This is the second article in the series about artificial neural networks. If you have not already done so, I recommend you read the first article, “Neural Networks: The Node“, before proceeding. It covers material that should be understood before attempting to tackle the topics presented here and in future articles in this series.

There are several properties that define the structure and functionality of neural networks: the network architecture, the learning paradigm, the learning rule, and the learning algorithm.

As I covered previously in “Introduction to Neural Networks,” artificial neural networks (ANN) are simplified representations of biological neural networks in which the basic computational unit known as an artificial neuron, or node, represents its biological counterpart, the neuron. In order to understand how neural networks can be taught to identify and classify, it is first necessary to explore the characteristics and functionality of the basic building block itself, the node.

This is going to be another one of my “selfish” posts – written primarily for me to refer back to in the future and not because I believe it will benefit anyone other than me. The idea is one that I always took for granted but had a hard time proving to myself once I decided to try.

Theorem: Suppose we have an M bit unsigned binary integer with value A. Consider the first (least significant) N bits with value B. Then:

Put another way, arithmetic with unsigned binary integers of a fixed length N is always performed modulo .

If you’ve got the word “power” in your name, you’d better believe expectations are going to be sky high for what you can do. The Power Rule in calculus brings it and then some.

The Power Rule, probably the most used rule when differentiating, gives us a drop dead simple way to differentiate polynomials. Specifically it says for that any polynomial term raised to the power with coefficient :

(1)

Apply this to every term in your polynomial, and you’ve got its derivative! Easy peasy. Let’s prove it.

So we recently covered the Pythagorean Theorem, and I am betting you’ve got at least one concern. No, I don’t mean about whether or not Planet X has begun cutting a destructive swath across our solar system. I mean in regards to calculating the length of the side of a triangle if you know the other two. Don’t have a right triangle? You may think you’re SOL. I am here to tell you there is hope. Enter the Law of Cosines.

Pythagoras of Samos
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It may very well be the second most famous equation of all time, outshone only by that braggart Einstein’s mass–energy equivalence equation. But for those of us that aren’t theoretical physicists, the Pythagorean Theorem is likely to play a fundamental role in many of the calculations we do whether we realize it or not.

Everyone knows the equation, but before we get into it’s proof, let’s re-cap what it means. Consider the following right triangle:

This triangle has sides and hypotenuse . The Pythagorean Theorem simply tells us then that the square of the length of the hypotenuse () is equal to the square of the length of side plus the square of the length of side ().  Well, duh. And after we learn it in middle school many of us take it as axiomatic, but it does have “theorem” in it’s name so I feel its worth revisiting it’s proof every once in a while to keep me on my toes.

GE2524 Multimeter

Question: Based on your previous post regarding voltmeters and internal resistance, I can safely assume that an ammeter must also have some internal resistance. What effect does this resistance have on current measurements, and what resistance value would minimize any error it introduces to the reading?

Answer: It is indeed true that an ammeter must have some internal resistance because like a voltmeter, it must draw current from the circuit in order to make its measurement. The similarities between ammeter and voltmeter internal resistances end there, however.

GE2524 Multimeter

Question: Does a voltmeter have any internal resistance, and if so, what, if any, affect does it have on voltage readings?

Answer: Even if you have no electrical experience at all, I would hope that your answer to the first part of the question was, “yes.” Otherwise this would turn out to be an incredibly short post.

Leonhard Euler

Besides being an obvious lady killer, Swiss mathematician Leonhard Euler gifted the world with some pretty important mathematical concepts, notational conventions, and formulas. I almost feel bad about the fact that I couldn’t even spell his name correctly until I was well into adulthood.

You are probably thinking, “Sure, he had a bitchin’ robe, and for an old-timey dude he was pretty good looking, but was he really all that?”

I can’t find any quotes from the ladies, but his peers, both contemporary and modern have had a few things to say about him and his work.

(Note: I lied. This will be my first “neural dump.” I began writing about Euler’s Formula, but felt what follows was worthy of its own post and a better foundation for what will follow when I tackle Euler.)

Complex numbers arose from the fact that there is no solution for in the equation in , the set of real numbers.

Early mathematicians being the devil-may-care mavericks that they were, were all like, “Screw it. Let’s just invent a new number. We’ll just call this number and say that the solution is .” Or, in other words, this new number they imagined up (see what I did there) is equal to .