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Category Archives: algebra
A simple proof of the quadratic formula
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
Fibonacci’s Problem of the Birds
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
Sigma notation ninja tricks 2: splitting sums
[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
Sigma notation ninja tricks 1: jumping constants
Almost exactly ten years ago, I wrote a page on this blog explaining bigsigma notation. Since then it’s consistently been one of the highesttraffic posts on my blog, and still gets occasional comments and questions. A few days ago, a … Continue reading
MaBloWriMo 30: Cyclic subgroups
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 cyclic, groups, Lagrange, MaBloWriMo, proof, subgroups
6 Comments
MaBloWriMo 29: Equivalence classes are cosets
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
Posted in algebra, group theory, proof
Tagged classes, cosets, equivalence, groups, Lagrange, MaBloWriMo, proof
Comments Off on MaBloWriMo 29: Equivalence classes are cosets
MaBloWriMo 28: Equivalence relations are partitions
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 equivalence, groups, Lagrange, MaBloWriMo, partition, proof, relation
2 Comments
MaBloWriMo 27: From subgroups to equivalence relations
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
Posted in algebra, group theory, proof
Tagged equivalence, groups, Lagrange, MaBloWriMo, proof, relation, subgroups
Comments Off on MaBloWriMo 27: From subgroups to equivalence relations
MaBloWriMo 26: Left cosets
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 cosets, groups, Lagrange, MaBloWriMo, proof, subgroups
1 Comment
MaBloWriMo 25: Subgroups
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 groups, Lagrange, MaBloWriMo, proof, subgroups
1 Comment