Category: General

In an earlier post, I argued that the definition of factorial (\(n!\)) is the number of ways that a number of n distinct objects can be arranged. I have recently been told that, no, the formal definition of factorial is as generally offered by sources such as Wolfram and Numberphile: The product of all integers from…

On a mathematics community on Google+, Michal Nalevanko asked the question (paraphrased here, including my assumptions): Let us say there is a lottery game in which twenty numbered balls are pulled from a pool of eighty. What is the probability that three or more numbers will be consecutive? It is assumed that the numbers are…

In a YouTube video, James Grime of NumberPhile makes the claim that the meaning of the factorial is \[n! = \prod_{i=1}^n i\] for n > 0, and proceeds to explain why 0! = 1 using a recursive proof. This echoes what Wolfram Mathworld has to say on the subject: “The factorial n! is defined for a…

I remember as a child studying fractions, being told that the top was called the numerator and that the bottom was called the denominator, for reasons that were not made clear to me at the time. In retrospect, it’s possible that I was told and that it just didn’t make any sense to me anyway,…

Background Mathematical conventions represent the linguistic aspect of mathematics. One of the strengths of modern mathematics is the way in which we can represent some fairly complex ideas in a shortened, rigorous symbol set. However, as a result of these abbreviations, there are some ambiguities that are generally settled democratically: Some group decides that the…

One particularly tricky aspect of number sense is being able to separate the abstract notion of value from more concrete visual representations of numbers, and the even more concrete notion of countability. For instance, some people get caught up on zero not being a value because there’s no point in counting zero of everything; after…

An interesting question on the G+ Mathematics community I co-moderate asked about the difference between “numbers” and “numerals”. We wound up discussing this at a party I was hosting (which shows the sort of nerds we are), and this post is born from those discussions and my further thoughts. It seems to me we have…

One basic concept in mathematics that I see students struggle with, and with which I struggled myself, is the notion of expressions. However, when we remove the mathematical component, we can see that expressions behave much like a concept that natural language speakers deal with fairly easily. So let’s start there, in natural language. You…

In my previous post, I tackled this problem: Try to express positive integers in terms of the sum of two or more consecutive positive integers. For instance, 3 = 1+2, 9 = 2+3+4, and so on. For which numbers 1 to 25 is it possible to do this? I incorrectly concluded that there were no…