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Another clever exploration by James Tanton

I really enjoy James Tanton's Math explorations because they tend to be easy to describe and rich in exploration value. Here's such an exploration:

The problem statement is very simple. Is there a way in which we can say that there are more triangular numbers than square numbers? If so, how do we compare the sizes of the two sets? Can we compute the ratio of triangular to square numbers where both T(n) and S(n) are less than an arbitrary constant? Can we generalize what we find for other polygonal numbers?

This is a great exploration!

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  1. using the same argument (much more easy as well) we may conclude that the amount of even numbers is half of the total numbers…

    A clue on why this reasoning is not valid: “don’t ever apply a well known result or technique of finite mathematics to the infinite wihtout checking what you are doing…”

    In this case the easy way back to the original asnwer (the correct one!) is easy… once you have finished and have seen that the ratio limit is square od 2…just ask yourself how muach is SQ(2)*infinite? yes you are right, it is infinite!

    So even after all this fancy calculations the answer still is the same: There are the same triangular numbers and square numbers..

  2. So … What is the balance between the role of the potential infinite and that of tha actual infinite? Of course, from a cardinality point of view, both sets are equinumerous and there is nothing more to say. But from the human experience – how many more triangular numbers am I likely to see in my lifetime as opposed to square numbers – the answer is sqrt(2) more. There is nothing wrong with analysing equinumerous sets further by asking about the distribution of their members. I think you want me to state my conclusion as: There are equally many in the actual infinite sense, but among any finite set of numbers, the triangular outweigh the squares (a potential infinite sense).

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