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I have prepared a new paper containing a catalogue of 72 summation formulas, integral formulas and iterative algorithms for Pi. The catalogue contains both classical and modern formulas, ranging from Archimedes’ 2200-year-old algorithm to intriguing formulas found by Ramanujan and the quadratic, cubic, quartic and nonic algorithms of Jonathan Borwein and Peter Borwein, the latter of which double, triple, quadruple and nine-times, respectively, the number of correct digits with each iteration.
The catalogue of formulas and iterative algorithm is followed by results of carefully designed computer implementations, which enable one to compare the relative speed of these formulas.
Continue reading PiDay 2020: A catalogue of formulas involving pi, with analysis
Yes, it is that time of year — Pi Day (March 14, or 3/14 in North American month/day date notation) is here.
So in honor of the occasion, I have constructed a new crossword puzzle — see below. This puzzle honors several of the key persons through history who have made significant contributions to the theory and computation of Pi.
This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).
Continue reading Pi Day 2020: A new crossword puzzle
Updated 23 July 2025
Introduction
A growing controversy over the multiverse and the anthropic principle has exposed a major fault line in modern physics and cosmology. Some researchers see the multiverse and the anthropic principle as inevitable, others see them as an abdication of empirical science. The controversy spans quantum mechanics, inflationary Big Bang cosmology, string theory, supersymmetry and, more generally, the proper roles of experimentation and mathematical theory in modern science.
The “many worlds interpretation” of quantum mechanics
Since the 1930s, when physicists first developed the mathematics behind quantum mechanics, researchers have found that this theory appears to
Continue reading Universe or multiverse? The war rages on
Do probability arguments refute evolution? Updated 30 September 2025 (c) 2025
Introduction
In the past few decades, modern science has uncovered a universe that is far vaster and more awe-inspiring than ever imagined before, together with a set of elegant natural laws that deeply resonate with the idea of a cosmic lawgiver. Along this line, 46% of Americans, including, interestingly enough, 54% of atheists, 55% of agnostics and 43% of nones, say that they experience a “deep sense of wonder about the universe” on at least a weekly basis [Masci2016].
In spite of these exhilarating developments, some writers, principally
Continue reading Do probability arguments refute evolution?
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Factorization and cryptography
Until a few decades ago, number theory, namely the study of prime numbers, factorization and other features of the integers, was widely regarded as the epitome of pure mathematics, completely divorced from considerations of practical utility. This sentiment was expressed most memorably by British mathematician G.H. Hardy (best known for mentoring Ramanujan and results on the Riemann Zeta function), who wrote in his book A Mathematician’s Apology (1941),
I have never done anything “useful”. No discovery of mine has made, or is likely to make, directly or indirectly, for good
Continue reading New factorization advances: Is your bank account safe?
[Note: Simons died 10 May 2024 at the age of 86. See this New York Times report for a nice synopsis of his life and work.]
Gregory Zuckerman, author of The Greatest Trade Ever, has published a new book highlighting the life and work of Jim Simons, who, at the age of 40, walked away from a very successful career as a research mathematician and cryptologist to try his hand at the financial markets, and ultimately revolutionized the field. Zuckerman’s new book is titled The Man Who Solved the Market: How Jim Simons Launched the Quant Revolution.
Upbringing and academic
Continue reading Jim Simons: The man who solved the market
Credit: IPCC
The facts of climate change
At this point in time, the basic facts of climate change are not disputable in the least. Careful planet-wide observations by NASA and others have confirmed that 2018 was the fourth-warmest year in recorded history. The only warmer years were 2016, 2017 and 2015, respectively, and 18 of the 19 warmest years in history have occurred since 2001. Countless observational studies and supercomputer simulations have confirmed both the fact of warming and the conclusion that this warming is principally due to human activity. These studies and computations have been scrutinized in great
Continue reading The scientific debate is over: it is time to act on climate change
IBM’s “Q” quantum computer; courtesy IBM
Google’s quantum computing achievement
For at least three decades, teams of researchers have been exploring quantum computers for real-world applications in scientific research, engineering and finance. Researchers have dreamed of the day when quantum computers would first achieve “supremacy” over classical computers, in the sense that a quantum computer solving a particular problem faster than any present-day or soon-to-be-produced classical computer system.
In a Nature article dated 23 October 2019, researchers at Google announced that they have achieved exactly this.
Google researchers employed a custom-designed quantum processor, named “Sycamore,” consisting of programmable quantum
Continue reading Quantum supremacy has been achieved; or has it?
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Credit: Ancient Origins
Introduction In a previous Math Scholar blog, we presented Archimedes’ ingenious scheme for approximating $\pi$, based on an analysis of regular circumscribed and inscribed polygons with $3 \cdot 2^k$ sides, using modern mathematical notation and techniques.
One motivation for both the previous blog and this blog is to respond to some recent writers who reject basic mathematical theory and the accepted value of $\pi$, claiming instead that they have found $\pi$ to be a different value. For example, one author asserts that $\pi = 17 – 8 \sqrt{3} =
Continue reading Pi as the limit of n-sided circumscribed and inscribed polygons
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Log_10 of the error of a continued fraction approximation of Pi to k terms
Approximation of real numbers by rationals
The question of finding rational approximations to real numbers was first explored by the Greek scholar Diophantus of Alexandra (c. 201-285 BCE), and continues to fascinate mathematicians today, in a field known as Diophantine approximations.
It is easy to see that any real number can be approximated to any desired accuracy by simply taking the sequence of approximations given by the decimal digits out to some point, divided by the appropriate power
Continue reading New paper proves 80-year-old approximation conjecture
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