The Diagonal Paradox

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Consider the stepped line . As goes to infinity, approaches the hypotenuse of the isosceles right triangle that has length . So the length of the limit of is . Is this right?

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No, the length of is 2 for all , so the length of the limit of is also 2.

This is known as the diagonal paradox. The resolution is that you cannot reverse the measurement of length and the approach to infinity, or in symbols, .

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Contributed by: Izidor Hafner (January 2016)
Open content licensed under CC BY-NC-SA


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Reference

[1] E. P. Northrop, Riddles in Mathematics, A Book of Paradoxes, Mineola, NY: Dover Publications, Inc., 2014 pp. 138–139.



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