Last time, we saw how Newton’s method works. Here, we’ll look at a question about why it might not work, which will lead to a deep examination of how iterative methods work in general, from which we will discover why Newton’s method is as good as it is. I have to say, as I read …
When Newton’s Method Fails – and Why It’s So Good When It Doesn’t! Read More »
Last week we looked at some basics about zero; now let’s look at whether zero is positive or negative, and then at the topic of the recent comment that triggered this series: whether zero is even or odd.
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.
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.
Last week we looked at what functions are; but many students wonder why it all matters. What makes them useful? What makes functions worth distinguishing from non-functions? Why do we make the distinction we do? We love “why” questions, because they make us think more deeply!
(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 »
(A new question of the week) We’ve looked at the concept of limit of a function from several perspectives, including why they are needed, and what the definition means. Here we have a more fundamental question, which applies to both functions and sequences: What do we mean when we say a value approaches some number? …
Last week we looked at a recent question about recurrence relations, and I realized it needs a companion article to introduce these ideas. So here we will look at some answers from Ask Dr. Math about the simpler case, including general methods, why they work, and applications.
One of the more common questions we’ve been asked is, How can the product of two negative numbers be positive? Between this post and the next, I’ll put together many of the answers we have given, starting here with examples from the “real world” (gradually getting more abstract), and next time we’ll look at proofs. …
Negative x Negative = Positive? Concrete Illustrations Read More »
(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …