Is this the Coolest Approximation for e? @WrathofMath

Uploaded February 2025 | Updated September 2026, 2 weeks ago

In 2004 Eric Friedman issued a challenge - to find the best approximation for some famous irrational numbers using the first n digits. Among these numbers was, of course, Euler's number e. One brave challenger, Richard Sabey, had an incredible showing - giving us an amazing approximation of e, accurate to over 18 septillion digits, using the digits 1-9. We explore the approximation today, and how to determine its accuracy without computing the massive numbers involved. This involved discussing the limit definition of e, as well as compound interest and power series. #maths #mathematics #math

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SOURCES
Eric's original problem page: erich-friedman.github.io/mathmagic/0804.html
mathworld.wolfram.com/eApproximations.html
math.stackexchange.com/questions/1945026/an-amazing-approximation-of-e
johndcook.com/blog/2014/03/30/amazing-approximation-to-e
en.wikipedia.org/wiki/Pandigital_number
Stewart, J. (2012). Calculus : early transcendentals. Belmont, Cal.: Brooks/Cole, Cengage Learning.

0:00 Intro
1:03 e Approximations
1:53 Friedman's Challenge
4:39 Sabey's Pandigital Approximation
5:33 The Issue
9:47 Rewriting
12:39 Showing the log is Near 1

Is this the Coolest Approximation for e?

$the math teacher can’t figure it out This seemingly simple geometry problem got a lot of traffic recently on r/askmath, as many tried to solve it, got it wrong, and got stuck. The OP’s wife, a math teacher, was unable to solve it - and this time that’s not because the teacher is incompetent but the problem is simply much harder than it looks. We’ll solve this variation of Langley’s Adventitious Angles using some clever geometric constructions and basic properties of triangles. #geometryproblems #mathchallenge Join Wrath of Math to get exclusive videos, lecture notes, and more: https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join Check out the coolest math clothes in the world: https://mathshion.com/ More Math Chats: https://www.youtube.com/playlist?list=PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO Outro music: https://youtube.com/shorts/tfifiSdTZRg Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● TikTok: https://www.tiktok.com/@wrathofmathedu ● X: https://x.com/wrathofmathedu SOURCES https://www.reddit.com/r/askmath/comments/1lq21qv/my_wife_math_teacher_cannot_figure_this_out https://www.reddit.com/r/askmath/comments/1lqy3vj/comment/n18lc4a/?context=3 https://mathwithbaddrawings.com/2015/06/10/trick-or-technique/ Tripp, Colin. “Adventitious Angles.” The Mathematical Gazette, vol. 59, no. 408, 1975, pp. 98–106. JSTOR, https://doi.org/10.2307/3616644. https://www.duckware.com/tech/worldshardesteasygeometryproblem.html Adventitious angles problem: the lonely fractional derived angle by Yong Kong and Shaowei Zhang MUSIC Special World 8 (Crown) - Super Mario 3D Land Athletic BGM (Super Mario World 2 - Yoshis Island) - Mario Kart World Musee Champignon Aquatic Ambience Torigoth - Night Ordon Village Dire Dire Docks Mario 64 End Theme 0:00 Intro 1:09 Hogwash Montage 3:19 Whats the Deal? 5:33 Adventitious 7:25 Solution 19:10 Conclusion$

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