# Euclid's formula and real numbers

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## Summary:

Generating the sides of a right triangle using all real numbers

## Main Question or Discussion Point

Recently I created a spreadsheet that generates Phythagorean triples. Curious, instead of using only positive integers for the values of m and n, I found that as long as m>n, the sides 2mn, msq + nsq, msq - nsq, still form the sides of a right triangle even though m and n are non-whole numbers. I assume this is no big deal and that this is already known in the mathematical community. Does anyone know of a proof that Euclid's formula works for the set of all real numbers, not just for integer numbers?

PeroK
Homework Helper
Gold Member
Summary:: Generating the sides of a right triangle using all real numbers

Recently I created a spreadsheet that generates Phythagorean triples. Curious, instead of using only positive integers for the values of m and n, I found that as long as m>n, the sides 2mn, msq + nsq, msq - nsq, still form the sides of a right triangle even though m and n are non-whole numbers. I assume this is no big deal and that this is already known in the mathematical community. Does anyone know of a proof that Euclid's formula works for the set of all real numbers, not just for integer numbers?
Algebra works for real numbers as well as whole numbers. The proof is the same, whether the sides are real numbers or whole numbers.

PS There are no real constraints on real Pythagorean triples. If $a, b$ are positive real numbers, then $a, b, \sqrt{a^2 + b^2}$ is a Pythagorean triple.

Last edited:

robphy
Homework Helper
Gold Member
You might find this 3Blue1Brown video interesting

jedishrfu
Mentor
Given any Pythagorean triple and multiplying by some real number scaling factor will generate a real number Pythagorean triple too. An infinite number of them.

You might find this 3Blue1Brown video interesting
Cool. Thanks for the video.