisfies triangle inequality. Any help would be appreciated! How many "super imaginary" numbers are there? How many "super imaginary" numbers are there? Numbers like $i$? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $i$ and rotation, but what about hyperbolic rotation? Like we have a complex number $$ z = a + bi $$ can describe a matrix $$ \begin{bmatrix} a &amp; -b \\ b &amp; a\end{bmatrix} $$ You can "discover" $i$ by doing (which is used for another discovery) $$ \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{bmatrix} c &amp; -d \\ d &amp; c\end{bmatrix} = \begin{pmatrix} ac - bd \\ ad + bc \end{pmatrix} $$ $$ (a + bi) \cdot (c + di) = ac + adi + bci + bdi^2 $$ From here on you can infer that $ i^2 = -1 $.  So what if we do the same thing, but a different matrix? $$ z = a + bh$$ can describe a matrix $$ \begin{bmatrix} a &amp; b \\ b &amp; a\end{bmatrix} $$ and we can discover it the same way $$ \begin{pmatrix} a \\ b \end{pmatrix} \cdot \begin{bmatrix} c &amp; d \\ d &amp; c\end{bmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix} $$ $$ (a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2 $$ From here we infer that $ h^2 = 1 $.  Also $$ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots  $$ $$ \begin{align} e^{xh} &amp; = 1 + \frac{xh}{1!} + \frac{(xh)^2}{2!} + \frac{(xh)^3}{3!} + \frac{(xh)^4}{4!} + \frac{(xh)^5}{5!} + \cdots \\ &amp; = 1 + \frac{xh}{1!} + \frac{x^2}{2!} + \frac{x^3h}{3!} + \frac{x^4}{4!} + \frac{x^5h}{5!} + \cdots \\ &amp; = \cosh{x} + h \cdot \sinh{x} \end{align} $$  How many more numbers like this are there? And does that mean that for each set of trigonometric functions there exists a number which can turn multiplication into a rotation using those trigonometric functions?  (Sorry if I got some things wrong) How to use Noto Color Emoji with lualatex? I want to use Noto Color Emoji with lualatex (package fonts-noto-color-emoji on my box). However, this fails as follows. How to use Noto Color Emoji with lualatex?  \documentclass{article} \usepackage{fontspec} \setmainfont{Noto Color Emoji} \begin{document} 😺 👅  \end{document}   Error:  $ lualatex emoji.tex  This is LuaTeX, Version 1.07.0 (TeX Live 2019/dev/Debian)   restricted system commands enabled. (./emoji.tex LaTeX2e &lt;2018-12-01&gt;  luaotfload | main : initialization completed in 2.224 seconds (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo)) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec.sty (/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3-code.tex) (/usr/share/texlive/texmf-dist/tex/latex/l3kernel/l3pdfmode.def))) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec-luatex.sty (/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty (/usr/share/texlive/texmf-dist/tex/latex/base/tuenc.def)) (/usr/share/texlive/texmf-dist/tex/latex/fontspec/fontspec.cfg))) (./emoji.aux) [1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./emoji.aux))  409 words of node memory still in use:    3 hlist, 1 vlist, 1 rule, 2 glue, 3 kern, 1 glyph, 5 attribute, 48 glue_spec , 5 attribute_list, 1 write nodes    avail lists: 2:15,3:4,4:1,5:21,6:1,7:13,8:1,9:6 &lt;/usr/share/fonts/truetype/noto/NotoColorEmoji.ttf ! error:  (file /usr/share/fonts/truetype/noto/NotoColorEmoji.ttf) (ttf): sfnt  table not found !  ==&gt; Fatal error occurred, no output PDF file produced! set only lower bound of a limit for ggplot "a while" vs "awhile" Is there a difference between "awhile" and "a while"? If there is, what is it? I've been wondering this for awhile, but now I actually need the answer. How to give different permissions to multiple groups of users? I installed Ubuntu 12.04 server and want to give full folder access say example: "Folder01" to group "Group01' and Limit "Group02" for read access only. How I can do this. Can I visit Chicago during a 5.5 hour layover in O'Hare? I've been using mac for years, but prefer to do software development on windows. I'm really used to is using the folowing shortcuts on windows very much:  &lt;CTRL&gt;  + INSERT: Copy &lt;SHIFT&gt; + INSERT: Paste &lt;SHIFT&gt; + DELETE: Cut   Of late, I've taken to doing even my development work on the Mac. However, as a matter of habit, I'm missing these shortcuts very much. Actually, I find these a lot more convenient than COMMAND + (C/V/X) combo.  Is there a way to reassign these three shortcuts at least?   I've look all over the internet and in the "Keyboards" system preferences but I did not find anything.  PS: If you're wondering where I find the INSERT keys on a mac book pro, I use a wireless keyboard. Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges How to make sure I'm not using any proprietary software after installation? I know I can check the currently installed non-free software with the following command (assuming aptitude is FOSS):  aptitude search '?installed (?section(restricted) | ?section(multiverse))'   And I can remove the "restricted" and "multiverse" sources (and not add any PPA's) to keep that list empty. I'm assuming from this point, whenever I apt install something, that something is either in the allowed sources or it is not found.  Is this enough? Does this make sure all software I'm running (in Ubuntu) is open-source? If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a normal distribution? My understanding is that this is true with the means of a large number of samples, but is it also true with medians?  If not, what is the underlying distribution of sample medians? How to get perfect UV sphere Mercator projection? I'm trying to unwrap a UV Sphere and I can't seem to get a perfect  projection. What process should I follow to get a perfect UV unwrap that looks like the black grid lines in this image?: How do I convert a hex string to an int in Python?   I may have it as "0xffff" or just "ffff". Functions vs. patterns Error -504 on appliation installation Yes. I've read all other questions regarding this problem. Like this one:  It's the same problem as this one but with more background information and an other app:   The details: My device is an Ulefone Be Touch 2 running Android 5.0.  My connection is perfect. To be sure I tried different networks, among them are 3G, 4G and wifi on ADSL, VDSL, cable and fiber. I've also used Tor and 3 different VPN providers in 3 countries on 2 continents. Let's say I ruled the network out.   A single app does not install: OpenKeychain.  Currently I have 126 apps installed. It does not install via desktop browser > play store > install button.    I have enough free space (40GB). The app is 7MB.   Re-adding Google account and clearing caches did not work.    Even a complete device wipe and re-install didn't work.     Downloaded the apk and tried manual installation, same error.   I also own a Nexus 7 (2013 wifi) and the app installs fine on that device. Shortcut for "null if object is null, or object.member if object is not null" Mint 18 will boot only to a black screen I have a 4GB DDR3 800Mhz recently I installed another 4GB. It turns out it's  DDR3 667MHz. Does it really matter I should own both rams 800MHz?    Slot1 shows recent RAM installation. ALSO I don't know if this is 1600MHz (it shows 800x2 and 667x2 for other) or just 800MHz ram.