A Rational Triangle Idea

Last week Sol at "Wild About Math" posted a link to the Wolfram Demo site related to a novel relationship that is not well known to students/teachers. For every rational number,q, on the open interval from 0 to 1, there is a Pythagorean Triple.

Actually the demo may oversell the idea a little, or perhaps I don't completely understand it, but I don't think it enumerates all the Pythagorean triples, but it is still a nice idea to introduce to students because many of the ideas can be proved with nothing beyond a good grasp of Alg II (for non-US folks, that's about a 14-16 year old students grasp of Algebra).

I had written about the relation back at the end of 2009. If you haven't read that one, and are not familiar with the idea, you might want to start there.

Anyway, Sol's post got me thinking again about how I would approach this idea with students. The first big idea is that if you pick a point on the y-axis between y=0 and y=1, with a rational y-coefficient, then the line through that point and the point (-1,0) will cross the unit circle in the first quadrant in a point (p,r) whose x and y coordinates are both rational. Since the point is on the unit circle, that means that p²+r² = 1. But since both p and r are rational, they can be written with a common denominator, Wolfram uses (a/c, b/c) and so (a/c)²+(b/c)²=1. If we multiply both sides of the expression by c², we get a²+b²= c² and so {a,b,c} is a Pythagorean triple.

So what might we ask our young charges to do with this? Well, first, do we believe that ANY rational on the positive y axis REALLY will produce a rational intersection on the unit circle? The proof requires no more than the solution of a quadratic. If the students are not up to challenging an open proof, have them start by confirming that some given rational y-intercepts will produce a rational intersection on the unit circle, and then go on to find the Pythagorean triple associated with that rational. For example, Wolfram's demo seems to associate the rational 1/2 with a (3,4,5) triangle... can they show why? Can you find the triple associated with 3/7?

I mentioned above that I didn't think this could produce ALL the Pythagorean triples... and we might want to ask our students if they think it could. One of my concerns about the Wolfram presentation is that the point (0, 1/3) is associated with the triple (8,6,10). Why not (4,3,5)??? how did they arrive at the non-primitive triple they chose, and not, for instance, (12,9,15) or ??? pick your favorite. I knew that they had used (3,4,5) in the first example (0, 1/2)...

In fact, would (9,12,15) show up later if I plowed through enough rational fractions? I suspected it wouldn't. First, the symmetry of the situation seemed to play against it. In simplest form, the x coordinate for any version of a (3,4,5) triple must be either 3/5 or 4/5; or at least so it seemed to me.

I think I would challenge my students to work backwards from a given triple and see if they could find the related rational y-intercept. Since we know (see my earlier blog) that the x-coordinate was (1-t²)/(1+t²), we could take any primitive triple, say (5,12,13) and set either the x value = 5/13 or 12/13 and get the two (and only two) values that would generate appropriate intersections on the unit circle. As we do this, we discover that the rational always turns out to be
where H is the hypotenuse of the triple, and L is the leg we chose to use for the x-coordinate ratio. For (5,12,13) we get the two rational values 2/3 (this is number 3 on the Wolfram demo) and 1/5 (number 7, but labeled (24, 10, 26) ) ... Bright students should be challenged to prove that any Primitive Pythagorean triple, will also be such that the ratio (H-L)/(H+L) give will produce a square of a rational number.

So maybe the Wolfram demo is just good enough (and just bad enough) to motivate our students, after all, they love exposing our flaws.

White Rabbit Mathematics

One of the things that amazes me, and I think most people who are attracted to math, is the mysterious way that different parts of math come together in unexpected ways. I tried to explain this to someone once using a literary analogy..."It is as if you were reading along in some great drama, or trying to understand the message in some grand poem, and suddenly the White Rabbit from Alice in Wonderland comes running through muttering, "Oh dear! Oh dear! I shall be too late!"

It is not the White Rabbit you see in math, but the effect is the same. Euler must have felt that feeling after he struggled to find the value of the series .. and finds that it turns out to be . Wait.... Pi is the ratio of the circumference to the diameter of a circle, but there are no circles in the sum of the squares of the reciprocals of the integers; and yet, there it is, the mathematical white rabbit coming seemingly from nowhere. Certainly none of the many mathematicians of great repute who had worked on the problem found (or expected) Pi to appear.

The normal distribution is another example; De Moivre takes the binomial probability distribution for flipping a coin and generalizes it toward an infinite number of flips, and POW, the normal or bell-shaped curve that is ubiquitous in intro stats. And what happens? Right there in the middle, the height of the normal curve at Z=0 is .39894... No, NO, NO, NOT JUST .39894.. but the .39894... that is exactly equal to .

Ok, so what brought this sudden rebirth of excitement about mathematical interrelationships? Well recently I came across a blog that referred to another blog that (as these things sometimes do) led me to a paper on just such a mathematical "white rabbit". The paper was about partitions of numbers as powers of two (1, 2, 4, 8, 16, etc..)

It began with a simple question, what is the number of ways to write a number n as a sum of powers of two if each value can be expressed no more than two times. For example, we could express 4 as 4, or as 2+2, or as 2 + 1 + 1 since each value is a power of two, and none appears more than twice. You couldn't use 1+1+1+1 since it appears more than twice. For n= 4 it turns out that the number of partitions, as shown above, is three. If we assume that there is one way to express zero, and one way to express one, and figure out the others we get a string like this


1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7,..

Ok, you don't see a white rabbit yet... but then someone ask you a different question. Is it possible to write out ALL the rational numbers in simplified form without repeating any of them. The answer is "Yes, of course, see the list above."

"What?", you ask, "How?", but there it is... The sequence of rational numbers is formed by taking each of the numbers to be the numerator, and using the number behind it to be the denominator. 1/1; 1/2; 2/1; 1/3; 3/2; ... and you never get a repeat, never get an unsimplified form, and you eventually get them ALL, the entire Infinite Set.....

No way you would expect that partitions of powers of two should give you the rational numbers in their entirety... there is (it would seem) nothing to relate the two questions... and yet... there it is. I think that is what makes math the most exciting area of study in the world.

Prove it you say? Nope, In truth I ain't man enough, but you can find the entire paper
Recounting the rationals, by Neil Calkin and Herb Wilf. Read their proof and Enjoy.