Let $n$ be a positive integer. Then the length $\ell(n)$ of $n$, that is, the number of digits in the decimal representation of $n$, is given by $$\ell(n) = \lfloor \log_{10}(n)\rfloor + 1.$$ How can one go about rigorously proving that this holds for all positive integers?
I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map. I'm having trouble distinguishing between the concepts of scedasticity and stationarity. As I understand them, heteroscedasticity is differing variabilities in sub-populations and non-stationarity is a changing mean/variance over time.   If this is a correct (albeit simplistic) understanding, is non-stationarity simply a specific case of heteroscedasticity across time? Could we send a man safely to the Moon in a rocket without knowledge of general relativity? How do I manage how many revisions snapd keeps of an application? When to use reinterpret_cast? I am little confused with the applicability of reinterpret_cast vs static_cast. From what I have read the general rules are to use static cast when the types can be interpreted at compile time hence the word static. This is the cast the C++ compiler uses internally for implicit casts also.  reinterpret_casts are applicable in two scenarios:   convert integer types to pointer types and vice versa  convert one pointer type to another. The general idea I get is this is unportable and should be avoided.     Where I am a little confused is one usage which I need, I am calling C++ from C and the C code needs to hold on to the C++ object so basically it holds a void*. What cast should be used to convert between the void * and the Class type?  I have seen usage of both static_cast and reinterpret_cast? Though from what I have been reading it appears static is better as the cast can happen at compile time? Though it says to use reinterpret_cast to convert from one pointer type to another? I know that in the case of Gaussian mixture, the "intuition" is that you're drawing from a PDF which itself is just a sum of weighted Gaussian PDFs.  I don't understand the intuition behind how the sum of two iid Gaussian RVs leads to a Gaussian RV. I know the maths behind it, but I'm having a hard time visualizing how the sum of two Gaussian RVs leads to a Gaussian RV (and not a mixture). I remember coming across this fact a while ago in a pdf somewhere, but I haven't been able to find it again. Can someone show me how to prove it? I would appreciate easier proofs.  EDIT: I'm very sorry, I forgot to actually state it.  Prove that $-1$ is not a quadratic residue modulo primes of the form $4k+3$.  Thanks! How do I get add-apt-repository to work through a proxy? Effective Refractive Index in Paper I am reading the paper, entitled "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings" by Eric B. Grann, M. G. Moharam, and Drew A. Pommet.   In the paper they mention this formula:        Do you know how it can be derived and where it comes from ? Failed upgrade from Ubuntu 19.10 to 20.04 "Please update the microcode" While installing the Ubuntu 20.04 updates during upgrade from 19.10 to 20.04, my screen suddenly went completely white and displayed a message saying  Please logout and try again.  But I could not locate my cursor and so I switched off my laptop. After restarting it again, it was showing an error message saying  Please update the microcode  Why did this happen? How can I solve this problem? Does Stack Exchange use caching and if so, how? Recently, I read this  about how they have some of their infrastructure laid out. I was curious to know, does Stack Exchange use caching (both at the application layer and/or database layer) and if so, how? I'm using a flow to set a field to a certain value (01, 02, 03) depending on the value of another field on the same record. I initially wanted to do it with process builder but because of the mapping it seems I would need three 'criteria nodes' or a formula field. Furthermore, it seems Salesforce is recommending using flows instead of process builder for before update actions as it is faster.  The problem with Flows is that I found a way to trigger it when the record is new or updated but not when the specific field is changed so it triggers every time the whole record is updated even when the field stay the same.  On Process Builder, you can use isChanged and then call the flow with specific record but that kinds of defeat the point of using flows for before Update.  So is there a possibility to do the same thing directly in flow ?  Thank you for the help ! Did Darth Sidious and Vader ever discover how to cheat death? In Episode 3 Revenge of the Sith Palpatine/Sidious strings Anakin along to the dark side with the promise that together the two of them will be able to re-discover Darth Plagueis' secret to cheating death.      "To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret."   From the end of Revenge of the Sith to their deaths in Return of the Jedi did Darth Sidious and Darth Vader ever manage to re-discover this secret? Pgfplots: Linewidth when using \ref in custom legend Reading grep patterns from a file To my understanding, "Control" can have two meanings in statistics.   Control group: In an experiment, no treatment is given to the member of the control group.  Ex: Placebo vs Drug: You give drugs to one group and not to the other (control),  which is also referred as "controlled experiment".  Control for a variable: Technique of separating out the effect of a particular independent variable. Some of the other names given to this techniques are, " accounting for", "holding constant", "controlling for", some variable.  For example: In a Football viewing study (like or not like),  you may want to take out the effect of gender as we think gender causes bias, that is, male may like it more than female.     So, my question is for point (2). Two questions:  How do you "control"/ "account for" variables, in general. What techniques are used? (In terms of regression, ANOVA framework).   In above example, does choosing male and female randomly constitutes control? That is, is "randomness" one of the techniques for controlling other effects? How to partition my hard drive for Ubuntu use? Till now I was windows user. From now on I want to use only Linux.   I have 500Gb HDD. How do I partition it properly? I read that there is no right or wrong way, but still, I am confused. I did something and I have primary partition mounted on / (160Gb) which I believe is a OS and 350Gb extended partition of which I have 4Gb of swap and 346Gb mounted on /home.  I got used to C:\ and D:\ partitioning, but I don't see file system in that way. I am lost. Where is what? How can I make C:\ partition for OS and D:\ for apps, movies, music, photos. Or what I want is Windows way and I have to get used to Linux way? What's the best way to season a cast iron skillet? I just purchased a new cast iron skillet. What's the best way to season it?   And if I need to re-season an old pan, is the process any different? Command to perform a recursive chmod to make all .sh files within a directory executable? Add chat rooms for Android app Being able to join chat rooms from the Android app would be a plus! It would truly make us connected on the go, instead of having to sit down at our desks and open a browser window. This will allow more users to fully utilize the chat system and join events no matter where they are.