Last week we looked at how to “cast out nines” to check arithmetic, and touched only briefly on its relationship with modular arithmetic and remainders. Here we’ll look at several explanations of why it works, aimed at different levels of students, with varying levels of success..
Last time we looked at how to find the volume of a frustum of a pyramid or cone. But sometimes what looks at first like a rectangular frustum actually isn’t. This case turns out to have a more general formula almost as nice as what we have for an actual frustum. We’ll discover that the …
We’ve looked in the past at volumes and surface areas of familiar geometric shapes like spheres, pyramids, and cones; but more can be done. If we cut parallel to the base of a pyramid or cone, the result is called a frustum (no, not a frustrum!). Let’s derive some formulas, which will be remarkably simple.
When we recently looked at the Chain Rule, I considered including two questions about its proof, but decided they would be too much. However, when a recent question asked about a different version of the same proof, I decided to post all three. It is a nice illustration of how a mathematician’s view of a …
Last time, we considered the Chain Rule for derivatives. This time, we’ll look at the product and quotient rules, focusing on how to keep the formulas straight, and make them easier to apply. We’ll look primarily at the quotient rule to start with, and then examine the product rule at the end.
A recent comment on the site raised questions about zero, beyond what we have discussed in the past about division by zero. Here we’ll look at basic questions about whether zero is actually a number at all, and then about multiplication by zero, which confuses a lot of people.
Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.
Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …
Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.
(A new question of the week) In an ellipse, \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with focal distance c, parameters a, b, and c all make natural sense, and it is easy enough to see why \(a^2 = b^2 + c^2\). But in the hyperbola, \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the equivalent relationship, \(a^2 + b^2 = c^2\), is not nearly as natural, nor …