Back in January, I discussed the issue of division by zero. There is a special case of that that causes even more trouble, in every field from arithmetic to calculus: zero divided by zero. I’ll look at several typical questions that we answered at different levels.
(A new question of the week) I want to look at a question that came in recently that is, in one sense, very simple, but at the same time is quite challenging. It was given to a 12-year-old whose father asked us about it, and requires some skill in thinking about non-routine problems.
(A new question of the week) It can be an interesting challenge to be presented with a formula and asked how it was derived. This becomes a bigger challenge when the formula is only approximate, so we have to figure out how to arrive at this particular approximation. But it is impressive when several different …
Distances to an Arc: Exact and Approximate Formulas Read More »
(A new question of the week) We have had a number of challenging questions about inequalities recently. I want to show one of those here, because it involved a useful discussion about how to prove them.
(An archive question of the week) Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. A mixture of tools will make the work easier. Here it …
Square roots commonly are irrational numbers, so that it is necessary to estimate them. Usually today, we use a calculator to find them. Many students, however, are curious about how they could do it without a calculator. Here I want to reverse the usual format of this blog, presenting a summary I have written that …
(An archive question of the week) Last time, as part of our series on estimation, we looked at some numerical methods for solving equations approximately. I mentioned the Method of False Position, but when I looked for more detailed expositions in our archive, I realized that in a sense it is really two different things, …
The problems students see in class are usually only those that can be solved by the methods they have been taught. Too many students conclude that algebra can solve anything! But the reality is that if you just wrote an equation at random, it probably could not be solved algebraically. When students ask us about …
Numerical Approximation Methods: When Algebra Doesn’t Work Read More »
I want to close out this series on multiplication with a very different kind of question. We have seen that multiplication of natural numbers can be modeled as a repeated sum of the multiplicand, taken the number of times indicated by the multiplier; and that the terms “multiplier” and “multiplicand” reflect only this model, not …
To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?