A poster on the Google Plus Mathematics community commented that one feature of the Golden Ratio Ï• is that adding one to Ï• yields the same value as squaring Ï• does. That is, \[\phi^2 = \phi + 1\] He was surprised that there would be such a number. While this is indeed an interesting attribute…
Author: Clio
Negative Bases
And now, for something silly. In general, number bases are expected to be positive integers greater than one. The most widely used are decimal (because we have ten fingers and ten toes), binary (how computer data is stored), hexadecimal (a more convenient way of writing binary), and octal (base eight), but, mathematically speaking, there’s no…
What Do Digits Mean, Anyway?
Puzzle I found this puzzle in the G+ Mathematics community, courtesy of Paul Cooper. Solve the final addition: 50 + 60 + 90 = 380 30 + 40 + 60 = 330 90 + 60 + 70 = 350 50 + 90 + 30 = 10 70 + 30 + 20 = 370 40 +…
A pair of probability problems
I’ve recently come upon two probability problems with counterintuitive solutions. One I’d seen before and dismissed because I didn’t understand the write-up (mea culpa); the other is new to me. Born on a Sunday Puzzle: You are introduced to a randomly selected family that happens to have two children. If one is a girl that…
All Lines are Congruent
A standard high school geometry textbook talks about congruence in terms of three types of objects: Line segments, angles, and polygons. Congruence is then defined in terms of measurable parameters: “Two figures are congruent if they have the same size and the same shape” (Carnegie’s Bridge to Algebra Student Text, 2008, p. G-9). Math Open Reference…
GeoGebra Tutorial: Golden Ratio / Power of a Point
Introduction In my previous post, I included this image, which I’d made in GeoGebra. The image satisfies the conditions of the problem: \(AD\) is tangent to \(\odot P\) and \(\overline{BC} \cong \overline{AD}\). In order to create this image, I created a dynamic GeoGebra image where A, B, P and the radius of P can be…
The Golden Ratio and the Power of a Point Theorem
The Golden Ratio By definition, the Golden Ratio is a ratio involving overlapping line segments. Given collinear points A, B, and C, such that B is between A and C, if the ratio between the two subsegments is the same as the ratio between the entire segment and the longer segment, then that ratio is…
Schrödinger’s Brat and 3-Door Monte
The Monty Hall problem persists in Internet mathematics discussions, as if its results are somehow spectacularly unique or mystifying. Here is the problem: You are on a game show and are presented with three doors. Behind one door is some wonderful prize, and behind the other two is a goat (or something else of negligible…
Proof of the Power of a Point Theorem
I had to dig for a bit to find a complete proof for each part of the Power of a Point Theorem, so I thought it would be useful to compile my own proof. The Power of a Point Theorem states: Given a point P and a circle C, any line through P that intersects…
Math Needs Better PR
I was recently reading a book on Greenfoot, a Java-based GUI intended for teaching programming to high schoolers and college underclassman. In the “Newton’s Lab” project, the writer assuaged the reader who might be leery of the mathematics in that particular project. Remember, the reader was told: Programming can do a variety of things, including…