Category Archives: infinity

The MacLaurin series for sin(x)

Posted on February 13, 2017 by Brent

In my previous post I said “recall the MacLaurin series for :” Since someone asked in a comment, I thought it was worth mentioning where this comes from. It would typically be covered in a second-semester calculus class, but it’s … Continue reading →

Posted in calculus, infinity, iteration | Tagged cos, derivative, infinite, MacLaurin, series, sin, sum, Taylor | 4 Comments

The Basel problem

Posted on February 6, 2017 by Brent

I wanted to follow up on something I mentioned in my previous post: I claimed that At the time I didn’t know how to prove this, but I did some quick research and today I’m going to explain it! It … Continue reading →

Posted in infinity, number theory | Tagged Basel, infinite, pi, Riemann, sum, zeta | 9 Comments

u-tube

Posted on January 10, 2012 by Brent

[This is the eighth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does "close to" mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading →

Posted in computation, convergence, infinity, iteration, modular arithmetic, number theory, programming | Tagged decadic, Haskell, idempotent, streaming, u | 2 Comments

A self-square number

Posted on January 4, 2012 by Brent

[This is the seventh in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic numbers, More fun with infinite decadic … Continue reading →

Posted in arithmetic, infinity, iteration, modular arithmetic, proof | Tagged decadic, idempotent, self, square | 12 Comments

More fun with infinite decadic numbers

Posted on October 28, 2011 by Brent

This is the sixth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic numbers). Last time I left you … Continue reading →

Posted in arithmetic, infinity, number theory | Tagged decadic, decimal, fractions, integers, representation | 4 Comments

Infinite decadic numbers

Posted on October 19, 2011 by Brent

To recap: we’ve now defined the decadic metric on integers by where is not divisible by 10, and also . According to this metric, two numbers are close when their difference is decadically small. So, for example, and are at … Continue reading →

Posted in arithmetic, convergence, infinity, number theory | Tagged decadic, negative, numbers | 7 Comments

Book review: Roads to Infinity

Posted on July 5, 2011 by Brent

What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this well-written and fascinating book, John Stillwell takes us on a tour through some of the answers to these … Continue reading →

Posted in arithmetic, books, computation, induction, infinity, logic, proof, review | Tagged infinity, John Stillwell, proof, roads | Comments Off

Rational numbers and decimal expansions

Posted on September 7, 2008 by Brent

As you may remember from school, rational numbers have a terminating or eventually repeating (periodic) decimal expansion, whereas irrational numbers don’t. So, for example, 0.123123123123…, with 123 repeating forever, is rational (in fact, it is equal to 41/333), whereas something … Continue reading →

Posted in infinity, iteration, number theory, pattern | Tagged decimal, periodic, rational | 9 Comments

Recounting the Rationals, part II (fractions grow on trees!)

Posted on January 7, 2008 by Brent

Today I’d like to continue my exposition of the paper “Recounting the Rationals”, which I introduced in a previous post. Recall that our goal is to come up with a “nice” list of the positive rational numbers — where by … Continue reading →

Posted in infinity, number theory, pattern, recursion, sequences | 22 Comments

Carnival of Mathematics #23: Haiku Edition

Posted on December 28, 2007 by Brent

Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be … Continue reading →

Posted in algebra, books, calculus, challenges, counting, fractals, geometry, infinity, links, meta, number theory, pascal's triangle, pattern, people, sequences, trig, video | 17 Comments