Category Archives: algebra

A computer-checked proof of the odd order theorem

Posted on November 11, 2012 by

Big news: a proof of the Feit-Thompson Theorem (also known as the “odd order theorem”) has been completely formalized and verified by a computer, using the Coq proof assistant! Wait, what? Huh? you’re probably thinking. Well, let me unpack that … Continue reading →

Posted in algebra, links, programming, proof | Tagged , , , , , , | 10 Comments

Sigmas and sums of squares

Posted on November 29, 2011 by

Commenter Rachel recently asked, How would you find the sum of ? See here for an explanation of sigma notation—in this case it denotes the sum Of course, for any particular value of we can just plug in values and … Continue reading →

Posted in algebra | Tagged , , , , | 6 Comments

Cassini’s identity

Posted on May 15, 2011 by

My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading →

Posted in algebra, fibonacci, induction, pattern, proof, solutions | Tagged , , | 8 Comments

A Fibonacci pattern

Posted on May 13, 2011 by

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading →

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences | Tagged , , | 4 Comments

Irrationality of pi: the integral that wasn't

Posted on February 11, 2010 by

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: method 2

Posted on June 11, 2009 by

A little while ago I wrote about the Babylonian method for approximating square roots with pencil and paper. In that post I noted that the Babylonian method is quite efficient, but annoying in some ways since it makes you deal … Continue reading →

Posted in algebra, computation | Tagged , | 8 Comments

Square roots with pencil and paper: the Babylonian method

Posted on May 18, 2009 by

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

Rational and irrational numbers

Posted on August 31, 2008 by

Most readers of this blog probably know what a rational number is: it’s a number that can be represented as a ratio (hence rational) of two integers. In other words, a fraction. Examples are 3/4, 99/2, and -20837/231, and even … Continue reading →

Posted in algebra, challenges, number theory, proof | Tagged | 11 Comments

Carnival of Mathematics #23: Haiku Edition

Posted on December 28, 2007 by

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 →

Posted in algebra, books, calculus, challenges, counting, fractals, geometry, infinity, links, meta, number theory, pascal's triangle, pattern, people, sequences, trig, video | 17 Comments

Perfect numbers, part III

Posted on November 27, 2007 by

This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the … Continue reading →

Posted in algebra, number theory, open problems, primes, solutions | 11 Comments