This problem was brought to my attention on G+, but I wasn’t satisfied with the solution presented. There are actually two versions, the one that was originally presented on G+ and the corrected one that matches the standard version. I’ll discuss the standard version first. Standard version: All seats on an airplane are assigned. However,…
Category: Mathematics
The History of Factorials: Kramp
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…
Infinity and String Theory
There’s a Numberphile video that’s hurting people’s brains. It claims to prove (several ways, including in a companion video) that \[\sum_{n=1}^{\infty} n = -\frac{1}{12}\] This is, of course, highly counterintuitive. The video itself is misleading in that the speakers refer to the sum of all natural numbers, when this is not in fact the sum…
Probabilities: Consecutive numbers
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…
The Quadratic Formula and the Shortcut
Background There are two common ways for finding roots of quadratic equations, that is, equations of the form \[ax^2 + bx + c = 0\] The one that’s usually taught first is a shortcut that works best when \(a = 1\) and two factors of \(c\) have a sum of \(b\): In fact, that’s exactly…
The square of a sum and the sum of cubes
Let’s start with a fun fact: Take the first \(n\) integers. Add them and then square the result; call this \(S\). Take the cube of each and add those numbers; call this \(C\). It will always be the case that \(S = C\). For instance, take \((1, 2, 3)\): \[(1 + 2 + 3)^2 =…
Factorials and the meaning of “is”
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…
Numerators and denominators
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,…
0.999… = 1 and Zeno’s Paradox
Overview One surprisingly difficult concept for many students of mathematics is understanding that 0.999… (more properly depicted as \(0.9\overline{9}\)), that is, a decimal with an infinite number of 9s, is equal to 1. There are various proofs of it, and various arguments against it. Below, I’m going to present a discussion of this problem in…
Negative numbers squared
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…