Interesting occurrence with 7th root of unity

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I've come across a property of that is rather interesting. Letting , we let , we will see that .

Instead of working directly with this difference, we square the expression for something more easily manipulated. . Using the equalities , etc., we can simplify this down to . Since the sum of the 7th roots of unity equals 0 and the sum only neglects the root , we see that . Thus . Amazing stuff, right?

Divide by 0 != universe explode

Post inspired by: The Last Denominator

I just want to let people know that there is no way that any (true) theorem about the field of real numbers (or any field) has a division by zero. In order to get to the concluding statement would require an element that is 1/0 (which I will call the inverse of zero) to be summoned. Since this element does not exist in a field other than the zero ring (in which case 0 = 1), it cannot be used. The step taken is invalid.

Now that I have shown that the usage of zero's inverse is invalid, I will show that no true statement can imply a theorem that involves zero's inverse. By the axioms of inference that logicians, philosophers and mathematicians have developed, A implies B is equivalent to (NOT A) OR B. Therefore since B = conclusion with zero's inverse, if A is true, then the implication is false. And so we conclude that only false statements can imply the existence of zero's inverse (with the exception of the extremely uninteresting case of the zero ring).

To those of you who do not know to what I refer to as a ring and field, here is the break down (first we need the concept of a group):
A group G is a set of elements with a very limited amount of structure imposed on them under one operation (call it *):
1) There exists an identity element, 1, such that 1*x = x for each element x in G.
2) There exists an inverse element 1/x such that 1/x * x = 1 for each x in G.
3) For two elements in G, x and y, x*y is also in G.
4) For three elements in G, x,y and z, (x*y)*z = x*(y*z). In other words, * is an associative operation.

A ring R is simply a set of elements that have a certain structure under two operations defined among the elements (call them + and *):
1) R is an abelian group under + (abelian just means that for x,y in R, x + y = y + x. In other words, + is commutative).
2) For two elements x, y in R, x*y is in R.
3) The distributive laws hold: x,y,z in R imply x(y + z) = xy + xz, and (x + y)z = xz + yz.

A field F is a special kind of ring that really has a lot of great qualities that allow us to prove a lot of cool things:
1) F is a ring.
2) All elements of F except the identity of + form an abelian group under *.

The "except the identity of +" is the key term. Since we commonly call the identity of +, 0, this means that 0 is not required to have an inverse (with regards to *) in F. We know that the real numbers form a field because every element other than 0 has a multiplicative inverse (namely 1/x), and every element has an additive inverse (-x), and you can't add or multiply two numbers and get an answer that isn't a real number. Since 0*x = 0 for any x in the reals, 0 does not have an inverse z such that 0*z = 1. Thus 0 does not have an inverse in the real numbers.

Okay? Now you can be sure that if you get a division by 0 that either the number is too small for your calculator to keep the correct precision and it just rounded to 0, you started off with a false statement, or you had an incorrect deductive step.

Unstructured Thoughts on Structures

As an undergraduate in mathematics, I know more math than you, but still less math than epsilon. It is for this reason that most all my opinions and thoughts on mathematics are much more romantic than they are useful. I have written papers on different areas that interested me, and I have gone to many different talks and read several papers. Could I come up with an hypothesis and prove it? Probably not, but given my current position in university, I am sure I could find guidance toward starting such an endeavor.
My goal with this blog is to increase readers' interest in mathematics and programming. I plan on talking about many random topics (but not truly random... a very fun investigation to do right there) that come to catch my interest, and I also aim to post solutions to different programming problems I come across. As long as my solutions to obscure problems are archived on the net, search engines will be able to index them and help a programmer in distress in the future.
I plan not to touch such topics as politics, philosophy or my personal life, but as an opinionated person without the motivation to maintain more than one blog, I make no promises about the future.

Back to the topic of this post, "Unstructured Thoughts on Structures." I am sure by now you have observed the unstructured thoughts, but what of these structures? Mathematics is all about creating abstractions to quantitatively and generally describe a problem. Structures, namely algebraic structures, are currently my focus of interest. Group theory, ring theory, field theory and Galois theory may ring bells for some of you. This is what I am talking about. This area of mathematics is so vastly different from anything else one studies before it that even mathematicians like to joke about it not being, "real math." A quote comes to mind from a source I cannot currently recall that is along the lines of, "Now we can solve this problem without any math at all; just group theory!" With this tidbit about structures to pique your interest, I will save more depth descriptions and discussions for later posts.

Welcome to my blog, friends of mathematics and computers.