pythagorean triples complex squares
While working on a math problem recently that involved Pythagorean Triples I spotted a connection with the complex numbers that I had never noticed previously. It is a trivial observation and I was surprised that in my many years as a mathematician I had never heard anyone mention this little fact. I am curious if this little fact is well known to others and I just happened to miss it or if it really is unknown to most folks.
A Pythagorean triple is a set of three integers like 3,4,5 that make a right triangle with integer sides. Being a right triangle the numbers fit the Pythagorean theorem, a^2 + b^2 = c^2 and sho nuff 9 + 16 = 25
Euclid has a proof that is presented in the Elements that all Pythagorean triples can be generated from two integers s, and t by means of these formulae
a = 2st
b = s^2 - t^2
c = s^2 + t^2
It is fairly easy to use algebra to square a and b using those formulae and see that these numbers do in fact form a Pythagorean triple. It is only slightly more difficult to go the other way and show that all triples are of this form.
So all of this is ancient history - known to Euclid.
Now let us take a complex number, c = a + bi. Let it be a complex integer, meaning that a and b are actually integers. Viewed from the complex plane, the complex number c represents a little right triangle, one leg is the real part, one leg is the imaginary part and the hypotenuse is the norm of the complex number c i.e.
|c| = sqrt(a^2 + b^2)
Clearly because of the square root sign the norm of c is not necessarily an integer and thus the triangle a,b,c is not necessarily a Pythagorean triple. However just square the complex number c
(a + bi)*(a + bi) = (a^2 - b^2) + (2ab)i
Notice anything about the form of a perfect square in the complex plane and Euclid's characterization of a Pythagorean triple? One leg is a difference of squares the other is twice the product of the two numbers.
They are one and the same. The Pythagorean triples are just exactly the perfect squares in the complex plane.
So for example (2 + i) is not a square and is not a Pythagorean triple. Its norm is sqrt(5). but if you square it, you get (3 + 4i) whose norm is 5.
That is the observation. Nothing difficult. Perfectly obvious. Every mathematician from Euclid on knows about right triangles and everyone since Gauss has been able to multiply two complex integers. It is about as easy as math gets. I was astonished that in years as a working mathematician I had never noticed this nor heard anyone mention this simple fact.
Yes, I know - you're thinking that the reason this never came up in conversation is because this observation is totally irrelevant and practically useless, but that has never stopped mathematics. Just ask any mathematician to tell you about paracompact Hausdorff spaces and listen to an amazing tirade of irrelevant and practically useless information.
The Pythagorean triples are just the perfect squares in the complex plane.
Just for grins Google 'Pythagorean triples "complex squares"' to see a Google search that returns but a single result. Hole in one. except that it is not about the result that I just mentioned.
I make no claim that any of this is original, I am only flabbergasted at my ignorance and am casting out to see if I was just asleep the day they covered this in high school or if this really is a rather obscure and little known even if trivial observation.
There is nothing left to say but to ask the survey questions
1) Did you know this before you read it here?
2) If you did know it, was it shown to you or did you discover it yourself?
Enjoy!
