Category Archives: algebra

A simple proof of the quadratic formula

Posted on December 23, 2019 by Brent

If you’re reading this blog you have probably memorized (or used to have memorized) the quadratic formula, which can be used to solve quadratic equations of the form But do you know how to derive the formula? Usually the derivation … Continue reading →

Posted in algebra, proof | Tagged , , , , | 3 Comments

Fibonacci’s Problem of the Birds

Posted on April 1, 2017 by Brent

I have been enjoying reading Keith Devlin’s new book, Finding Fibonacci. I’ll write more about the book later. For now, I just wanted to share a nice problem I learned about which Leonardo Pisano, aka Fibonacci, included in his book … Continue reading →

Posted in algebra, arithmetic, challenges, people | Tagged , , | 6 Comments

Sigma notation ninja tricks 2: splitting sums

Posted on March 18, 2017 by Brent

[Previous posts in this series: jumping constants] Trick 2: splitting sums I’ve written about this before, but it’s worth spelling it out for completeness’ sake. If you have a sum of something which is itself a sum, like this: you … Continue reading →

Posted in algebra, arithmetic | Tagged , , , , , | 3 Comments

Sigma notation ninja tricks 1: jumping constants

Posted on March 11, 2017 by Brent

Almost exactly ten years ago, I wrote a page on this blog explaining big-sigma notation. Since then it’s consistently been one of the highest-traffic posts on my blog, and still gets occasional comments and questions. A few days ago, a … Continue reading →

Posted in algebra, arithmetic | Tagged , , , , | 1 Comment

MaBloWriMo 30: Cyclic subgroups

Posted on December 1, 2015 by Brent

Today, to wrap things up, we will use Lagrange’s Theorem to prove that if is an element of the group , the order of evenly divides the order of . So we have a group and an element . In … Continue reading →

Posted in algebra, group theory, proof | Tagged , , , , , | 6 Comments

MaBloWriMo 29: Equivalence classes are cosets

Posted on November 30, 2015 by Brent

Today will conclude the proof of Lagrange’s Theorem! Recall that we defined subgroups and left cosets, and defined a certain equivalence relation on a group in terms of a subgroup . Today we’re going to show that the equivalence classes … Continue reading →

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MaBloWriMo 28: Equivalence relations are partitions

Posted on November 29, 2015 by Brent

Today we’ll take a brief break from group theory to prove a fact about equivalence relations, namely, that they are the same as partitions. A partition is a pretty intuitive concept: you take a big set, and cut it up … Continue reading →

Posted in algebra, group theory, proof | Tagged , , , , , , | 2 Comments

MaBloWriMo 27: From subgroups to equivalence relations

Posted on November 28, 2015 by Brent

Again, let be a group and a subgroup of . Then we can define a binary relation on elements of , called , as follows: if and only if there is some such that . That is, for any two … Continue reading →

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MaBloWriMo 26: Left cosets

Posted on November 27, 2015 by Brent

Let be a group and a subgroup of . Then for each element we can define a left coset of by . That is, is the set we get by combining (on the left) with every element of . For … Continue reading →

Posted in algebra, group theory, proof | Tagged , , , , , | 1 Comment

MaBloWriMo 25: Subgroups

Posted on November 26, 2015 by Brent

So in the remainder of the month, we’ll prove that in any group , the order of each element must evenly divide the order (size) of the group. I said in an earlier post that this is called Lagrange’s Theorem; … Continue reading →

Posted in algebra, group theory, proof | Tagged , , , , | 1 Comment