My personal experience with Neural Networks is that everything became much clearer when I started ignoring full-page, dense derivations of backpropagation equations and just started writing code. Thus, this tutorial will contain very little math (I don't believe it is necessary and it can sometimes even obfuscate simple concepts). Since my background is in Computer Science and Physics, I will instead develop the topic from what I refer to as hackers's perspective. My exposition will center around code and physical intuitions instead of mathematical derivations. Basically, I will strive to present the algorithms in a way that I wish I had come across when I was starting out.
"...everything became much clearer when I started writing code."
You might be eager to jump right in and learn about Neural Networks, backpropagation, how they can be applied to datasets in practice, etc. But before we get there, I'd like us to first forget about all that. Let's take a step back and understand what is really going on at the core. Lets first talk about real-valued circuits.