A Thought Provoking Paradox about Numbers


The Following is Entirely Paraphrased from Douglas Hofstadler in his book "I am a Strange Loop"...


There are people who focus on finding ways to express numbers using the least amount of syllables possible. Put aside for moment how much you think these folks need better hobbies and try to grasp what this means.
For instance, the number 144 written out as One-Hundred-Forty-Four contains 6 syllables. But you could express the same number by merely saying One-Forty-Four, which is 4 syllables. Even better is Twelve Squared.
The point is that that language permits you to convey the same number many ways, some of which require less energy. It’s a question of how efficient you can express a concept.
With small numbers, it’s easy Things get more interesting with large numbers, mainly because it gets very difficult when they do not breakdown easily.
100,000 is easily expressed as Ten-to-the-Fifth. Even 100,001 has a neat solution given its close proximity, as it can be expressed as Ten-to-the-Fifth plus One.
But what about a number like 171,228? This is not so easy. The least amount of syllables I’ve been able to express this with is 7.
Can you imagine the time required to determine the least amount of syllables to express much longer numbers like: 778,494,845,211,987? Yet people do waste their time doing this, and they’ve discovered a very interesting paradox.
The Smallest Number Requiring More than Thirty Syllables
What these folks with too much time on their hands have discovered is that nearly every number can be expressed in less than thirty syllables. This is because most numbers, when broken down, are found to contain multiples of smaller numbers that can be put together.
Thus, there is an ongoing search to find the smallest number requiring more than 30 syllables. Let’s call this number X.
But herein lies the paradox. Whatever number X turns out to be, the number itself can therefore be expressed as the smallest number requiring more than 30 syllables, which is a statement made up of only 15 syllables.
Therefore, the smallest number requiring more than thirty syllables can always be expressed using less than 30 syllables. More specifically, the moment you find X, by labeling it correctly, it can no longer be X.
So if X isn’t it, then the next number higher than the previously found X unable to be expressed with less than 30 syllables must be the answer. Let’s call this new number Y. But since X could be expressed in less than 30 syllables, this new number Y now becomes the the smallest number requiring more than thirty syllables. 15 Syllables.
And around and around we go.



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