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…
Author: Clio
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…
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…
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 zero not being a value because there’s no point in counting zero of everything; after…