# Category Archives: infinity

## The MacLaurin series for sin(x)

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 , , , , , , , | 4 Comments

## The Basel problem

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 , , , , , | 9 Comments

## u-tube

[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

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## A self-square number

[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 , , , | 12 Comments

## More fun with infinite decadic numbers

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 | | 4 Comments

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 , , | 7 Comments

## Book review: Roads to Infinity

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 , , , | Comments Off on Book review: Roads to Infinity

## Rational numbers and decimal expansions

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 , , | 9 Comments

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

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

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