The first part of this is not my work, but I offer an intriguing extension.
This appeared on Twitter recently. (source)
Despite its apparent notional confusion, it is a true statement. Since both sides are positive, you can square both sides without producing extraneous results. Doing so proves the statement.
It’s a lovely, but curious piece of arithmetic trivia. A more mathematical question:
Does this pattern hold for any other numbers?
Thomas Oléron Evans has a proof on his ‘blog here in which he solves the equation under the assumptions that a, b, and c are natural and is any fraction in its most reduced form. Doing so leads to the equation
where A is any natural number larger than 1. Nice.
While the derivation is more complicated for middle and upper school students, proof that the formula works is straightforward.
A>0, so all terms are positive. Square both terms, find a common denominator, et voila!
Using Evans’ assumptions, the formula is inevitable, but any math rests on its assumptions. I wondered if there are more numbers out there for which the original number pattern was true.
Using Evans’ formula, my very first thought was to violate the integer assumption. I let and grabbed my Nspire.
Checking the fractional term, I see that I also violated the “simplest form” assumption. Converting this to a fractional form to make sure there isn’t a decimal off somewhere down the line, I got
So it is true for more than Evans claimed.
I don’t have time to investigate this further right now, so I throw it out to you. How far does this property go?