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Category Archives: convergence
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
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
What does “close to” mean?
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 absolute value, Cauchy, distance, limit, sequence
3 Comments
Irrationality of pi: the integral that wasn't
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 inequality, integral, irrational, Niven, pi
8 Comments
Square roots with pencil and paper: the Babylonian method
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 Babylonian, method, pencil and paper, square root
10 Comments
Predicting pi: pretty graphs and convergents
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 approximation, convergents, graphs, pi
5 Comments
Predicting Pi: solution
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 approximants, approximation, floor, pi, sequence
5 Comments
The Mandelbrot Set
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
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