Category: Algebra

This is my translation of Gerono’s 1877 proof listing all the possible solutions (x, y) for the equation \(y^2 = x^3 + x^2 + x + 1\). “Solutions to questions posed in The New Annals: Question 1177.” MM. Gerono, Nouvelles annales de mathématiques. Journal des candidats aux écoles polytechnique et normale, second series, volume 16…

A poster on the Google Plus Mathematics community commented that one feature of the Golden Ratio ϕ is that adding one to ϕ yields the same value as squaring ϕ does. That is, \[\phi^2 = \phi + 1\] He was surprised that there would be such a number. While this is indeed an interesting attribute…

Introduction and Terms Recently, a post on the G+ Mathematics community involved how to determine \(x\) and \(y\) when: \[3x + 5y = 12 \\ x + y = 2\] This is generally referred to as simultaneous equations or a system of equations. As a general rule, for such a problem to be solvable, you…

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…

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 traditional way of teaching the multiplication of binomials is FOIL: First, Outside, Inside, Last. For instance: \[(x + 3)(2x – 5) = (x)(2x) + (x)(-5) + (3)(2x) + (3)(-5) \\ = 2x^2 -5x + 6x – 15 \\ = 2x^2 + x – 15\] This works well enough for binomials, but for more complicated…

Factoring a quadratic equation involves finding two linear equations whose product is the quadratic equation. This is an example where mathematics teachers often act as if (a) there is one method of solving and (b) there is one solution. The AC Method The “one method of solving” strategy usually goes something like this: If the…

Edit: The bit about the larger prime numbers was due to an error in my VBA programming, but it lead to a better understanding of the problem. Don’t take this article as “final”, is the point. This week’s puzzle in my Mathematics Reasoning class: Try to express positive integers in terms of the sum of…