Recent content by aikismos

The question of pollution is broad. For instance, how you combust has a major effect on your byproducts, so it depends on your method. For instance, a leaky, old diesel engine will leak more pollution than a state-of-the-art, high quality gasoline engine with a sophisticated ECM and pollution...

Well, by definition, you are talking about a metaphor when you are talking shadows and projections in the basic sense extended to constructing a tesseract as a 3-D object; after all shadows and projections as we understand them intuitively are exercises of visualization in three-dimensional...

According to Dictionary.com a dimension has the following definitions (emphasis mine): "Mathematics. a property of space; extension in a given direction: A straight line has one dimension, a parallelogram has two dimensions, and a parallelepiped has three dimensions.1. the generalization of...

You're looking for permutations of a multiset (a set in which redundant elements are allowed). (See https://en.wikipedia.org/wiki/Permutation#Permutations_of_multisets.)

As all dimensions are mathematical counts, another interesting aspect of dimensionality in mathematics is the use of fractional dimensions. (See https://en.wikipedia.org/wiki/Fractal_dimension).

https://en.wikipedia.org/wiki/Anode Anode sort of means current flows away, but originally when charge was explored, it was presumed that the positive particles moved (only later was it discovered that in an electrical circuit, it was the negatively charged electron that moved). It's also a...

@Yomi, Thanks for responding to my post in such an articulate manner to clarify the relationship between some of these ideas. And your first post to boot! Okay, so I knew most of that (in a somewhat disjoint fashion) mostly from poking around haphazardly. I do have a reason for my...

Well, the question of the nature of 'use' and 'more' is debatable. But it bears reminding ourselves that the use of 0 to represent off is arbitrary. You can use a perfectly isomorphic Boolean algebra using 1 and 2 instead... :D

We do modular math without zero every day. ..., 1 o'clock, 2 o'clock, ... , 11 o'clock, 12 o'clock, ...

As far as I understand it, ## 2^{p-1}−1≡0\ (mod\ p) ## is a corollary to Fermat's Little Theorem, and is a subset in the case only where ## p ## doesn't divide ## a ##. As such, I believe the question still stands?

Wouldn't be better to say that f(x) = 3x + 2 defines a function, not a relation? In math, if it has an equal sign, as far as I understand, it's an equation.

I'm trying to wrap my mind around the notion of the difference of forms. Since the AKS primality test is logically equivalent to Fermat's ## 2^{p-1} - 1 ≡ 0\ (mod\ p) ##, is it fair to say that the main advantage of the form of the former is that it essentially splits up a single division...

To add to Geofleur's points, the point of stating an equation in functional notation adds some extra information when stating the equivalence relation (in this case an arithmetic equality). The moment you declare an equation a function, you're talking about a relationship between the domain and...

@JJ1979 The title: Set Theory and the Structure of Arithmetic by Hamilton and Landin (Allyn and Bacon) no ISBN, but Library of Congress Catalog: 61-15038 Using a sufficiently axiomatized set theory (naive), the three volume set establishes all sorts of proof by building up the natural numbers...

I have the first of a three-volume set of U of I, that uses naive set theory to construct arithmetic, etc. You have any interest in a pointer to that?