Hello stackexchange users! I am somewhat confused on $\sqrt{-1}$. I believe the answer should be no solution, or inconsistent, or at least something in that nature. Am I correct or am I most likely wrong? Any help or insight would be deeply appreciated
At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?  Why is the concept even useful?
How do I contact other users? Decoupling caps, PCB layout I guess I've been somewhat ignorant when it comes to the finer details of pcb layout. Lately I've read a couple of books that try their best to lead me on the straight and narrow. Here is a couple of examples of a recent board of mine, and I have highlighted three of the decoupling caps. The MCU is a LQFP100 package and the caps are 100nF in 0402 packages. The vias connect to ground and power plane.     The top cap (C19) is placed according to best practices (as I understand them). The other two are not. I haven't noticed any problems. But then again the board has never been outside the lab.  I guess my question is: How big a deal is this? As long as the tracks are short, does it matter?  The Vref pins (reference voltage for the ADC) also have a 100nF cap across them. Vref+ comes from an onboard TL431 shunt regulator. Vref- goes to ground. Do they require special treatment like shielding or local ground?    EDIT    Thanks for great suggestions! My approach has always been to rely on an unbroken ground plane. A ground plane will have the lowest possible impedance, but this approach may be too simplistic for higher frequency signals. I've made a quick stab at adding local ground and local power under the MCU (The part is an NXP LPC1768 running at 100MHz). The yellow bits are the decoupling caps. I'll look into paralleling caps. The local ground and power are connected to the GND layer and the 3V3 layer where indicated.  The local ground and power are made with polygons (pour). It's going to be a major rerouting job to minimize the length of the "tracks". This technique will limit how many signal tracks can be routed under and across the package.  Is this an acceptable approach? Why can I not place a pregnant woman to work but others work while pregnant? I am playing Fallout Shelter, have 3 kids in my shelter, all going well, thought I would make one more kid with big strength numbers and decent agility/perception (kids take on SPECIAL stats from their parents). All the women who are pregnant work in my shelter, apart from this one. She can not me put to work, can not be moved from the living shelter, can not be moved to a job via the clipboard short cut.  TL;DR One of the NPC females who is pregnant can not be put to work, can not be moved from the living shelter room, she just walks about as if she is carrying a mammoth inside her. I have NOT moved the shelter, have not let the dad die, have not done anything at all. "Open in browser" does nothing when Android app is set as default show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$. Given two numbers $a$ and $b$, where $a$ is co-prime to $b$,     Show that for any number $c$, if $a|c$ and $b | c$ then $ab| c$. Is the reverse also true? In other words, if $ab |c$ and $a$ is co-prime to $b$, then do we have $a | c$ as well as $b|c$? Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks I am firmly convinced he is innocent and his accusation is the result of the police conspiracy.     I am firmly convinced he is innocent and his accusation is the result of the police's conspiracy.    Which sentence do you think is a better choice? I've been modeling an F-15, but when I rotate the flaps on Y-Axis the Flaps offset from the wing and they look like they are about to fall out, so I end up correcting it by rotating in the X and Z-axis to align them correctly.  However when I rotate them back from the Y-Axis they look distorted again because I did rotations on the X and Z axis. I tried riding it, no help. I saw something about custom orientation to fix it but that was not clearly explained and it was done on an ideal plane that was (0,0,0); mine is pre-rotated on all axis.   I'll take scripts if needed and I'm using Blender 2.8.  When I rotate it on the Y-Axis:    Fixing it using X and Z rotation:    Rotating it back on the Y distorted it again:    Here's the wing with ailerons and flaps: How to peel hard boiled eggs easily? This is one of the most annoying duties in the kitchen, for me.  What are yours tips and tricks to peel hard boiled eggs easily? Suppose someone from "Harry's world" married a Muggle, and they want to escort their child. Can the Muggle parent get through to the Hogwarts platform? It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it? (no) Properties in Java? How to upgrade to MySQL 8.0? I have a Drupal site 8.6.10 on an Ubuntu server 18.04 with MySQL 5.7.25  I want to update MySQL to version 8.0  Here are the steps I followed :   I backed up my database. I uninstalled MySQL from my server with the commands :  $ sudo systemctl stop mysql $ sudo apt remove mysql-* $ sudo apt purge mysql-* $ sudo apt autoremove $ sudo dpkg -l | grep mysql | grep ii  I added the repository with the command :  $ wget https://repo.mysql.com//mysql-apt-config_0.8.12-1_all.deb  I installed the package with the command :  $ sudo dpkg -i mysql-apt-config_0.8.12-1_all.deb            My problem :   I install MySQL with the command :  $ sudo apt update $ sudo apt install mysql-server    When I do step 5, it installs MySQL 5.7 why does not it install version 8 ? What's the difference between "established" and "founded" In this sentence: "CARE was ...... in 1945 to help people after World War II."    A. founded  B. built C. established D. constructed    The answer is C. But why don't we use "founded" in this case, I think they have the same meaning. Can you help me to give a reason for that? How to "add existing frameworks" in Xcode 4? I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the following limit: $$\lim_{x\to\infty}{\frac{F_n}{F_{n+1}}}=?$$ Where $F_n$ is the nth Fibonacci number? The kernel and range of the powers of a self-adjoint operator In Photoshop I've taken Magic Wand as far as I can, I think. Now I'm left with this grayish edge around the coffee saucer. How do I remove it? I've tried altering tolerance in Magic Wand but don't seem able to get it. Any suggestions please? Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$? Is is possible to move all windows (or all not minimized windows) form one workspace to another?   I know I can move one window to other workspace with Shift+Ctrl+Alt+arrow, but it will move only that one focused window.