When not to treat dy/dx as a fraction in single-variable calculus?
Can I ever go wrong if I keep thinking of derivatives as ratios?
Greatest comeback in 2nd leg match There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?  Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.  However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.  This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful! What is the main difference between StringBuffer and StringBuilder? Is there any performance issues when deciding on any one of these? Is there any Android XML documentation? Is it possible to add new alphabets to unicode-math? Producing .bbl file with Lyx and JabRef How do I produce the .bbl file required for arXiv using JabRef for bibtex and Lyx? There's more gravitational force in our galaxy (and others) than can be explained by counting stars made of ordinary matter. So why not lots of dark planetary systems (i.e., without stars) made of ordinary matter? Why must we assume some undiscovered and unexplained form of matter? I am trying to install reaver 1.4, but when I run ./configure ; make I get this message at the end  int ret_val = EXIT_FAILURE, r = 0;   I have searched a fair bit and cannot find an answer, any help would be appreciated. I apologize if this has been answered and I've just failed to find it. How do I make Firefox start always maximized? I am setting up Ubuntu with gnome for my grandma. How can I make firefox start ALWAYS maximized. How can I remove launcher drive icons? I am looping over the features of a shapefile containing multiple polygons. If  certain criteria are met, I want to save the current feature in a new shapefile.  I am doing this using GDAL 1.10.0 with Python 2.7.3  The code runs, creates the files I want, calculates the area correctly. In the shapefile that is created, the attribute table contains the propoer fields ('name' and FID). Two things are wrong:   No polygons are visible (in ArcMap). This seems to me particularly weird, because the polygons seem to me proper, in the sense that they are closed (and the area is calculated properly).  The coordinate system seems off (in Arc Catalog there appears in the bottom -20,9667 0,0000 (are these degrees??)), despite the fact that I am creating a projection file (they are in German national grid 3)   Any hints as to what the problems could be? I tried to put the relevant excerpt of the code below:  from osgeo import gdal, gdalnumeric, ogr, osr, gdalconst  ### prepare shapefile into which to save driver = ogr.GetDriverByName('ESRI Shapefile') path_matched = r'.../.../path' datasource = driver.CreateDataSource(path_matched) lyr_matched = 'matched_%s' % cur_id  # from above # Remove output shapefile if it already exists name_shpfile = lyr_matched + '.shp' if os.path.exists(os.path.join(path_matched, name_shpfile)):     driver.DeleteDataSource(os.path.join(path_matched, name_shpfile)) layer = datasource.CreateLayer(lyr_matched, geom_type=ogr.wkbPolygon) #layer.ResetReading() field_name = ogr.FieldDefn("name", ogr.OFTString) layer.CreateField(field_name)  ### prepare file in which I want to write the areas of matched polygons path_area = path_matched file_area = 'matched_area_%s.txt' % (cur_id) fobj_area = open(os.path.join(path_area, file_area), 'w')  ### original shape file containing all polygons shp = ogr.Open(shp_all)  # os fobj from above lyr = shp.GetLayer()  # loop over polygons in original shameful for FID in featList:     feat = lyr.GetFeature(FID)      # this is from the shapefile      cur_name = feat.GetField(1)      # ... work on conditions ...      if condition = true:         myPoly = ogr.Geometry(type=ogr.wkbPolygon)         cur_poly = feat.GetGeometryRef()         myPoly.AddGeometry(cur_poly)         cur_spat_ref = cur_poly.GetSpatialReference()          cur_area =  cur_poly.GetArea()         cur_area_string = "%s \t %f \n" % (gw_name_EZG, cur_area)         fobj_area.write(cur_area_string)          featur_dfn = layer.GetLayerDefn()         feature_abc = ogr.Feature(featur_dfn)         feature_abc.SetGeometry(myPoly)         feature_abc.SetField("name", cur_name)         feature_abc.SetFID(FID)         layer.CreateFeature(feature_abc)         myPoly.Destroy()         feature.Destroy()   # generate projection file prj_filename = lyr_matched + '.prj' file = open(os.path.join(path_matched, prj_filename), 'w') targetSR = cur_spat_ref targetSR.MorphToESRI() file.write(targetSR.ExportToWkt()) file.close()   datasource.Destroy() Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$ Let $x$ and $y$ be integers not congruent to $0$ modulo $p$ where $p$ is a prime. Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$.   I thought about proving this algebraically and by contradiction. So we say $x^2+y^2 = pz$ and have to show this is impossible. $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$? The topological space $X=\mathbb{N}\times \mathbb{Q}$ with subspace topology of $\mathbb{R}^2$ the set $S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?  $S^c=\phi\times\{{1\over n}\}^c$ right? but I know $\mathbb{Q}$ is neither open nor closed in $\mathbb{R}$, I am confused here, could any one help me to answer this question?  the boundary of $S$ is $\bar{S}\setminus $ int S. as int S is $\phi$ so boundary is $\mathbb{N}\times \mathbb{R}$? Which is stronger, a rope without knots or a rope with knots? Suppose I have two identical ropes, one of which I manually tie some knots in. If I use them to hang clothes, which one is more durable?      Personally I think that rope with knots will be more durable, but I can't come up with a satisfactory reason. Here are some options I thought of, not sure which is the right one.   There was an I/O error reading from the pipe. The process writing to the other end of the pipe died with a failure. All processes who could write to the pipe have closed it. The write buffer of the pipe is full. The peer has closed the other direction of the duplex pipe. Writing failed because there are no processes which could read from the pipe. A system call returned the EPIPE error, and there was no error handler installed. Based on the question , I got curious. Is there a guide or resource that describes how the price of a ticket is determined?  When you search you often see curious effects like cheaper tickets for a flight from A to B with a layover compared to a direct flight. This seems counter-intuitive from an outsider's perspective since you fly twice. Also one way tickets more expensive than 2 ways.  It can be a short set of rules. Something that helps someone to get started with finding good prices. I know these can be very complex and they probably change over time and over airline interests but there is probably a simplified set of rules that can be enunciated. Prove if f(z) is entire function s.t $|f(z)|\leq M|z|$ then $f(z)$ Is polynomial of degree $1$ Let $f(z)$ be an entire function s.t $|f(z)|\leq M|z|$ ; $M&gt;0$ is a constant then $f(z)$ Is polynomial of degree 1   My try $f(z)$ is entire so it can be written as a $\sum a_n z^n$  $$|\sum a_n z^n|\leq \sum |a_n||z|^n\leq M|z|$$  So $n=1$  Seems that the move $\sum |a_n||z|^n\leq M|z|$ is problematic Unable to play MP3 or MP4 files in Rhythmbox or Movie Player I am unable to play any MP3 or MP4 files. When I try to open any MP3 or MP4 file, both show an error, a Python 2.7 plug-in is required. What methods of ‘clearfix’ can I use? Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$. $f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)&lt;0$.  Attempt: Let us define a function $h(x) = g(x) g~'(x)$. Then $$h'(x) =  g(x)g''(x) + \left( g'(x) \right)^2 \tag 1$$ If we prove that for some $c \in (-3,3), h~'(c) &lt; 0,$ then $$g(c)g''(c) &lt;0 \tag 2$$ Also, $$\left|f(0)\right| &lt; 1 \implies f'(0) \in (-3,-2\sqrt 2 ) \cup (2\sqrt 2,3) $$ Could someone please advise me how do I move forward from here. Thank you very much for your help in this regard.