Reals

Articles and videos

Infinite series
- Ramanujan Summation by James Grime (May 1^st, 2016) ► A quick presentation of Ramanujan Summation.
- Infinite Sums | Geometric Series | Explained Visually by "Think Twice" (February 1^st, 2018) ► A visual explanation of the geometric series formula.
- Alternating series #1 | Visual solution | by "Think Twice" (August 7^th, 2018) ► £[\sum_{n=0}^{∞}\left(-\frac{1}{2}\right)^n=\frac{2}{3}£].
- Blackpenredpen vs. Dr. Peyam on convergent series (uncut, unscripted) by Steve Chow and Peyam Ryan Tabrizian (December 12^th, 2018) ► Some questions on series convergences.
- Infinite Series - Numberphile by Charlie Fefferman (April 2^nd, 2019) ► Some very well-known series.
- Alternating series #2 | Visual solution | by "Think Twice" (May 27^th, 2019) ► £[\sum_{n=0}^{∞}\left(-\frac{1}{3}\right)^n=\frac{3}{4}£].
- Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+... by Burkard Polster (December 24^th, 2019) ► Proving £[\frac{\pi}4 = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}£] by using Jacobi’s theorem on sums of two squares.
- The Answer Is Too Good by Presh Talwalkar (January 2^nd, 2020) ► £[\frac{e}{\sqrt{e}}\cdot\frac{\sqrt[3]{e}}{\sqrt[4]{e}}\cdot\frac{\sqrt[5]{e}}{\sqrt[6]{e}}\cdots=2£].
- Power Tower with @3blue1brown by Grant Sanderson and Tom Crawford (July 29^th, 2020) ► Some musing about the £[x^{x^{x^{x^{x^{x^{…}}}}}}=2£] equation.
- Sum of n/2^n by Peyam Ryan Tabrizian (November 18^th, 2020) ► £[\sum_{i=1}^\infty\frac{n}{2^n}=2£].
- The Problem With Infinite Summations On YouTube by Matt Parker (February 3^rd, 2022) ► How to compute the sum of a geometric series.
- The Finite Difference Method by James Grime (June 7^th, 2022) ► An explanation of Newton’s Forward Difference Formula.
- ↪Follow-Up: Finite Difference Method by James Grime (June 13^th, 2022) ► Some feedback about the previous video, the most interesting being Gilbreath’s conjecture.
- Oraux X-ENS - 03 - Calcul d'une somme by "Maths*" (September 20^th, 2022) ► Computing £[\sum_{n=1}^{∞}\frac{(-1)^n ln(n)}{n}£].
- The Reciprocal Prime Series (this proof should be taught in calculus!) by Trefor Bazett (November 1^st, 2022) ► A proof that the sum of the reciprocals of primes diverges.
- The most interesting iterated root problem I have seen! by Michael Penn (September 6^th, 2023) ► This computation is not rigorous (we need to prove that the series converges) and there is a mistake at the end.
- The harmonic series
  - Deux (deux ?) minutes pour l'escargot de Gardner by Jérôme Cottanceau (March 5^th, 2015) ► A paradox based on the divergence of the harmonic series.
  - Math in the Simpsons: Apu's paradox by Burkard Polster (October 2^nd, 2015) ► A proof that the harmonic series diverges.
  - The mystery of 0.577 - Numberphile by Tony Padilla (October 5^th, 2016) ► The harmonic series, Nicole Oresme’s divergence proof, the ant on a rubber rope, and Euler–Mascheroni constant.
  - ↪0.577 (extra footage) - Numberphile by Tony Padilla (October 7^th, 2016) ► The continuation of the previous video.
  - A harmonic series with only primes. by Michael Penn (July 18^th, 2020) ► A proof that the sum of the reciprocals of the primes is diverging.
  - 700 years of secrets of the Sum of Sums (paradoxical harmonic series) by Burkard Polster (November 21^st, 2020) ► Some well-known and some new facts about the harmonic series.
  - The Euler Mascheroni Constant by Peyam Ryan Tabrizian (December 2^nd, 2020) ► A proof of the existence of the constant.
  - Extending the Harmonic Numbers to the Reals (August 22^nd, 2021) ► How to extend the harmonic series to reals and to get the digamma function.
- Basel problem
  - Euler's real identity NOT e to the i pi = -1 by Burkard Polster (August 11^th, 2017) ► Basel problem and how Euler solved it.
  - Why is pi here? And why is it squared? A geometric answer to the Basel problem↑ by Grant Sanderson (March 2^nd, 2018) ► Using geometry to prove Basel equation, this is not a rigorous proof, but the method is stunning.
- The sum of all integers
  - One minus one plus one minus one - Numberphile by James Grime (June 25^th, 2013) ► Some ways to compute that £[1-1+1-1…£] gives £[\frac{1}{2}£].
  - ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 by Tony Padilla (January 9^th, 2014) ► A "proof" than the sum of integers is £[-\frac{1}{12}£].
  - Why -1/12 is a gold nugget by Edward Frenkel (March 18^th, 2014) ► Some discussion about the fact that the sum of integers can be treated as £[-\frac{1}{12}£], but this is very unclear since no mathematical details are given.
  - L'incroyable addition 1+2+3+4+...=-1/12 - Micmaths by Mickaël Launay (November 21^st, 2014) ► Some explanation about the formula.
  - Sum of Natural Numbers (second proof and extra footage) by Edmund Copeland and Tony Padilla (January 11^th, 2015) ► Another "proof".
  - What does it feel like to invent math? by Grant Sanderson (August 14^th, 2015) ► £[1+2+4+8+…=-1£] and 2-adic metric.
  - Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. by Burkard Polster (April 22^nd, 2016) ► The fact that Ramanujan worked on the subject and Cesaro convergence.
  - Numberphile v. Math: the truth about 1+2+3+...=-1/12 by Burkard Polster (January 13^th, 2018) ► Some bashing about Numberphile video and a much more mathematical explanation of the formula (supersum, the Eta function, analytic continuation).
  - the most famous Ramanujan sum 1+2+3+...=-1/12 by Steve Chow (December 28^th, 2018) ► £[\sum_{n=0}^{∞}n\stackrel{R}{=}-\frac{1}{12}£] (Ramanujan summation).
  - Not -1/12 by Steve Chow (December 31^st, 2018) ► "Proving" £[\sum_{n=0}^{∞}n=-\frac{1}{8}£].
  - La Somme des entiers positifs fait-elle vraiment -1/12? (Benoit Rittaud) by Benoit Rittaud (October 17^th, 2019) ► The video explains that Ramanujan summation is special, but it does not give any details about it.
  - L'énergie du vide quantique, -1/12 et l'effet Casimir by David Louapre (December 17^th, 2021) ► A simple explanation why some physicists use £[1+2+3+…=\frac{1}{12}£].
  - ↪L’effet Casimir et la série divergente 1+2+3+4+… by David Louapre (December 17^th, 2021) ► Some little additional information.
  - The Return of -1/12 - Numberphile by Tony Feng (February 15^th, 2024) ► Yet another version of this story.
  - Does -1/12 Protect Us From Infinity? - Numberphile by Tony Padilla (February 16^th, 2024) ► Numberphile restarts the thread of £[\sum_{n=0}^{∞}n\stackrel{R}{=}-\frac{1}{12}£], but, this time, the explanations are less shitty than 10 years ago.
- Riemann rearrangement theorem
  - Riemann's paradox: pi = infinity minus infinity by Burkard Polster (July 9^th, 2016) ► Riemann rearrangement theorem.
  - The best A – A ≠ 0 paradox by Burkard Polster (September 9^th, 2023) ► Some facts about the alternating harmonic series.
Irrational numbers
- Can We Combine pi & e to Make a Rational Number? | Infinite Series by Kelsey Houston-Edwards (April 13^th, 2017) ► When Joel David Hamkins and Terence Tao look at the question "If I exchange infinitely many digits of π and e, are the two resulting numbers transcendental?"
- Visualising irrationality with triangular squares by Burkard Polster (April 14^th, 2018) ► Using tile constructions to prove the irrationality of the square roots of 2, 3, 5 and 6.
- The golden ratio spiral: visual infinite descent by Burkard Polster (May 11^th, 2018) ► A similar video, this time using infinite spiral as a rational test. The link between infinite spirals and continued fractions is explained.
- Classic Proof that Irrational to Irrational Power Can Be Rational (+an extra proof) by "LetsSolveMathProblems" (July 23^rd, 2018) ► The title says it all.
- Irrational Roots by Burkard Polster (December 24^th, 2018) ► The integral root theorem and the rational root theorem.
- ★ Le choc de la découverte des Irrationnels (avec Logos) - La Classification des Nombres #7 by Tristan Audam-Dabidin, Keshika Dabidin, "Osiris", and "Apeiron" (May 27^th, 2022) ► How the Pythagoreans stumbled upon irrational numbers, some proofs of the irrationality of some numbers, and how irrationality was mastered later on.
- Rational or Not? This Basic Math Question Took Decades to Answer. — It’s surprisingly difficult to prove one of the most basic properties of a number: whether it can be written as a fraction. A broad new method can help settle this ancient question. by Erica Klarreich (January 8^th, 2025) ► Apéry’s proof that ζ(3) is irrational and the new results of Vesselin Dimitrov, Yunqing Tang, and Frank Calegari’s new results.
Transcendental numbers
- Transcendental numbers powered by Cantor's infinities by Burkard Polster (April 22^nd, 2017) ► Trying to get an understanding of transcendental numbers by using Cantor’s diagonal.
- Liouville's number, the easiest transcendental and its clones (corrected reupload) by Burkard Polster (June 20^th, 2017) ► A proof that Liouville’s number is transcendental.
- The PROOF: e and pi are transcendental by Burkard Polster (September 7^th, 2018) ► The proofs are not fully detailed, but the path is clearly explained.
- Trajectoires #18: Nombres Transcendants, avec Michel Waldschmidt by "FibreTigre" and Michel Waldschmidt (November 12^th, 2019) ► The history and the current status of the research on transcendental numbers.
- How Math Achieved Transcendence — Transcendental numbers include famous examples like e and π, but it took mathematicians centuries to understand them. by David S. Richeson (June 27^th, 2023) ► The subtitle says it all.
Fibonacci and the Golden Ratio
- Little Fibs - Numberphile by Colm Mulcahy (June 2^nd, 2016) ► A card trick based on Fibonacci numbers.
- ↪Little Fibs (extra footage) - Numberphile by Colm Mulcahy (June 2^nd, 2016) ► The continuation of the previous video.
- Infinite fractions and the most irrational number by Burkard Polster (July 30^th, 2016) ► Continued fractions and why the Golden Ration can be considered as the most irrational number.
- The fabulous Fibonacci flower formula by Burkard Polster (August 20^th, 2016) ► An explanation why Fibonacci numbers appear in flowers.
- Réglons une bonne fois pour toute cette histoire de nombre d'or by Jérôme Cottanceau (July 29^th, 2017) ► The title says it all.
- Tuto Maths & Cuisine : Fibonacci et le nombre d'or - Micmaths by Mickaël Launay (November 6^th, 2017) ► Explaining some simple mathematics while cooking.
- Phi and the TRIBONACCI monster by Burkard Polster (December 9^th, 2017) ► Some impressive formulas with ϕ and a quick presentation of the similar formulas with Tribonacci numbers.
- The Golden Ratio (why it is so irrational) - Numberphile by Ben Sparks (May 8^th, 2018) ► The usual explanation of why the golden ratio is the most irrational number.
- Le nombre d'or - Automaths #05🚫 by Jason Lapeyronnie (June 16^th, 2018) ► The golden and silver triangles.
- Random Fibonacci Numbers - Numberphile by James Grime (March 8^th, 2020) ► The random Fibonacci sequence and Viswanath’s constant.
- ↪Random Fibonacci Numbers (extra) - Numberphile by James Grime (March 8^th, 2020) ► The continuation of the previous video and Embree–Trefethen constant.
- La suite de Fibonacci et le nombre d'or - Myriogon #2 by Mickaël Launay (March 17^th, 2020) ► Computing the closed-form expression of the Fibonacci sequence.
- Secrets of the Fibonacci Tiles - 3B1B Summer of Math Exposition↑ by Eric Severson (August 23^rd, 2021) ► Some clear demonstrations of some formulas in the Fibonacci sequence.
- The Truth About Fibs - Numberphile by Marcus du Sautoy (October 26^th, 2022) ► The use of Fibonacci sequence for music and poetry.
- Aux origines du nombre d'or - Deux (deux ?) minutes pour... by Jérôme Cottanceau (July 21^st, 2023) ► The history of the golden ration fad.
- Way beyond the golden ratio: The power of AB=A+B (Mathologer masterclass) by Burkard Polster (August 3^rd, 2024) ► Burkard Polster shows some equations, similar to the golden ratio ones, with the chords of a heptagon.
- A Fascinating Frog Problem - Numberphile by Tom Crawford (September 30^th, 2024) ► Yet another video on Fibonacci… come on!
- Helicone math and the phyllotactic microscope: a secret key to nature and numbers by Burkard Polster (January 4^th, 2025) ► The usual fact that humans see spirals when a rotation is almost a rational ratio of 360°.
- Matt Parker
  - Brady Numbers - Numberphile by Matt Parker (September 22^nd, 2014) ► The ratio of consecutive elements for any sequence defined by £[a_{n+1}=a_n+a_{n-1}£] converges toward Φ.
  - Golden Proof - Numberphile by Matt Parker (September 22^nd, 2014) ► As usual with Numberphile, this is not a proof, but only the calculation of the limit.
  - Lucas Numbers - Numberphile by Matt Parker (September 22^nd, 2014) ► The title says it all.
  - ↪Golden Ratio and Fibonacci Numbers (extra bit) by Matt Parker (September 23^rd, 2014) ► The continuation of the previous video.
  - Golden Ratio BURN (Internet Beef) - Numberphile by Matt Parker (September 6^th, 2018) ► Some facts about the Fibonacci sequence and Lucas numbers.
  - Lucas Numbers and Root 5 - Numberphile by Matt Parker (September 6^th, 2018) ► The formula defining the limit of the ratio of consecutive elements for any sequence.
  - Complex Fibonacci Numbers? by Matt Parker (July 2^nd, 2020) ► This is not about the Fibonacci series, but about drawing a complex function.
- Metallic ratios
  - Beyond the Golden Ratio | Infinite Series by Gabe Perez-Giz (January 26^th, 2018) ► Some properties common to the metallic ratios.
  - The Silver Ratio - Numberphile by Tony Padilla (May 11^th, 2018) ► A simple presentation of metallic ratios.
- The plastic ratio
  - The Plastic Ratio - Numberphile by Edmund Harriss (March 15^th, 2019) ► A presentation of the plastic ratio and of Padovan numbers.
e
- Math in the Simpsons: e to the i pi by Burkard Polster (November 20^th, 2015) ► A basic introduction to Euler’s identity and Euler’s formula.
- e to the pi i for dummies↓ by Burkard Polster (December 24^th, 2015) ► Showing graphically the £[e^{iπ}=-1£].
- e (Euler's Number) - Numberphile by James Grime (December 19^th, 2016) ► An introduction to e by looking at Bernouilli’s compound interest.
- e (Extra Footage) - Numberphile by James Grime (December 25^th, 2016) ► Using binomial theorem to prove that £[e = \sum{\frac{1}{n!}}£].
- The number e explained in depth for (smart) dummies by Burkard Polster (March 30^th, 2017) ► The power series of the exponential function, how to compute the n first digit of e, e is irrational, the exponential function is the derivative of itself, the surface below the hyperbole between 1 and e is 1, Euler’s formula.
- A proof that e is irrational - Numberphile by Edmund Copeland (January 24^th, 2021) ► Fourier’s proof that e is irrational.
π
- Le nombre Pi by Jean Brette (2003) ► Some very basic information about π and the computation Archimedes has done to evaluate it.
- Pi and the size of the Universe - Numberphile by James Grime (February 20^th, 2013) ► How many π digits are useful and the history of calculating them.
- Pi and Four Fingers - Numberphile by Simon Singh (October 31^st, 2013) ► The Simpsons and how to compute π decimals.
- Pi is Beautiful - Numberphile by James Grime (January 3^rd, 2014) ► Drawing pictures using π as a sequence of random numbers.
- Pi and the Mandelbrot Set - Numberphile by Holly Krieger (October 1^st, 2015) ► π appears in the escape time.
- ↪Pi and Mandelbrot (extra footage) by Holly Krieger (October 1^st, 2015) ► The continuation of the previous video.
- When Pi is Not 3.14 | Infinite Series | PBS Digital Studios by Kelsey Houston-Edwards (January 5^th, 2017) ► The value of π depending on the used distance.
- Pi hiding in prime regularities by Grant Sanderson (May 19^th, 2017) ► Using Gaussian primes to demonstrate that the Gregory series converges toward π/4.
- Pi is IRRATIONAL: animation of a gorgeous proof by Burkard Polster (December 23^rd, 2017) ► Lambert’s proof of π’s irrationality.
- Pi is IRRATIONAL: simplest proof on toughest test by Burkard Polster (February 3^rd, 2018) ► A simpler but less natural proof of π’s irrationality.
- How pi was almost 6.283185... by Grant Sanderson (March 14^th, 2018) ► How we ended up with π representing half the perimeter of a circle of radius 1, and why this definition is not important.
- The Wallis product for pi, proved geometrically by Grant Sanderson (April 20^th, 2018) ► The title says it all.
- Approximating Pi ( Monte Carlo integration ) | animation by "Think Twice" (June 10^th, 2018) ► The title says it all.
- Euler's infinite pi formula generator by Burkard Polster (May 2^nd, 2020) ► How to generate some π formulas from Euler’s infinite product formula for sine.
- Why π^π^π^π could be an integer (for all we know!). by Matt Parker (February 27^th, 2021) ► Why it is currently impossible to prove that £[π^{π^{π^π}}£] is not an integer.
- Le Palais de la Découverte fête π by Robin Jamet, Sabrina Coudry, and Mickaël Launay (March 14^th, 2021) ► Some more or less well-known facts about π.
- The Discovery That Transformed Pi by Derek Muller and Alex Kontorovich (March 16^th, 2021) ► Newton’s formula to compute π was much more powerful than the previous methods using polygons.
- New World Record! 100 Trillion digits of π. by Matt Parker and Emma Haruka Iwao (June 15^th, 2022) ► An interview of Emma Haruka Iwao who improved her previous record.
- Using an Out-of-Control Car to Calculate π. by Matt Parker (March 10^th, 2023) ► Yet another silly way to evaluate π.
- Powell’s Pi Paradox: the genius 14th century Indian solution by Burkard Polster (May 6^th, 2023) ► Madhava’s estimation of π and using it to explain Powell’s discovery about π digits vs. Leibniz’ formula digits.
- The biggest hand calculation in a century! [Pi Day 2024] by Matt Parker (March 13^th, 2024) ► Calculating 139 digits by hand.
- New Recipe for Pi - Numberphile by Tony Padilla (July 19^th, 2024) ► Some series converging toward π and the new one discovered by Arnab Priya Saha and Aninda Sinha.
- ↪New Pi Formula (the extra physics bit) - Numberphile by Tony Padilla (July 20^th, 2024) ► Some little information about the real content of the paper.
- ↪Pi-oneers (interview with Sinha & Saha) - Numberphile by Arnab Priya Saha and Aninda Sinha (July 22^nd, 2024) ► An interview of the two authors.
- Collisions
  - Pi and Bouncing Balls - Numberphile by Edmund Copeland (March 12^th, 2012) ► π appears in a collision system. It is a pity the computations are skipped.
  - The most unexpected answer to a counting puzzle by Grant Sanderson (January 13^th, 2019) ► The same.
  - ↪Why do colliding blocks compute pi? by Grant Sanderson (January 20^th, 2019) ► A first method to prove the result presented in the previous video.
  - ↪How colliding blocks act like a beam of light...to compute pi. by Grant Sanderson (February 3^rd, 2019) ► A second method to prove the same, this one uses an analogy with optics.
  - ↪We calculated pi with colliding blocks by Matt Parker, Grant Sanderson, and Steve Mould (March 13^th, 2025) ► Trying to perform the experiment in the real world.
  - Aurélien Alvarez - Billard à deux billes et une bande↓ by Aurélien Alvarez (May 10^th, 2019) ► This is only the problem solved by Grant Sanderson with no information.
  - How Pi Connects Colliding Blocks to a Quantum Search Algorithm — A curious physicist has discovered an unexpected link between theoretical block collisions and a famed quantum search algorithm. by Grant Sanderson (January 21^st, 2020) ► The same calculation appears in Grover’s algorithm.
  - Pi se tape contre le mur (March 18^th, 2021) ► Yet another presentation, this one is in French, shorter, and of lesser quality.
- Indiana Pi Bill
  - How Pi was nearly changed to 3.2 - Numberphile by James Grime (March 12^th, 2013) ► When someone tried pass a bill defining a method to square the circle.
  - Le mathématicien nul de l'Indiana - Flash 09 - e-penser by Bruce Benamran (December 27^th, 2019) ► The same story.
Real numbers
- Why do we "complete the square"? by Presh Talwalkar (October 2^nd, 2015) ► A geometric visualisation of the quadratic formula.
- e^pi vs pi^e (no calculator) by Steve Chow (July 1^st, 2016) ► Determining what is the bigger: e^π or π^e?
- believe in the math, not wolframalpha by Steve Chow (August 18^th, 2017) ► Computing manually £[\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}£].
- the end-of-year exponential equation! by Steve Chow (December 31^st, 2017) ► Finding the solution of £[x^{x^{x^{2017}}}=2017£] not with calculus but with observation.
- A COUNTER-INTUITIVE CALCULUS LIMIT by Steve Chow (January 8^th, 2018) ► £[\lim_{x\to\infty}{(1+\frac{1}{x})^xx-ex}=-\frac{e}{2}£].
- The Viral Balloon Puzzle - The REAL Answer Explained (Using Ph.D. Level Math)⇊ by Presh Talwalkar (July 3^rd, 2018) ► Some delirium using Lambert W function and Knuth’s up-arrow notation.
- yes, I can do ln(e+ln(e+ln(e+...))) in UNDER one minute by Steve Chow (December 3^rd, 2018) ► £[ln(e+ln(e+ln(e+…)))=-e-W(-e^{-e})£].
- solving the tetration equation x^x^x=2 by using Newton's Method by Steve Chow (December 11^th, 2018) ► Numeric resolution of £[x^{x^x}£] using Newton’s method.
- 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle? by Burkard Polster (August 24^th, 2019) ► A presentation of the cubic formula and how it can be built.
- Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula) by Burkard Polster (October 26^th, 2019) ► From power sums to the Euler–Maclaurin formula.
- ALL solutions to x^2=2^x by Steve Chow (October 29^th, 2019) ► Solving £[x^2=2^x£].
- A Golden Answer by Presh Talwalkar (November 19^th, 2019) ► Solving £[4^x+6^x=9^x£].
- life changing quadratic formula by Peyam Ryan Tabrizian (December 7^th, 2020) ► Another way to solve a quadratic equation.
- AI maths whiz creates tough new problems for humans to solve — Algorithm named after mathematician Srinivasa Ramanujan suggests interesting formulae, some of which are difficult to prove true. by Davide Castelvecchi (February 3^rd, 2021) ► Some little information about the Ramanujan Machine.
- Comment écrire les nombres ayant une infinité de décimales ? by David Louapre (August 27^th, 2021) ► Some miscellaneous facts about numbers: continued fractions, Khinchin’s constant, definable numbers…
- can you solve this “impossible” trig problem? by Peyam Ryan Tabrizian (December 14^th, 2022) ► Solving £[81^{sin(x)^2}+81^{cos(x)^2}=30£].
- Ramanujan's easiest hard infinity monster (Mathologer Masterclass) by Burkard Polster (June 24^th, 2023) ► Showing that £[e^{\frac{x^2}{2}} \cdot \sqrt{\frac{ \pi }{2}} = (\frac{x^1}{1}+\frac{x^2}{1 \cdot 2}+\frac{x^3}{1 \cdot 2 \cdot 3}+\frac{x^4}{1 \cdot 2 \cdot 3 \cdot 4}+\ldots) + \frac{1}{x + \frac{1}{x + \frac{2}{x + \frac{3}{x + \frac{4}{x + \ddots}}}}}£].
- The Prime Constant - Numberphile by Matt Parker (October 6^th, 2024) ► This video is just about the fact that any monotonic series of integers can be encoded as a real number, with the prime constant being the one for primes.
Hyperreal numbers
- The Infinitesimal Monad - Numberphile by Carol Wood (September 4^th, 2015) ► The title says it all.
- ↪Infinitesimal Monad (extra footage) by Carol Wood (September 7^th, 2015) ► The continuation of the previous video.
- The Opposite of Infinity - Numberphile by James Grime (September 8^th, 2015) ► Calculus and hyperreal numbers.
Surreal numbers
- Surreal Numbers (writing the first book) - Numberphile by Donald Knuth (June 27^th, 2016) ► Donald Knuth speaks about his discussion with John Conway and his writing of the book for 6 days.
- ↪Surreal Numbers (Don Knuth Extra Footage) - Numberphile by Donald Knuth (June 28^th, 2016) ► The continuation of the previous video.
- John Conway: Surreal Numbers - How playing games led to more numbers than anybody ever thought of by John Conway (July 14^th, 2016) ► John Conway speaks about surreal numbers and Cantor.
- HACKENBUSH: a window to a new world of math by Owen Maitzen (August 22^nd, 2021) ► The presentation of Hackenbush, combinatorial game theory, and surreal numbers; the beginning is understandable, the end is crazy.
Benford’s Law
- Benford's Law - How mathematics can detect fraud! by James Grime (April 9^th, 2011) ► A presentation of Benford’s Law and a proof that the only distribution that is scale invariant must be the logarithmic distribution.
- Number 1 and Benford's Law - Numberphile by Steve Mould (January 20^th, 2013) ► Yet another presentation of Benford’s Law.
- ↪Brady's Videos and Benford's Law - Numberphile by Brady Daran (January 21^st, 2013) ► Is Benford’s law applicable to YouTube statistics?
- Powers of 2 - Numberphile by Dmitry Kleinbock (October 5^th, 2015) ► The first digit of the powers of 2 is regulated by Benford’s Law.
- Why do Biden's votes not follow Benford's Law? by Matt Parker (November 11^th, 2020) ► Benford’s law and other statistical tests to check data validity.
- ↪Do these scatter plots reveal fraudulent vote-switching in Michigan? by Matt Parker (November 15^th, 2020) ► Debunking an incorrect statistical analysis.
- ↪Why was Biden's win calculated to be ONE IN A QUADRILLION? by Matt Parker (December 18^th, 2020) ► Still debunking some statistical mistakes.
- La loi de Benford | Voyages au pays des maths | ARTE by Denis van Waerebeke (October 2^nd, 2021) ► A basic presentation of Benford’s Law.