Geometric and Algebraic Meaning of Determinants

A recent question led me to look back in the Ask Dr. Math archives for questions about the definition and deeper meaning of determinants. Next week, we’ll see another old question for additional background, followed by the new question.

What are Length and Width?

One of the recent discussions I showed last week dealt with the meaning of length, and I promised more about that. Here we will look at some older questions about the ambiguity of words  like length, width, depth, and height.

Clarifying Definitions: Triangle, Rectangle, Circle

(A new question of the week) Several recent questions involved details about definitions of geometrical objects, so I thought I’d group them together, because each is relatively short. We’ll be looking at the definitions of triangles (do we need to say “exactly three sides”?), rectangles (did Euclid use an exclusive definition?), and circles (can the …

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Prime Numbers: What About Negatives?

We’ve looked at the basic idea of primes, then at where 0 and 1 fit in. But what about negative integers? Can they be prime? If so, how does that affect the definition? And can you factorize a negative number if you don’t have negative primes?

Prime Numbers: What About 0 and 1?

Last week we looked at the definitions of prime and composite numbers, and saw that 1 is neither. The same is true of 0. What, then, are they? That raises some deep questions that we’ll look at here.

Prime Numbers: What and Why

I’ll begin a short series of posts on prime numbers with several questions on the basics: What are prime (and composite) numbers, and why do they matter?

How Imaginary Numbers Became “Real”

Last week we started a series on complex numbers, looking at how we introduce the concept. This time I want to look more at the actual history of the idea, leading to how mathematicians were able to define complex numbers without saying “Just suppose …”.

Fine Points on Polygons and Polyhedra

Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?

Degenerate Polygons

We’ve been looking at degenerate figures, starting with the most interesting case, degenerate conic sections. But other things can also be degenerate, so we should take a look at some of these, which perhaps arise even more often. We’ll examine triangles that aren’t triangles, rectangles that aren’t rectangles, and bigger polygons – or smaller polygons! …

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