Intersecting Powers and Roots
Here is an interesting little question. Its answer is simple, and not hard to see just by graphing examples; yet the algebra is easy to get wrong, as we’ll see several times. And subtle errors deserve study.
NQOTW
Adjoints and Inconsistency: A Questionable Test
(A new question of the week) Last week we looked at how the adjugate matrix can be used to find an inverse. (This was formerly called the [classical] adjoint, a term that is avoided because it conflicts with another use of the word, but is still used in many sources.) I posted that as background …
Determinants, Adjugates, and Inverses
(A new question of the week) Looking for a new topic, I realized that a recent question involves determinants, and an older one provides the background for that. We’ll continue the series on determinants by seeing how they can be used in finding the inverse of a matrix, and how something called the adjugate matrix …
How Can 3×3 Determinants Give Both Area and Volume?
(A new question of the week) A recent question asked for the connection between two different ways to use determinants geometrically: to find the area of a triangle, and to find the volume of a pyramid (or the area of a parallelogram and the volume of a parallelepiped). Last time we looked at what a …
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Diluting a Solution: Math vs. Reality
Here is a little question about making a formula to dilute a solution; we’ll see how to do the algebra, and also how what we teach in math classes isn’t quite real.
Function Transformations as Composition
(A new question of the week) We have discussed transformations of functions and their graphs at length, but a recent question suggested a slightly different way to think about them.
Subtleties in a Radical Limit Problem
(A new question of the week) I find it interesting to observe the process of problem-solving, particularly for proofs: how we discover a solution initially, and then how we turn that into a final answer. Sometimes we can see the main idea in a flash, but the process of writing it as a formal proof …
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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Experimenting with Triangles and Circles
(A new question of the week) I intended to fit three problems into last week’s post, but the third was too interesting to shorten, so I’m posting it separately. The problem itself is not hard, but in looking for a more direct solution, we extend it, discovering (through geometry software) more general facts, which lead …
Two Triangle and Circle Problems
(A new question of the week) Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.