Posts Tagged ‘Number theory’
“Why, sometimes I’ve believed as many as six impossible things before breakfast”*…
Imaginary numbers were long dismissed as mathematical “bookkeeping.” But now, as Karmela Padavic-Callaghan explains, physicists are proving that they describe the hidden shape of nature…
Many science students may imagine a ball rolling down a hill or a car skidding because of friction as prototypical examples of the systems physicists care about. But much of modern physics consists of searching for objects and phenomena that are virtually invisible: the tiny electrons of quantum physics and the particles hidden within strange metals of materials science along with their highly energetic counterparts that only exist briefly within giant particle colliders.
In their quest to grasp these hidden building blocks of reality scientists have looked to mathematical theories and formalism. Ideally, an unexpected experimental observation leads a physicist to a new mathematical theory, and then mathematical work on said theory leads them to new experiments and new observations. Some part of this process inevitably happens in the physicist’s mind, where symbols and numbers help make invisible theoretical ideas visible in the tangible, measurable physical world.
Sometimes, however, as in the case of imaginary numbers – that is, numbers with negative square values – mathematics manages to stay ahead of experiments for a long time. Though imaginary numbers have been integral to quantum theory since its very beginnings in the 1920s, scientists have only recently been able to find their physical signatures in experiments and empirically prove their necessity…
Learn more at “Imaginary numbers are real,” from @Ironmely in @aeonmag.
* The Red Queen, in Lewis Carroll’s Through the Looking Glass
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As we get real, we might spare a thought for two great mathematicians…
Georg Friedrich Bernhard Riemann died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
Andrey (Andrei) Andreyevich Markov died on this date in 1922. A Russian mathematician, he helped to develop the theory of stochastic processes, especially those now called Markov chains: sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. (For example, the probability of winning at the game of Monopoly can be determined using Markov chains.) His work on the study of the probability of mutually-dependent events has been developed and widely applied to the biological, physical, and social sciences, and is widely used in Monte Carlo simulations and Bayesian analyses.
“No part of mathematics is ever, in the long run, ‘useless’.”*…
The number 1 can be written as a sum of distinct unit fractions, such as 1/2 + 1/3 + 1/12 + 1/18 + 1/36…
Number theorists are always looking for hidden structure. And when confronted by a numerical pattern that seems unavoidable, they test its mettle, trying hard — and often failing — to devise situations in which a given pattern cannot appear.
One of the latest results to demonstrate the resilience of such patterns, by Thomas Bloom of the University of Oxford, answers a question with roots that extend all the way back to ancient Egypt.
“It might be the oldest problem ever,” said Carl Pomerance of Dartmouth College.
The question involves fractions that feature a 1 in their numerator, like 1/2, 1/7 or 1/122. These “unit fractions” were especially important to the ancient Egyptians because they were the only types of fractions their number system contained: With the exception of a single symbol for 23, they could only express more complicated fractions (like 3/4) as sums of unit fractions (1/2 + 1/4).
The modern-day interest in such sums got a boost in the 1970s, when Paul Erdős and Ronald Graham asked how hard it might be to engineer sets of whole numbers that don’t contain a subset whose reciprocals add to 1. For instance, the set {2, 3, 6, 9, 13} fails this test: It contains the subset {2, 3, 6}, whose reciprocals are the unit fractions 1/2, 1/3 and 1/6 — which sum to 1.
More exactly, Erdős and Graham conjectured that any set that samples some sufficiently large, positive proportion of the whole numbers — it could be 20% or 1% or 0.001% — must contain a subset whose reciprocals add to 1. If the initial set satisfies that simple condition of sampling enough whole numbers (known as having “positive density”), then even if its members were deliberately chosen to make it difficult to find that subset, the subset would nonetheless have to exist.
“I just thought this was an impossible question that no one in their right mind could possibly ever do,” said Andrew Granville of the University of Montreal. “I didn’t see any obvious tool that could attack it.”…
Bloom, building on work by Ernie Croot, found that tool. The ubiquity of ways to represent whole numbers as sums of fractions: “Math’s ‘Oldest Problem Ever’ Gets a New Answer,” by Jordana Cepelewicz (@jordanacep) in @QuantaMagazine.
* “No part of mathematics is ever, in the long run, ‘useless.’ Most of number theory has very few ‘practical’ applications. That does not reduce its importance, and if anything it enhances its fascination. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role.” – C. Stanley Ogilvy, Excursions in Number Theory
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As we recombine, we might send carefully-calculated birthday greetings to Ulugh Beg (or, officially, Mīrzā Muhammad Tāraghay bin Shāhrukh); he was born on this date in 1394. A Timurid sultan with a hearty interest in science and the arts, he is better remembered as an astronomer and mathematician.
The most important observational astronomer of the 15th century, he built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429– considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia. In his observations he discovered a number of errors in the computations of the 2nd-century Alexandrian astronomer Ptolemy, whose figures were still being used. His star map of 994 stars was the first new one since Hipparchus. Among his contributions to mathematics were trigonometric tables of sine and tangent values correct to at least eight decimal places.
“If the doors of perception were cleansed everything would appear to man as it is, infinite”*…
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise…
Infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.
Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number ℵ0 (“aleph-zero”).
But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.
Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.
Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from all the different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality ℵ1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.
His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely ℵ1 real numbers. In other words, the cardinality of the continuum immediately follow ℵ0, the cardinality of the natural numbers, with no sizes of infinity in between.
But to Cantor’s immense distress, he couldn’t prove it.
In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.
To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.
The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove. As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.
These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.
In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.
In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.
They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.
Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question…
Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible.
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In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said.
Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d long pursued the proof, Ralf Schindler of the University of Münster in Germany, and described his insight. “It was completely incomprehensible to me,” Schindler said. But eventually, the duo turned the phantasm into solid logic.
Their proof, which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Asperó and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true…
There are an infinite number of infinities. Which one corresponds to the real numbers? “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.”
[TotH to MK]
* William Blake
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As we contemplate counting, we might spare a thought for Georg Friedrich Bernhard Riemann; he died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.
“The urge to gamble is so universal and its practice so pleasurable that I assume it must be evil”*…
Gambling has existed since antiquity, but in the past 30 years it’s grown at a spectacular rate, turbocharged by the internet and globalisation. Problem gambling has grown accordingly, and become particularly prevalent in the teenage population. Even more troublingly, a study in 2013 reported that slightly over 90 per cent of problem gamblers don’t seek professional help. Gambling addiction is part of a suite of damaging and unhealthy behaviours that people do despite warnings, such as smoking, drinking or compulsive video gaming. It draws on a multitude of cognitive, social and psychobiological factors.
Psychological and medical studies have found that some people are more likely to develop a gambling disorder than others, depending on their social condition, age, education and experiences such as trauma, domestic violence and drug abuse. Problem gambling also involves complex brain chemistry, as gambling stimulates the release of multiple neurotransmitters including serotonin and dopamine, which in turn create feelings of pleasure and the attendant urge to maintain them. Serotonin is known as the happiness hormone, and typically follows a sense of release from stress or fear. Dopamine is associated with intense pleasure, released when we’re engaged in activities that deserve a reward, and precisely when that reward occurs – seeing the ball landing on the number we’ve bet on, or hearing the sound of the slot machine showing a winning payline.
For the most part, gambling addiction is viewed as a medical and psychological problem, though this hasn’t resulted in widely effective prevention and treatment programmes. That might be because the research has often focused on the origins and prevalence of addiction, and less on the cognitive premises and mechanisms that actually take place in the brain. It’s a controversial area, but this arguable lack of clinical effectiveness doesn’t appear to be specific to gambling; it applies to other addictions as well, and might even extend to some superstitions and irrational beliefs.
Can a proper presentation of the mathematical facts help gambling addiction? While most casino moguls simply trust the mathematics – the probability theory and applied statistics behind the games – gamblers exhibit a strange array of positions relative to the role of maths. While no study has offered an exhaustive taxonomy, what we know for sure is that some simply don’t care about it; others care about it, trust it, and try to use it in their favour by developing ‘winning strategies’; while others care about it and interpret it in making their gambling predictions.
Certain problem gambling programmes frame the distortions associated with gambling as an effect of a poor mathematical knowledge. Some clinicians argue that reducing gambling to mere mathematical models and bare numbers – without sparkling instances of success and the ‘adventurous’ atmosphere of a casino – can lead to a loss of interest in the games, a strategy known as ‘reduction’ or ‘deconstruction’. The warning messages involve statements along the lines of: ‘Be aware! There is a big problem with those irrational beliefs. Don’t think like that!’ But whether this kind of messaging really works is an open question. Beginning a couple of decades ago, several studies were conducted to test the hypothesis that teaching basic statistics and applied probability theory to problem gamblers would change their behaviour. Overall, these studies have yielded contradictory, non-conclusive results, and some found that mathematical education yielded no change in behaviour. So what’s missing?…
Catalin Barboianu, a gaming mathematician, philosopher of science, and problem-gambling researcher, asks if philosophers and mathematicians struggle with probability, can gamblers really hope to grasp their losing game? “Mathematics for Gamblers.”
For a deeper dive, see Alec Wilkinson’s fascinating New Yorker piece, “What Would Jesus Bet? A math whiz hones the optimal poker strategy.”
For cultural context (and an appreciation of the broader importance of the issue), see “How Gambling Mathematics Took Over The World.”
And for historical context, see (one of your correspondent’s all-time favorite books) Peter Bernstein’s Against the Gods: The Remarkable Story of Risk.
[image above: source]
* Heywood Hale Broun
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As we roll the dice, we might spare a thought for Srinivasa Ramanujan; he died on this date in 1920. A largely self-taught mathematician from Madras, he initially developed his own mathematical research in isolation: according to Hans Eysenck: “He tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered.” Seeking mathematicians who could better understand his work, in 1913 he began a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognizing Ramanujan’s work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that “defeated me completely; I had never seen anything in the least like them before.”
Ramanujan made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. During his short life, he independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae, and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Nearly all his claims have now been proven correct.
See also: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater,” and enjoy the 2015 film on Ramanujan, “The Man Who Knew Infinity.”
“He told me that in 1886 he had invented an original system of numbering”*…
The rational numbers are the most familiar numbers: 1, -5, ½, and every other value that can be written as a ratio of positive or negative whole numbers. But they can still be hard to work with.
The problem is they contain holes. If you zoom in on a sequence of rational numbers, you might approach a number that itself is not rational. This short-circuits a lot of basic mathematical tools, like most of calculus.
Mathematicians usually solve this problem by arranging the rationals in a line and filling the gaps with irrational numbers to create a complete number system that we call the real numbers.
But there are other ways of organizing the rationals and filling the gaps: the p-adic numbers. They are an infinite collection of alternative number systems, each associated with a unique prime number: the 2-adics, 3-adics, 5-adics and so on.
The p-adics can seem deeply alien. In the 3-adics, for instance, 82 is much closer to 1 than to 81. But the strangeness is largely superficial: At a structural level, the p-adics follow all the rules mathematicians want in a well-behaved number system…
“We’re all on Earth and we work with the reals, but if you went [anywhere] else, you’d work with the p-adics,” [University of Washington mathematician Bianca] Viray explained. “It’s the reals that are the outliers.”
The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory… which is itself at the heart of computer science, numerical analysis, and cryptography: “An Infinite Universe of Number Systems.”
* Jorge Luis Borges, Labyrinths
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As we dwell on digits, we might send carefully-calculated birthday greetings to Klaus Friedrich Roth; he was born on this date in 1925. After escaping with his family from Nazi Germany, he was educated at Cambridge, then taught mathematics first at University College London, then at Imperial College London. He made a number of important contribution to Number Theory, for which he won the De Morgan Medal and the Sylvester Medal, and election to Fellowship of the Royal Society. In 1958 he was awarded mathematics’ highest honor, the Fields Medal, for proving Roth’s theorem on the Diophantine approximation of algebraic numbers.
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