Uploaded December 2025 | Updated September 2026, 2 weeks ago

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In this short, we use a geometric representation and the Archimedean property to show why 0.111... equals 1/9. A consequence is that 0.999… = 1. This fact tends to be a bit controversial in the mathematics classroom, but the only way to make sense of an infinite sum is to treat it as a limit. Thus, we can use the term equality when discussing infinite sums like 0.999... and there is no number this sum could equal besides 1.

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This animation is based on a proof by James Tanton from the March 2008 issue of The College Mathematics Journal page 106. (jstor.org/stable/27646594).

To see a longer video with a similar fact (and perhaps more justification at the end), see
youtu.be/BGsJjHbjcnA

Here is a playlist with other geometric sums dissection proofs:
youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw

#infiniteseries #paradox #manim #math #mathshorts #visualproof #proofwithoutwords

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