Polynomials: A Matter of Degrees
Last time we examined why polynomials are defined as they are. This time, let’s look at some tricky aspects of the concept of “degree”, mostly involving something being zero.
Definitions
Polynomials: Why Are Terms What They Are?
A question last week (Hi, Zahraa!) led me to dig up some old discussions of how we define a polynomial (or monomial, or term) and, specifically, why the exponents have to be non-negative integers. Why can we only multiply, and not divide by, variables? Since we’ve been looking at polynomials, let’s continue.
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 …
Clarifying Definitions: Triangle, Rectangle, Circle Read More »
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 …”.
Partial Fractions: Complex and Trivial Cases
(A new question of the week) We discussed four years ago how to make a partial fraction decomposition of a rational function, and why it can always be done; a question from mid-May brings up two side issues: when you can factor the denominator, and whether a trivial decomposition, which takes no work at all, …