Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions) @Mathologer

Uploaded January 2021 | Updated September 2026, 2 weeks ago

Okay, as it says in the title of this video, today's mission is to figure out how many ways there are to make change for one googol, that is 10^100 dollars. The very strange patterns in the answer will surprise, as will the explanation for this phenomenon, promise.

0:00 Intro
1:15 Chapter 1: curious count
9:05 Chapter 2: googol
14:10 Chapter 3: infinite change
28:41 Acknowledgements

A very nice Mathematica file created by Andrew Neitsch in 2005 covers pretty much every aspect of change making mathematics. andrew.neitsch.ca/publications/m496pres1.nb
Here is a pdf version of this file: andrew.neitsch.ca/publications/m496pres1.nb.pdf
What I cover in the last part of this video is pretty much fleshing out and animating the section "Coin change revisited". All this is part of to Andrew's answer to a post on math.stackoverflow stackoverflow.com/questions/1106929/how-to-find-all-combinations-of-coins-when-given-some-dollar-value

The visual algebra approach to calculate the number of ways to make change at the very beginning of this video was inspired by this article G. Pólya, On Picture-Writing, Am Math Monthly 63 (1956), 689-697. jstor.org/stable/2309555

Concrete mathematics by Graham, Knuth and Patashnik, the book I mentioned at the end of the video does the whole analysis for the coin set 1, 5, 10, 25, 50 (so no dollar coins).

A complete list of all the different ways to make change for a dollar appears in this post maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-dollar

The book "Generatingfunctionology" by Herbert Wilf, is a great intro to generating functions :) https://www2.math.upenn.edu/~wilf/DownldGF.html

Ron Graham to who this video is dedicated did a couple of videos with Numberphile. So if you'd like to see him in action, check out those videos. A lot of other interesting articles about Ron Graham can be found on his wife's (also a math professor) website. http://www.math.ucsd.edu/~fan/ron/

As usual the music in the video is from the free YouTube audio library: No. 2 Remembering her by Ester Abrami, Morning Mandolin by Chris Haugen, First time experience and T'is the season by Nate Blaze

Today's t-shirts: google "only half evil t-shirt".

Enjoy!

Burkard

Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions)

$The best A – A ≠ 0 paradox This video is about a new stunning visual resolution of a very pretty and important paradox that I stumbled across while I was preparing the last video on logarithms. 00:00 Intro 00:56 Paradox 03:52 Visual sum = ln(2) 07:58 Pi 11:00 Gelfonds number 14:22 Pi exactly 17:35 Riemanns rearrangement theorem 22:40 Thanks! Riemann rearrangement theorem. https://en.wikipedia.org/wiki/Riemann_series_theorem This page features a different way to derive the sums of those nice m positive/n negative term arrangements of the alternating harmonic series by expressing H(n) the sum of the first n harmonic numbers by ln(n) and the Euler–Mascheroni constant. That could also be made into a very nice visual proof along the lines that I follow in this video https://youtu.be/vQE6-PLcGwU?si=iWTasKqo_JFn4etG&t=1321. Gelfonds number e^π being approximate equal to 20 + π may not be a complete coincidence after all: @mathfromalphatoomega Theres actually a sort-of-explanation for why e^π is roughly π+20. If you take the sum of (8πk^2-2)e^(-πk^2), it ends up being exactly 1 (using some Jacobi theta function identities). The first term is by far the largest, so that gives (8π-2)e^(-π)≈1, or e^π≈8π-2. Then using the estimate π≈22/7, we get e^π≈π+(7π-2)≈π+20. I wouldnt be surprised if it was already published somewhere, but I havent been able to find it anywhere. I was working on some problems involving modular forms and I tried differentiating the theta function identity θ(-1/τ)=√(τ/i)*θ(τ). That gave a similar identity for the power series Σk^2 e^(πik^2τ). It turned out that setting τ=i allowed one to find the exact value of that sum. (@kasugaryuichi9767) I dont know if its new, but its certainly not well known. To quote the Wolfram MathWorld article Almost Integer: This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to why e^π-π≈20 is true has yet been discovered. Ratio of the number of positive and negative terms Its interesting to look at the patterns of positive & negative terms when rearranging to Pi. We always only use one negative term before we switch. The first ten terms on the positive side are: 13, 35, 58, 81, 104, 127, 151, 174, 197, 220,... If you look at the differences between terms, you get: 22, 23, 23, 23, 23, 24, 23, 23, 23, 23, 23, 23, 23, 24,... The reason for this is that Gelfonds number is approximately equal to 23. It turns out that if an arrangement of our series has the sum pi, then the ratio of the numbers of positive to negative terms in the finite partial sums of the series converges to Gelfonds number. This is just one step up from what I said about us being able to get arbitrarily close to pi by turning truncations of the decimal expansion of Gelfonds number into fractions. Similarly for other target numbers. For example, to predict what the repeating pattern for e is, you just have to calculate e^e :) @penguincute3564 thus ln(0) = negative infinity (referring to +0/1-) Bug report: At the 1:18 mark, I say minus one sixth when I should have just said one sixth. Music: Silhouettes only-piano by Muted T-shirt: Pi Day Left Vs Right Brain Pie Math Geek T-Shirt https://tinyurl.com/3e3p5yeb Enjoy! Burkard$

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$Eulers infinite pi formula generator Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Walliss infinite product formula, Lord Brounckers infinite fraction formula, Eulers Basel formula and its infinitely many cousins. And we do this starting with one of Eulers crazy strokes of genius, his infinite product formula for the sine function. This video was inspired by Paul Levries one-page article Eulers wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here https://link.springer.com/journal/283/34/4 Very pretty. If you are a regular and some of what I talk about in this video looks familiar thats not surprising since weve visited this territory before in Eulers real identity NOT e to the i pi = -1: https://youtu.be/yPl64xi_ZZA 0:00 Intro 1:49 A sine of madness. Eulers ingenious derivation of the product formula for sin x 7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*.../1*3*3*5*5*... 9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+... 11:50 Brounckers infinite fraction formula for pi: 4/pi = ... 18:31 Eulers solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+... 21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+... , etc. Music (all from the free audio library that YouTube provides to creators): https://www.youtube.com/audiolibrary/music?nv=1 Take me to the Depth (chapter transitions) Fresh fallen snow Morning mandolin English country garden Enjoy! Burkard$ $The PROOF: e and pi are transcendental Today’s video is dedicated to introducing you to two of the holy grails of mathematics, proofs that e and pi are transcendental numbers. For the longest time I was convinced that these proofs were simply out of reach of a self-contained episode of Mathologer, and I even said so in a video on transcendental numbers last year. Well, I am not teaching any classes at uni this semester and therefore got a bit more time to spend on YouTube. And so I thought why not sink some serious time into trying to make this “impossible” video anyway. I hope you enjoy the outcome and please let me know in the comments which of the seven levels of enlightenment that make up this video you manage to conquer. Even if you just make it to the end of level one it will be an achievement and definitely worth it :) 0:00 Intro 1:48 Enter Transcendence 2:12 Level 1 - e - irrational 6:40 Level 2 - e - quadratic irrational 10:03 Level 3 - e - master proof 12:47 Level 4 - e - gamma 26:14 Level 5 - pi - Lindemann’s trick 29:59 Level 6 - pi - transcendence 35:35 Level 7 Today’s video would have been impossible without the help from my analyst colleague and friend Marty Ross who did most of the heavy lifting in terms of identifying and adapting the least crazy transcendence proofs for e and pi in existence for this video. Here is Marty’s writeup of the infinite series based proof that e is not a quadratic irrational http://www.qedcat.com/notes/e%20not%20quadratic.pdf and here is his formal write-ups of the proofs for the transcendence of pi and e that this video is based on http://www.qedcat.com/notes/e%20+%20pi%20transcendental.pdf. Marty’s version of the proofs is based on this paper by Steinberg and Redheffer https://projecteuclid.org/download/pdf_1/euclid.pjm/1103051870 which in turn has its origins in a proof by David Hilbert http://www.cs.toronto.edu/~yuvalf/Hilbert%20Ueber%20die%20Transzendenz%20der%20Zahlen%20e%20und%20pi.pdf. Apart from Marty I’d also like to thank MIchael Fraklin for his help with recording this episode of Mathologer, as well as Danil Diitriev who as usual will take care of the preparing amazing Russian subtitles for this video. The mysterious Buddhabrot fractal was discovered by Melinda Green. For an intro to this strange mathematical creature check out my Mandelbrot video. http://youtu.be/9gk_8mQuerg Finally, here is a playlist of all my videos on irrational and transcendent numbers. http://www.youtube.com/playlist?list=PLmNp3NTX4KXK6-xgFJdMAbavzgQaKjRzY Enjoy :) Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)$

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