There's a problem with data when I try to add  It's specifically happening when the no. of digits after decimals is four.  num1 = 1.5 num2 = 6.6758 sum = float(num1) + float(num2) print('The sum of {0} and {1} is {2}'.format(num1, num2, sum))   The sum of 1.5 and 6.6758 is 8.175799999999999
Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen?
Prove that a positive polynomial function can be written as the squares of two polynomial functions Finding the null space of symmetric matrix generated by outer product Let $p, q \in \mathbb{R}^n$ such that $\|p\| = \|q\| = 1$ and define $$A = pq^T + qp^T$$ Find the null space of $A$.    I am not having very much luck. I have managed to show that $p + q$ and $p - q$ are eigenvectors, but I can't seem to connect those two. I know that since $A$ is generated by an outer product, it is of rank one. That implies that the rank of the null space is $n - 1$, but I can't seem to get any further. Can anyone help? Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction. Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ Determine the set of points $z$ that satisfy the condition $|2z|&gt;|1+z^2|$  I tried to redo this problem and got to this point  $|2z|&gt;|1+z^2|$ $\Rightarrow$ $2|z|&gt;1+|z^2|$ $\Rightarrow$ $2|z|&gt;1+z\overline z$  Let $z=x+iy$ then   $$2\sqrt{x^2+y^2}&gt;1+(x+iy)(x-iy)$$  $$2\sqrt{x^2+y^2}&gt;1+(x^2+y^2)$$  $$0&gt;1-2\sqrt{x^2+y^2}+(x^2+y^2)$$  $$0&gt;(1-\sqrt{x^2+y^2})^2$$  since $x,y$ are real number , so there is no solution for this inequality? In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points... but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.  I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?  Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.  I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology? Subgroups of Symmetric groups isomorphic to dihedral group Is $D_n$, the dihedral group of order $2n$, isomorphic to a subgroup of $S_n$ ( symmetric group of $n$ letters) for all $n&gt;2$? I keep hearing a lot about functors in C++. Can someone give me an overview as to what they are and in what cases they would be useful? C++, array of objects without Problem with tcolorbox package How to increase JAVA Heap Size? Is starting a group with 5 newbies (DM included) possible? Some friends and I want to start playing D&amp;D (5e), but none of us, DM included (not me) have played before to any great extent. The DM and I have some cursory experience with 4e, but have never been able to play to any great extent. Is this advisable, or should we play with existing groups first? We live in a relatively remote area, making it hard to find experienced players. Number of non-isomorphic groups of order 21 Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa? What is a word to describe inefficient communication? Example 1 - When reviewing résumés I don't want someone to use 1.5 pages to describe one job.   Example 2 - When a person uses excessive description to answer a simple question. What are the regional Pokémon? I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in? Reference: What is variable scope, which variables are accessible from where and what are "undefined variable" errors? What happens to waves when they hit smaller apertures than their wavelenghts? I was wondering this for quite a long time now. Let's say you have a water wave (like ripples, not the ones you see during tsunamis) with wavelength 10 m. Imagine you put a boundary with an opening of 1 m. Will diffraction be observed? According to my research, no. But, then, what would be seen? How to make sure an application keeps running on Linux I'm trying to ensure a script remains running on a development server. It collates stats and provides a web service so it's supposed to persist, yet a few times a day, it dies off for unknown reasons. When we notice we just launch it again, but it's a pain in the rear and some users don't have permission (or the knowhow) to launch it up.  The programmer in me wants to spend a few hours getting to the bottom of the problem but the busy person in me thinks there must be an easy way to detect if an app is not running, and launch it again.  I know I could cron-script ps through grep:  ps -A | grep appname   But again, that's another hour of my life wasted on doing something that must already exist... Is there not a pre-made app that I can pass an executable (optionally with arguments) and that will keep a process running indefinitely?  In case it makes any difference, it's Ubuntu. What currency does the galaxy deal in besides credits? In APM and the Clone Wars series, among other media, there are frequent instances where Republic credits are not accepted. These are typically places outside of the Republic and especially in the criminal underworld, Republic or otherwise.  If Anakin wants to pay Hondo to deliver some missiles illegally and not ask any questions, and Hondo does not accept credits, then what does Anakin pay him off with? I'm not able to get access to $wpdb