1. ## Pythagorean Triples

I've always seen it stated that a Pythagorean Triple is three positive integers $\displaystyle (a, b, c)$ where $\displaystyle a^2+b^2=c^2$.

Why the restriction that $\displaystyle (a, b, c)$ are positive?

Similarly with Euclid's formula for generating such triples giving the restriction $\displaystyle m>n$ for...

$\displaystyle a=m^2-n^2$
$\displaystyle b=2mn$
$\displaystyle c=m^2+n^2$

If we let $\displaystyle m<n$ we generate the same |absolute| values with -ve $\displaystyle a$

According to the restrictions $\displaystyle (-a, b, c)$ is not a Pythagorean Triple but the result

$\displaystyle a^2+b^2=c^2$

still holds true.

Are the restrictions merely convention or is there another reason?

Thanks

2. ## Re: Pythagorean Triples

Just a guess, but perhaps only positive solutions are considered Pythagorean triples because they correspond to the sides of right triangles.

3. ## Re: Pythagorean Triples

I suppose that could be it.

Thanks Petek

4. ## Re: Pythagorean Triples

with a right triangle, the other 2 angles are always acute. one can consider the sine and cosine of obtuse angles, which leads to a triangle that is no longer right (one has to consider a right triangle that makes up the "complementary angle").

one can even consider extending sine and cosine to "angles greater than 180 degrees", but then the angle you are measuring is "outside the triangle" (equivalently, you can consider "negative angles").

while all these notions can be done in a way "that makes sense", they don't really introduce any new information (we still wind up considering some right triangle with acute angles). the greek mathematicians who pioneered geometry, thought of numbers as being magnitudes, our modern notion of number makes them more like vectors (magnitude = absolute value, direction = sign).

so when we draw -3 and 3 on "the number line" (this is essentially linear algebra in one dimension), we get 0.

for the greeks, a line segment 3 units in one direction, and a line segment 3 units in the other direction (starting from a common point), formed a line segment 6 units long.

for US, "+" is a fundamentally different KIND of thing.

as far as properties of integers (as opposed to natural numbers) go, it turns out (fortunately), that multiplying by -1 (changing sign) doesn't "really change anything":

sure, we can factor 6 as (-2)(-3), but it's not the "sign out front" that tells us the factors of 6, it's the primality of 2 and 3. abstractly, we say that -1 is a unit (it has an inverse, namely -1). when we study "factoring", "units may as well be 1". for example, x - 1 = (1/4)(4x - 4), but one typically doesn't think of 4x - 4 as a "factor" of x - 1 (in the rational numbers, 4 is a unit, it has an inverse).

only considering positive numbers streamlines statements about numbers (we don't have to include "cases" or things like "up to a change in sign", or put plus/minus signs in everything (which goobers up the algebra)).

5. ## Re: Pythagorean Triples

Thanks Deveno, it appears you are saying it's merely convention then.
The reason for my OP is that I have found a result which, in Euclid's formula, requires (unconventionally) $\displaystyle m<n$ and couldn't think of any mathematical reason this wasn't acceptable.