(A new question of the week) This week we have a short discussion of a question that takes a basic concept one step further: How do you graph an equation on the plane, that contains only one variable? It’s a simple question when applied to linear equations, but takes on new dimensions when we generalize …
(A new question of the week) We were recently asked to check work on an interesting little question about parallel vectors, and I was almost convinced that there was no solution … until I realized there was one! How was it missed? How can we avoid doing that? That’s our goal today.
We’ve been looking at the commutative, associative, and distributive properties of operations, starting at an introductory level. But why are these properties important? Why do they have names in the first place? And what other operations have them?
In working on last week’s post, I realized that I haven’t yet covered the general idea of properties, such as commutative, associative, and distributive. Here I’ll collect some introductory answers on that topic. Next week, we’ll dig a little deeper.
(A new question of the week) This week, we’ll look at two recent questions about how parentheses (brackets) are used, how they relate to the properties we use in algebra that let us add or drop them, and the related concept of factoring a polynomial. They are examples of how student questions can touch on …
Parentheses and the Associative and Distributive Properties Read More »
(An archive question of the week) A recent question asked about a well-known problem about stacking books (or cards, or dominoes) so that the top one extends beyond the base, giving a link to one of many explanations of it – but one, like many, that doesn’t quite fill in all the details. Doctor Rick …
(A new question of the week) Extraneous roots can not only confuse the final solution to a problem; they can also make it harder to solve in the first place if you don’t deal with them early. Here is a relatively complicated rational equation, two questions about its solutions, and several ways to make it …
A Rational Equation, With and Without Extraneous Roots Read More »
(A new question of the week) Some problems can be done either by algebra or by basic arithmetic methods and some creativity; and although algebra generally makes work easier by making it routine, sometimes special-purpose thinking (once you have thought it!) can be quicker. Here we have a problem where a creative method didn’t quite …
(A new question of the week) We often see polynomials in a simplistic way, imagining that any function whose graph resembles a polynomial is a polynomial. Much as an attempt to mimic random data often lacks essential properties of genuine randomness, so what we intend to be a polynomial often is not. As we observe …
We’ve looked at positive integer powers of complex numbers, and imaginary powers of e. Now let’s look at roots of complex numbers, which will bring us to the idea that led to this series, namely the roots of unity.