Since nobody knows if

Sqrt(5)+Sqrt(22+Sqrt(20))

is equal or not eqaual to

Sqrt(11+Sqrt(116))+Sqrt(16-Sqrt(116)+2*Sqrt(55-5*Sqrt(116)))

this appeal to our modesty is in order.

I have read this in the Scientific American 40 years ago, but I forgot what exact expressions where there. Thanks to Google I have re-discovered them. And that there was no resolution in meantime as well.

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I’m confused. It feel like when you look at something “poorly worded” just before you realize there was a pun. Maybe I’m missing something?

One confusing thing is that the second expression as unbalanced parentheses.

Then, looking at the math, for the first expression I get this:

>>> (5.0 ** 0.5) + ((22.0 + (20.0 ** 0.5)) ** 0.5)

7.381175940895658

For the second expression with the 11, 16, and 55 there are either two or three sums, depending on how the paren confusion is resolved.

The first sum with 11 in it is definitely

>>> ((11.0 + (116.0 ** 0.5)) ** 0.5)

4.6658685809042035

If we have three sums then the other two components of the second expression (that have 16 and 55) give:

>>> (16.0 – (116.0 ** 0.5)) ** 0.5

2.286847258942099

>>> (2 * (55 – (5.0 * (116.0 ** 0.5)))) ** 0.5

1.5154879931262806

The sum of these is 4.66 + 2.28 + 1.51 == 8.45 which is so far above 7.38 that there could be no confusion.

What if the paren is resolved to lump the 16 and 55 into the same term? Then we have:

>>> ((16.0 – (116.0 ** 0.5)) + (2 * (55 – (5.0 * (116.0 ** 0.5)))) ** 0.5) ** 0.5

2.5971442737855885

Then 4.6658 + 2.5971 == 7.2629 which is vaguely close to 7.3811 but still pretty far off. It isn’t like numerical solutions for these expressions grind on and on giving the same digits despite a proof of why being elusive… they just are different expressions as far as I can see?

I feel like a plane might be wooshing over my head. What am I missing?

I have omitted the rightmost bracket and I stand corrected about this, thank you!

Other than that you are completely wrong. Both results are 7.3811759408956575…

As far as we can see, both results are equal, but nobody is able to actually algebraically transform one expression into another one.

See the following link, where you can try the equality:

http://web2.0calc.com/

Wolfram alpha thinks that these are both roots of x^4 – 54x^2 – 40x + 269, which suggests that it shouldn’t be too hard to prove that they are equal. Verify that they are roots by substituting in, and then estimate their sizes to prove that they are in fact the same root.

It might be true that you can’t get from one to the other purely by algebraic manipulations though. Sort of like Tarski’s high school algebra problem.

The question, whether every two formulae yielding the same number, are in fact translatable from one to another using some finite set of rules, is interesting.