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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 secondsemester calculus class, but it’s … Continue reading
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
utube
[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 selfsquare 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 decadic, idempotent, self, square
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
Tagged decadic, decimal, fractions, integers, representation
4 Comments
Infinite decadic numbers
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
What is infinity? What is proof? These are two of the biggest questions mathematicians have grappled with over the years. In this wellwritten 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 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
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