Category Archives: infinity

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

Posted in computation, convergence, infinity, iteration, modular arithmetic, number theory, programming | Tagged , , , , | 2 Comments

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 | Tagged , , , , | 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 , , | 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 , , ,

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