Mixed Number Curiosity

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 \sqrt(a+\frac{b}{c}) = a \cdot \sqrt( \frac{b}{c}) under the assumptions that a, b, and c are natural and \frac{b}{c} 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 A=1.1 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?


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