Proof

False Proofs: Geometry

We have been looking at some classic “false proofs” or “fallacies”, where a seemingly valid proof shows something clearly false to be true. The goal is to learn from these, how to distinguish a valid proof from an error. In a post from last year, What Role Should a Figure Play in a Proof?, I …

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False Proofs: Complex Numbers

Last time we looked at some false proofs, which are often used to help students understand what does and does not constitute a valid proof, and in particular, to remind them to be careful in algebraic proofs, looking for issues like division by zero and taking square roots. This time, we’ll look at two similar …

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Why Do We Need Proofs?

One aspect of mathematics that students often struggle with, particularly in geometry (which traditionally has been where proof is introduced), is writing proofs. Why do we need to learn about proofs? Why are proofs needed in the first place? Here are a few answers we’ve given to these questions. Why does math need proofs? First, …

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More on Uncountable Irrationals

(An archive question of the week) While I was researching for the post on uncountable sets, I ran across a discussion that didn’t quite fit, but raises interesting questions about how countable and uncountable sets can fit together. How can the rational numbers be countable, but the irrational numbers, which are closely intertwined with them, …

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Frequently Questioned Answers: Uncountable Infinities

We could continue forever discussing questions whose answers are frequently questioned; but let’s finish by looking at infinity itself. The concept is impossible to fully grasp, because we are finite, and all of our experience is finite. Mathematicians have worked out ways to deal with infinity, though, and the results are often counter-intuitive. That means …

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More on 0.999…

(An archive question of the week) In collecting questions and answers about 0.999… for the last post, there were two that were too long to include, but that dig more deeply into issues that some of the standard answers tend to gloss over. So here, I want to look at those two answers, both of …

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Frequently Questioned Answers: 0.999… = 1

Having looked at two common questions in probability that are often challenged, let’s turn to the realm of numbers. Non-terminating decimals are inherently problematic, and one particular example causes difficulty for many, even after they fully accept the mathematics of it. Our FAQ page on this topic, at 0.9999… = 1, is very brief, and …

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