# Mathematics

## “Two Kinds” of Zero: Same But Not The Same?

I recently got into a protracted discussion in which the other person insisted that the fact that the character 0 is used in place value notation is merely a place holder is evidence that zero is not a number, but…

## Euclid’s proof of infinite primes

It has been known since at least Euclid’s time that there are an infinite number of prime numbers. Here is his basic proof: Imagine that there is a finite set of prime numbers, P. Let N be the product of…

Mathematics

I’m exploring if it’s possible to create a function in GeoGebra that would take an integer as input and create a simplified radical as output. For instance, it would take $$20$$ as input and return $$2\sqrt{5}$$ as output. I don’t…

## Indeterminate vs. Undefined

Here’s something that seems to confuse many people: $\frac{1}{0} \text{ is undefined}\\ \frac{0}{0} \text{ is indeterminate}$ If some number, any number at all, divided by zero is undefined, then why isn’t zero divided by zero likewise undefined? And what does…

Mathematics

## The Six Basic Trigonometric Functions

I read an article today on the six basic trigonometric functions, and I thought there was a particularly important insight that I wanted to present in my own words. When I was in school, we learned the six basic trigonometric…

## How Many Factors?

A post on G+ Mathematics asks: “How many of the positive divisors of 8400 have four or more positive divisors?” A divisor, or a factor, is an integer which evenly divides another integer; in other words, it is the opposite…

Mathematics

## Modeling in GeoGebra

Introduction In this entry, I’m going to demonstrate the use of GeoGebra to estimate a value for a fairly tricky trigonometry problem, then illustrate how to find the value using trigonometry and an appeal to WolframAlpha. In so doing, I…

## Angles and Congruence

Congruence As I discussed in an earlier post, there are two basic definitions of geometric congruence that are presented to students. The first is based on measurement: Definition 1. Two objects are congruent if all of their measurements are the…

Mathematics

## Programming, Mathematics, and Language

I’ve been struggling for a while now to find a way to frame and articulate the answer to what seems like a simple question: “What is mathematics?” At the same time, I’ve been seeking to layout the similarities and differences…

## The Golden Ratio and Generalizing Quadratics

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…

## 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…

Mathematics

## 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…