What’s the Difference? How the Meaning Gets Twisted
A recent question about the precise meaning of “difference” led me to some past discussions of the word.
Algebra
Is There More Than One Standard Form for an Equation?
A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.
A Cubic Challenge
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …
Internal and External Division of a Segment
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
Dividing a Segment: Get the Order Right!
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …
What If: Inventing “Pseudo-Complex” Numbers
We’ll work through a textbook exercise that encourages students to discover what it’s like to invent a new number system, as well as why some ideas work but others do not. The topic: What would happen if we changed the definition of the imaginary unit i so that its square is 1 rather than -1?
A System of Exponential Equations
What do you do when you are given a problem that starts with a “lie” and ends with a wrong answer? We’ll go in several directions with this problem, a system of two exponential equations in two variables.
Fun With a Functional Equation
Looking back at interesting questions I had to skip over when there were too many to choose, I found this interesting discussion of a functional equation.
Proving a Radical Expression Is Rational
It can be tricky deciding how to approach a proof; this problem, whose answer requires going in a very different direction than you might expect, provides some interesting insights into the nature of proof. The proof itself, in fact, is far less interesting than the process of getting there!
Finding the Length of a Rolled-up Carpet
One of the more frequent questions we had on Ask Dr. Math was about how to find the length of material (carpet, paper, wire, etc.) on a roll, knowing only the inner and outer diameters and something else: the thickness of the material, or the number of turns, or the original size of the roll. …