Infinite fractions and the most irrational number
Why Hacker News recommends it
Recommended as a clear, visual explanation of continued fractions and why phi is hardest to approximate by fractions.
What it is
Mathologer uses continued fractions to show why the golden ratio is the most irrational number, comparing it with root 2, e and pi and pointing to where this shows up in nature.
Recommended by 3 HN commenters across 3 different months, 2019 to 2026.
What Hacker News said
Recommends rrauenza · 2021-10 · on “Where transcendental numbers hide in everyday math”
I was going to post this Mathologer video ( [this video] ) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi. But its still a really interesting video!
Recommends jerf · 2022-10 · on “The Mystery of 355/113 (2010)”
"same with golden ratio (repeating 1s)" Mathologer has a good video in which he uses this to justify calling the golden ratio the "most irrational" number because of this expansion: [this video] The video also goes over the details of the pi expansion.
Recommends uoaei · 2023-08 · on “Decimal to Fraction”
Pretty good Mathologer video about continued fractions: [this video] One neat trivium I like to quiz the mathematically-minded on is: what number is the most irrational , i.e., toward which number do rational approximations converge the slowest ? The answer can be determined from continued fractions and is discussed in the video near the end.
Cites std_throwaway · 2019-08 · on “Golden Ratio”
The golden ratio is the most irrational number: [this video] This is why it creates some distinctive patterns when applied to circular/spiral shapes.
Cites AceJohnny2 · 2026-01 · on “Golden Ratio using an equilateral triangle inscribed in a circle”
Edit: years of searches and minutes after I post this I found [this video] thanks to using "continued fraction" in my search instead of "infinite series" X( Original: Tangentially, for a few years I've been looking for a Youtube video, I think by Mathologer [1], that explained (geometrically?) how the Golden Ratio was the limit of the continued fraction 1+1/(1+1/(1+1/(...))). Anyone know what I'm talking about? I know Mathologer had a conflict with his editor at one point that may have sown chaos on his channel. [1] [LINK]