Definitions

Properties as Axioms or Theorems

To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?

Derivative of Arcsin: From the Definition

(A new question of the week) In Monday’s post about fallacies in calculus, one of them used the definition of the derivative (or rather, misused it). Today we’ll look at a short question about applying that same definition, that came in last month.

What is a Fraction, Really?

(An archive question of the week) One of the things I have learned as a Math Doctor is that it can be dangerous looking up a definition online. Sources vary — not because they are wrong, but because definitions depend on context, so you can easily find what appear to be contradictions because they refer …

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What Do Exponents Mean?

(New Question of the Week) We recently had a long discussion about a very common question from a somewhat different perspective: What do exponents (zero, negative, fractional, …) actually mean? The hard part, in the end, was to decide what “mean” means. What does it mean to define something in math? I will pick out the main thread of the …

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What is a Diamond?

(Archive Question of the Week) Having discussed various issues involving categorizing shapes, let’s take a look at a very different shape question, which didn’t fit into the last post. Is “diamond” appropriate? The word “diamond” is not a formal mathematical term; some people take it as equivalent to “rhombus”, while others equate it to “kite”, …

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What Is a Trapezoid? More on Inclusive Definitions

A month ago, I wrote about classifying shapes, discussing inclusive and exclusive definitions, and variations in different contexts. I promised to return to the subject, moving on to the specific issue of trapezoids, and some other related topics. Now is the time.

Is Area of a Square a Circular Argument?

(New Question of the Week) I love it when students want to know why something has to be the way it is, and are not satisfied just being told to use a formula. Last month, Shunya asked this kind of question, which gave me a chance to refer to our archive and go beyond it.

Why Isn’t Slope Run Over Rise?

Definitions are interesting in several ways. Sometimes they are essentially arbitrary; other times there is a very good reason for them, and understanding that reason can be helpful in understanding and using them. But they are usually taught just as something to memorize. Let’s think about why slope is defined as it is, and not …

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