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Do Complex Pythagorean Triples Have Geometric Meaning?


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Real Pythagorean triples are numbers like 3, 4, and 5.

 

If a, b, and c are three positive real numbers and [math]a^{2}+b^{2}=c^{2}[/math], then a, b, and c can be used as the sides of a right triangle. So real Pythagorean triples have geometric meaning.

 

But what if a, b, and c are not real numbers?

For example:

[math](-13+6i)^{2}+(6+22i)^{2}=(3+18i)^{2}[/math]

 

If three complex numbers form a Pythagorean triple (and they are not real numbers) do they have a geometric meaning?

 

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Do you mean squares as you have indicated or multiplication by the complex conjugate, it makes a difference.

 

For example your first bracket

 

(-13 + 6i) (-13 + 6i) = (132 -156i - 62)

 

but

 

(-13 + 6i) (-13 - 6i) = (132 + 62)

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Yes, I do mean squares.

If we multiply by the complex conjugate then one side of the equation doesn't equal the other side, right?

 

(169+36)+(36+484)=(9+324)

is false.

 

But if we square the two complex numbers on the one side and add them together, then they do equal the square of the complex number on the other side.

 

(169-156i-36)+(36+264i-484)=(9+108i-324)

is true.

 

Unless, of course, I've made some mistake in the math.

 

My question is, does it mean anything Geometrically?

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Well multiplication of two complex numbers can be viewed as a stretching and rotation of the vectors for each on an Argand diagram.

 

So each square will produce a new vector representing a new complex number which when added to the other on in the normal way in an argand diagram produces a third in a vector triangle.

 

Sorry i can't so any drawing whilst I am still in Scotland.

Edited by studiot
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Whether or not you have the means of making drawings, thanks for your reply.

Yes, what you said about stretching and rotation is right. So, using my example, when we multiply (3+18i) by itself, the original vector, which had an angle measure of about 80.5 degrees is rotated another 80.5 degrees, so the resultant has a measure of about 161 degrees. Also, the original vector had a magnitude of about 18.2, and the resultant is stretched out to a magnitude of about 333. So the vector (3+18i) does have a geometric meaning, and when you square a single complex number, the resulting square has a geometric meaning that is related to the original vector.

A Pythagorean equation is something different though. We have three numbers. If the numbers are real then they have a geometric meaning relating to right triangles. But what if the numbers are not real? There are solutions to the equation, and they look interesting (to me at least) but do they mean anything geometrically?

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I would not use the term resultant, as that is reserved for the addition of two vectors.

 

But you have added two of these stretched, rotated vectors, each representing the square of a complex number.

This does have a geometric resultant in the proper sense of closing the triangle in the complex plane and I think ( though I am not sure) that the closing resultant is the complex number squared on the other side of your equation.

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You're right, the resultant is the sum of two vectors. I don't know the term for the product of two complex number vectors. How about if I use the words "product vector"?

 

I understand the next sentence. We're talking about two product vectors.

 

Then I think I understand the first part of the next sentence. Since the two product vectors are both vectors themselves, when we add them, we get a "resultant in the proper sense", that is, the sum of two vectors.

 

Now for the next part. If you add any two vectors together, you will get a third vector, the resultant, that will "close the triangle" in the complex plane. Is that what you were saying?

 

That is true, but I don't think that really answers the question. If you take any two vectors and add them together and show that they equal another vector, you don't necessarily have a Pythagorean equation. Do the vectors in a complex Pythagorean equation have any geometrical meaning beyond the meaning that you get when you add any two complex numbers together?


I've been thinking about my last post. If you take any complex number it does have a square root that is another complex number, so any equation x+y=z is necessarily a complex Pythagorean equation.

 

Thinking about that I think that answers my question.

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I don't know the term for the product of two complex number vectors. How about if I use the words "product vector"?

 

Yeah, product is good since the output is also a vector you could go further outer product or vector product might be better.

 

 

Thinking about that I think that answers my question.

 

 

 

:)

 

 

Even if I can't, draw yourself a few diagrams.

They help.

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This is what I'm thinking might be the answer to my question:

 

When you are talking about Pythagorean triples, you are usually talking about whole numbers. Pythagorean triples have a geometric meaning because they can be used for the sides of a right triangle, but the fact that they are whole numbers doesn't have anything to do with this geometric meaning. Is that right? I suppose you could say that 1+2=3 is an irrational Pythagorean equation.

 

I think Pythagorean triples are interesting.

When I first started thinking about complex Pythagorean triples, I was thinking of numbers of the type a+bi where 'a' and 'b' are both integers. I think you might use the term "complex integers", right? I think solutions to the Pythagorean equation that are complex integers are interesting, and I suppose they might have some meaning or significance, (although I don't know what that might be) but I'm guessing that they don't have any geometric meaning, because any three complex numbers x, y, and z, where x+y=z can be made to be the solution to a complex Pythagorean equation. The only thing that might make complex Pythagorean triples special is the fact that they are complex integers, and that doesn't have anything to do with Geometry. Any comments?

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