At some point, walking along the snow-flocked train tracks in the winter evening’s half-light, I became aware. By this I mean: All that I was was now. I had no past to dwell within, chiding myself, steeped in regrets. I had no future lingering in the wings like a Dickens villain, ready to set me…
Author: Clio
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 =…
The Grandfather Tree
I never thought I would give you permission to cut down that giant oak tree in the back of the property, the one that had led my grandfather to buy this land in the first place. It had been tall even when he’d seen it as a young man only a year into his first…
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,…
the daze of winter
i saw the daze of winter fading from the eyes of the downtrodden as a fire had been kindled afresh in the dying embers of august’s barbecue a bare-footed retinue flexed their toes in the muddy spring stretched their arms yawned out their souls and dug in again backs laden with the vexing hope of…
In the mirror
I saw my face in the mirror, in passing, and it was someone else. I didn’t recognize the eyes, or the hair, or the point of the nose. But that wasn’t it, because I never do. There was something different. A light, a candle flame, that used to flicker. It was gone. I stopped to…
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…