Intuition

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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More Than 100 Percent?

One of the questions we looked at in our recent survey of percent change problems involved percentages over 100%, which often confuse students. How can anything be more than 100%? Let’s look at a couple questions about that issue. No such thing? Take this question from 1999: More Than 100 Percent Please help to settle …

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When Math Doesn’t Make Sense

(Archive Question of the Week) One of my favorite questions, from 2001, asked about how to convince a skeptical friend, when a clear mathematical result goes against their intuition. Why should they believe the math? It led me into thoughts about the relationship of intuition to math, whether (and when) math can be trusted, and …

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Should Rare Events Surprise Us?

(Archive Question of the Week) We often receive questions saying something like, “My wife has the same birthday as my son-in-law, whose dog has the same name as my wife’s brother, … ,” and asking us to calculate how likely it is that this would happen. Sometimes it is simply impossible to answer (without, say, looking up …

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