A Modest Appeal to Modesty

Since nobody knows if


is equal or not eqaual to


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.


4 thoughts on “A Modest Appeal to Modesty

  1. JenniferRM says:

    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)


    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)


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


    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


    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:

  2. Oscar Cunningham says:

    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.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s