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

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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

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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

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Sigma notation ninja tricks 1: jumping constants

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

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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

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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

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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

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