Proof

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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Polygon Coordinates and Areas

We’ve been collecting techniques for finding areas of polygons, mostly using their side lengths. We started with triangles (Heron’s formula), then quadrilaterals (Bretschneider’s formula and Brahmagupta’s formula), and the fact that the largest possible area is attained when the vertices lie on a circle. We’ll look at one more way to find area, using coordinates …

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Building Patterns and Sequences

In the past (last May and November), we discussed ways to find patterns or sequences in numbers, sometimes leading to a formula. This included an example where the sequence turned out not to be just a provided list of numbers, but a process that generated the numbers. I want to focus on that type of …

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L’Hôpital’s Rule: What and Why

The next few posts will look at a powerful technique for finding limits in calculus, called L’Hôpital’s Rule. Here, we’ll introduce what it is, and why it works. In the next post we’ll examine some harder cases. Indeterminate forms The method we will be discussing is used to find limits that have an indeterminate form. …

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