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Number Palindromes of Differing Bases

Audience:

Tags: number-theoryinfinite-series

See how “number palindromes” relate to the convergence of an infinite series, Pascal’s Triangle, and the Triangular Numbers!

In this video, I discuss the answers to questions like:

Does changing the base of a set of numbers (like the Triangular Numbers) reveal more palindromes?
Is there a predictable pattern for finding (some) palindromes?
Does the sum of the reciprocals of every palindromic number in every base converge?
Is there a way to make palindromic numbers without needing … uh … numbers?

A number palindrome is a number that is read the same forwards and backwards. The palindromic numbers in this video are in integer bases $B \geq 2$ . This video is the product of around 6 months of research and 40 hours of animation and programming, so I really hope you enjoy!

This video was entirely animated and edited in Blender.

Footnotes and other extra information:

(Footnote 5) 11:14 Taking the limit of a function $f(x)$ as $x \to n$ means finding the number that $f(x)$ approaches as $x$ gets infinitely close to $n$ . When $n=\infty$ , this means finding the value that $f(x)$ approaches as $x$ gets infinitely big. For a more in-depth explanation on limits and other Calculus topics throughout this video, see @3blue1brown’s Essence of Calculus series.
The reciprocal of an integer n is equal to $\frac{1}{n}$
For time reasons, I never got to explain in the video that there are an infinite amount of palindromes in every base. This can be seen in the set $\{1_B, 11_B, 111_B, 1111_B, \dotsc\}$ that contains every number that is only made of 1’s in a given base, of which there are infinitely many.

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5.75 Overall score*

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Comments

Simple and nice animations that immediately grab your attention. An interesting topic, well presented. I’d add some relaxing background music and a few more interesting hooks at the beginning of the video (a preview, perhaps?). Try to reduce the quantity of computation in a single slide. Nice video.

Thank you for covering the Rule of Product! The connecting line segments show the product, something I thought about how to clearly show and how it isn’t addition.

I’m not sure exactly about that triangular numbers section, what do they have to do with this? Their formulas, and all.

The derivation of formulas could use some work overall, constructing them instead of sliding on to the screen.

5.5

The part about the (B-1)th triangular numbers being a palindrome in base B is the most interesting part of the video.

It seems strange to bring up Pascal’s Triangle just to introduce the triangular numbers as a particular entry, since the triangular numbers already have a direct geometric interpretation.

The question of whether the sum of reciprocals of palindromes converges seems arbitrary to me. The “mirror” method also seems belabored for how obvious it is. It is strange that you add two numbers in the middle for the mirror method when the number of digits is even. Why add any middle digits at all?

The video starts with an interesting premise: what happens if we look for palindromes in different bases? However, it didn’t feel like there was a good payoff at the end, or if there was one then it was lost on me. I think it would’ve been possible to avoid using actual series tests and gone for an intuitive explanation with the geometry you had already set up (for the purpose of reaching a wider audience). The algebra was also difficult to follow and could likely have been omitted, just showing the set up and the result. The visuals were very well put together and had some personality which I appreciated. This is nitpicky, but the transition at 9:05 was just a bit rushed for me, I would’ve appreciated another second or two on the previous screen.

3.1

Good video, clear conscise and relatively easy to follow. Seemed like there was some things you explained in depth, and other things you kind of glossed over. The biggest issue I had with the video was there was no motivation, why would even mathematicians care about these numbers, or whether the recpicals converge or diverge, and what does that have to do with Pascal’s triangle? It felt like a video with no point and not much payoff.

Not a fan of the infinite grid start for Pascals triangle. And shearing the triangle always confuses me more then it helps. But overall a great video!

6.8

It’s a good first video. I find the pacing a bit unbalanced; just as much time and depth are devoted to topics for absolute beginners as to more advanced ones. Everything is very clear and rigorous—very “polished.” The topic is very specific, and the question posed regarding convergence is interesting; it would make a good homework exercise for students.