Pythagorean Theorem Converse Proof

I've already given a proof for the Pythagorean Theorem here; it's a fundamental result in high school mathematics that most high school math teachers work with. The converse of the Pythagorean Theorem, however, gets a lot less attention. That's a shame, because the proof isn't difficult and students are supposed to learn the difference between a statement and its converse.

I've posted a proof to the converse of the Pythagorean Theorem on the Handouts page. The proof uses the diagram (shown above). The tex file for the diagram along with the cropped PDF output can be found on the Graphics page.

Problem: combinatorics

Start with a grid of whatever dimensions you want; for example, 4 by 6.

How many squares are contained in the grid? How many rectangles?

To create your own grids, start with the tex file for graph paper and adjust it as you see fit. I've added this to the Problems page.

 

Sage Interact: Goldbach's Conjecture

If you talk about proofs with your math classes then you have to talk about mathematical conjectures as well. Conjectures are results that mathematicians suspect might be true. One of the oldest and most famous is Goldbach's Conjecture: Every even integer can be written as the sum of two primes.

Of course, thinking a result might be true and proving that result to be true are 2 completely different things but computers are a useful tool that can check whether the conjecture is reasonable. As the Wikipedia link indicates, the problem goes back to 1742 and the result is also known as the Strong Goldbach Conjecture. That link also mentions that computers have verified the conjecture for values of n up to $latex n\leq1.609\times10^{18}$.

So computers have shown there is no counterexample to Goldbach's Conjecture up to $latex n\leq1.609\times10^{18}$; this still doesn't prove the result. Despite their amazing speed, computers can help us find a counterexample but can never prove conjectures that assert something is true/false for all integers. That's where mathematicians come in. I put together a Sage Interact manipulative that gives you two options: verify that Goldbach's Conjecture works for a specific (even) number or verify Goldbach's Conjecture for all the even numbers less than or equal to it.

The code for the manipulative is posted on the Python/Sage page; this is the output I got from running it on the Sage Cell Server:

I've put the output on the Sage Output page.

Japanese Line Functions (Amidakuji)

They're known as Amidakuji, Ghost Leg, Japanese Ladder Games, ladders, and maybe some other names, too. When my students first asked me about them they didn't know the name and I was unable to find them using the internet. I ended up referring to them as Japanese line functions. Several years later I eventually stumbled upon them; I've worked "Japanese line function" into the title to help anyone else who's having difficulty searching for these interesting structures.

In the US we flip a coin to decide on two outcomes, Amidakuji can be used to do the same thing when there are many outcomes. It's even found its way into video games, at least the basic structure of it. The rules are kind of loose at the websites I've seen and if you hunt around you'll find examples without horizontal rungs. I'm not sure about that last site. Here's how to construct them: draw a finite number of vertical lines followed by a finite number of horizontal lines drawn according to these rules:

A horizontal line starts at one vertical line and ends at the next vertical line.

Horizontal lines never connect the endpoints of the vertical lines.

Here's an example:

I've written up a more detailed explanation of how they are created along with the rules of movement. The tex file and PDF output are posted on the Handouts page. Four Amidakuji diagrams (3 from the article and 1 extra demonstrating another path in the example) are found on the Graphics page.

The basic idea is easy; there's a video explaining them here (until about the 3 minute mark). Likewise, here's a great website that illustrates the paths that take you from the top to bottom. Note that refreshing the page results in another Amidakuji, a very nice feature.

Whatever you decide to call them, you can tell from my description "Japanese line functions" that Amidakuji is an example of a function that you can see. In fact, every Amidakuji is a 1-1 function. The 1-1 nature is supposed to come from observing that (starting with just vertical lines) each horizontal "rung" added swaps the outcome of the letters on the vertical lines the "rung" connects. More about that later. If you're teaching functions, spend some time on Amidakuji; students like it because it's concrete and unlike anything they've seen before.

 

The origins of zero

I still don't know what classes I will teach for the upcoming school year so I'm spending more time looking at the history of mathematics. I was reading about the origins of zero and it's not as straightforward as you might think. First you have to figure out what you mean by zero: using 0 (or some other symbol) as a placeholder but not a number, using some "zero like" symbol as a number, or using the symbol 0 as a number. The idea of zero as a placeholder is generally accepted to go back to around 350 BCE though, as this article from Scientific American indicates, there is evidence that leads some to think it might go back to the Sumerians around 3000 BCE. The idea of 0 as a number is traced, according to the article, back to India around 450 ADE.

From India, the development our number system was picked up by the Persian and Arabic mathematicians around 825 and by the 10th century the decimal point was added. The number system is then brought to the west by Fibonacci in 1200 through his book "Liber Abacci".  Initially the Greeks weren't sure whether zero was a number but by 130 they had a "zero like"-symbol which functioned as zero so I've revised my earlier post which included 0 as one of their numbers.