Category Archives: convergence

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 , , , , | 2 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 , , | 7 Comments

What does “close to” mean?

Posted on September 22, 2011 by Brent

Continuing from last time, consider the (normal, decimal) number with an infinite number of 3’s after the decimal point. Now, you probably know that this represents . But why? How do we define what such an infinite sequence of digits … Continue reading →

Posted in convergence, number theory | Tagged , , , , | 3 Comments

Irrationality of pi: the integral that wasn't

Posted on February 11, 2010 by Brent

And now for the punchline! Today we’ll show that, for large enough values of , completing the proof of the irrationality of . First, let’s show that is positive when . We know that is positive for . But I … Continue reading →

Posted in algebra, calculus, convergence, famous numbers, proof, trig | Tagged , , , , | 8 Comments

Square roots with pencil and paper: the Babylonian method

Posted on May 18, 2009 by Brent

Everyone knows how to add, subtract, multiply and divide with pencil and paper; but do you know how to find square roots without a calculator? (Incidentally, I highly recommend reading The Feeling of Power by Isaac Asimov, a short story … Continue reading →

Posted in algebra, computation, convergence, iteration, number theory | Tagged , , , | 10 Comments

Predicting pi: pretty graphs and convergents

Posted on January 16, 2009 by Brent

Recall the challenge I posed in a previous post: given the sequence of integers , what can you learn about (assuming you didn’t know anything about it before)? The answer, as explained in another post, is that you can learn … Continue reading →

Posted in convergence, famous numbers, pattern, sequences | Tagged , , , | 5 Comments

Predicting Pi: solution

Posted on October 23, 2008 by Brent

Now for the solution to the question in my previous post, which asked what you can learn about , given the sequence of integers . Nick Johnson commented: Well, the obvious thing one can learn given just |(10^n)r| is the … Continue reading →

Posted in convergence, pattern, sequences, solutions | Tagged , , , , | 5 Comments

The Mandelbrot Set

Posted on May 8, 2006 by Brent

For those of you already familiar with the Mandelbrot Set, I suppose this will be like visiting with an old friend. For those of you who aren’t — you’re in for a treat! Okay, you say, that looks pretty cool … Continue reading →

Posted in convergence, fractals, infinity, iteration | 1 Comment

Convergence

Posted on March 20, 2006 by Brent

Let’s dig a little deeper behind the solutions to Challenges #1 and #2. What on earth does it mean for an infinite expression to have a “value”? Well, as noted in the solution to Challenge #1, what we’re really talking … Continue reading →

Posted in convergence, infinity | 3 Comments