Divided by Zero
divisbyzero.com · Personal blog · on HN since 2010
David Richeson, a mathematics professor, writes about accessible math topics, proofs, and recreational projects, including some done in Excel.
What HN saysHN links it for math explanations, such as p-adic numbers and finite differences of polynomials, which commenters discuss.
32HN people shared it
8linked it in comments
46stories submitted
824story points
1 year agolast shared
2010people per year2026
Most-shared posts
- 1What are p-adic numbers? (2008)4 people · 6 points · first shared 2013 · discussion on HN
- 2Mathematical surprises3 people · 42 points · first shared 2010 · 3 comments on HN
- 3Making the Mandelbrot Set with Excel3 people · 11 points · first shared 2024 · discussion on HN
- 4Tennenbaum's visual proof that the square root of 2 is irrational3 people · 2 points · first shared 2011 · discussion on HN
- 5Make a "Magic Eye" image using Excel2 people · 117 points · first shared 2024 · 56 comments on HN
- 6An application of graph theory to architecture2 people · 3 points · first shared 2011 · discussion on HN
- 7What is the difference between a theorem, a lemma, and a corollary?2 people · 3 points · first shared 2019 · 2 comments on HN
- 8Fold-and-Cut Hat and Spectre Tiles2 people · 2 points · first shared 2024 · discussion on HN
- 9The Unknotting Number Is Not Additive1 people · 178 points · first shared 2025 · 78 comments on HN
- 10Turing's topological proof that every written alphabet is finite (2010)1 people · 138 points · first shared 2024 · 114 comments on HN
- 11Proof Without Words: Gregory’s Theorem1 people · 97 points · first shared 2018 · 26 comments on HN
- 12The math behind a neat calculator trick1 people · 74 points · first shared 2010 · 17 comments on HN
- 13Make Your Own Pythagorean Cup1 people · 48 points · first shared 2018 · 22 comments on HN
- 14Maypole Dance of Braid Like Groups (2009)1 people · 37 points · first shared 2025 · 4 comments on HN
- 15The Most Imaginary Number Is Real1 people · 22 points · first shared 2019 · 3 comments on HN
What HN people wrote when they linked it
tarkin2 · 2010-07-24
Bit of a logical play, the dilemmic fallacy to be precise, but, alternatively:
If they're competent, leaving them a lot of money won't do them any harm. If they're not, they need his money.
[edit] here's the classical example of this change in perspective: http://divisbyzero.com/2008/11/30/euathlus-and-protagoras/
srean · 2025-06-09
I never knew Hokusai had interest in these.
Does anyone know if he had seen any Islamic tiling art ? Or for that matter if Escher had seen any of his.
Group theory / symmetry runs deep through these. One [0] of the circular ones made me think of braid(group)s. Not sure what the * operator would be in that case.
Speaking of braid groups Richeson had a very interesting blog post on the braid like groups of maypole dances.
Let me find and submit separately (spoiling a Hokusai story with another story, no matter how interesting, would be a personal sacrilege). Found it [1][2].
[0] https://dl.ndl.go.jp/pid/1899550/1/10/
[1]https://news.ycombinator.com/item?id=44225324
[2]https://divisbyzero.com/2009/05/04/the-maypole-braid-group/
version_five · 2023-04-19
Something I found interesting, once you understand the proof of why there are "more" real numbers than integers, it becomes easy to see that the p-adic numbers[0], which can have infinite digits to the left of the decimal, but finite digits to the right, have the same cardinality as the real numbers (there are more of them than integers).
This was unintuitive to me when I first thought about it, because I pictured a whole number (no fractional part) even with infinite digits, to be in the natural numbers, but in fact it's not. Or put another way, whole numbers with infinite non-terminating non-repeating digits are not natural numbers
[0] https://divisbyzero.com/2008/11/24/what-are-p-adic-numbers/
nanis · 2023-01-19
> Recently, somebody shared on Twitter a new proof
As far as I can tell there is no link to this tweet in this post which is bad form to say the list. The post has multiple links promoting author's other posts, but not to the raison d'etre for this post.
PS: Apparently, the author is Director of Research, Computer Laboratory, University of Cambridge which makes this lack of proper citation even more perplexing.
PPS: Compare the intro paragraph of this post to one that properly cites related work[1] with actual links to the referenced works:
>> Yesterday I came a across a new (new to me, that is) proof of the irrationality of \sqrt
{2}. I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.
[1]: https://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrational…
eternalban · 2021-04-09
David Richeson dives into the math behind it:
https://divisbyzero.com/2018/02/13/finite-differences-of-polynomials/
thih9 · 2020-12-13
> I disagree that or is ambiguous
There are examples of inclusive “or” in natural language. When the waiter asks: “Would you like cream or sugar?”, it’s not meant to imply that you should pick only one; it’s fine to answer “both” (example from: https://divisbyzero.com/2009/09/27/is-or-an-inclusive-or-or-an-exclusi… ).
faehnrich · 2014-11-05
This looks very similar to this
http://divisbyzero.com/2010/03/14/an-application-of-graph-theory-to-ar…
dvanduzer · 2011-11-19
Do a little more research on Stanley Tennenbaum. This discovery happened around the 1950s. Tennenbaum didn't drop out of mathematics so much as he dropped out of institutional mathematics. More here, including a post from one of his children in the comments: http://divisbyzero.com/2009/10/06/tennenbaums-proof-of-the-irrationali…
Stanley's up in heaven now.