## 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…

Mathematics

## 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…

Mathematics

## 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…

## 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…

## 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…

Mathematics

## 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…

## 10101 and 11011 are never prime

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…

## Just forget my Dear Aunt Sally

The purpose of a mnemonic is to make something easier to remember. Roy G. Biv represents the major colors of the spectrum (Red, Orange, Yellow, Green, Blue, Indigo, Violet); it has the weakness that most people tend to think of…

## Multiplying Polynomials

The traditional way of teaching the multiplication of binomials is FOIL: First, Outside, Inside, Last. For instance: \[(x + 3)(2x – 5) = (x)(2x) + (x)(-5) + (3)(2x) + (3)(-5) \\ = 2x^2 -5x + 6x – 15 \\ =…

Mathematics

## Hero’s Formula and Mirror Triangles

Here’s a problem with an interesting solution. You’re given two triangles, T1 and T2. The sides of T1 are 25, 25, and 30. The sides of T2 are 25, 25, and 40. Which has the greater area? The impulsive answer…

Mathematics