Is space continuous or discrete? Is it smooth, without gaps or discontinuities, or granular with a limit on how small a distance can be? What about time? Can time be repeatedly divided into smaller periods without any limit, or is there a shortest interval of time? We don’t know the answers. There is much we do not know about physical reality: is the universe finite or infinite? Are space and time arbitrarily divisible? Does our number system represent physical space and time? [TM215 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘All Numbers Great and Small’
Posts Tagged 'Number Theory'
All Numbers Great and Small
Published July 15, 2021 Irish Times Leave a CommentTags: Number Theory, Physics
The Simple Arithmetic Triangle is full of Surprises
Published June 3, 2021 Irish Times Leave a CommentTags: Arithmetic, Number Theory
Pascal’s triangle is one of the most famous of all mathematical diagrams, simple to construct and yet rich in mathematical patterns. These can be found by a web search, but their discovery by study of the diagram is vastly more satisfying, and there is always a chance of finding something never seen before [TM212 or search for “thatsmaths” at irishtimes.com].
Pascal’s triangle as found in Zhu Shiji’s treatise The Precious Mirror of the Four Elements (1303).
Continue reading ‘The Simple Arithmetic Triangle is full of Surprises’
Goldbach’s Conjecture: if it’s Unprovable, it must be True
Published March 4, 2021 Irish Times 1 CommentTags: Euler, Logic, Number Theory
Image © easycalculation.com
The starting point for rigorous reasoning in maths is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. The Greek mathematician Thales is credited with introducing the axiomatic method, in which each statement is deduced either from axioms or from previously proven statements, using the laws of logic. The axiomatic method has dominated mathematics ever since [TM206 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Goldbach’s Conjecture: if it’s Unprovable, it must be True’
Derangements and Continued Fractions for e
Published December 31, 2020 Occasional Leave a CommentTags: Combinatorics, Number Theory
We show in this post that an elegant continued fraction for 
Continue reading ‘Derangements and Continued Fractions for e’
Arrangements and Derangements
Published December 24, 2020 Irish Times Leave a CommentTags: Combinatorics, Number Theory
Six students entering an examination hall place their cell-phones in a box. After the exam, they each grab a phone at random as they rush out. What is the likelihood that none of them gets their own phone? The surprising answer — about 37% whatever the number of students — emerges from the theory of derangements.
Will RH be Proved by a Physicist?
Published December 10, 2020 Occasional Leave a CommentTags: Analysis, Number Theory
The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the critical line, 
The non-trivial zeros of 
eigenvalues of a hermitian operator.
Is there a Riemann operator? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?
This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!
The p-Adic Numbers (Part 2)
Published October 29, 2020 Occasional Leave a CommentTags: Number Theory
Kurt Hensel (1861-1941)
Kurt Hensel, born in Königsberg, studied mathematics in Berlin and Bonn, under Kronecker and Weierstrass; Leopold Kronecker was his doctoral supervisor. In 1901, Hensel was appointed to a full professorship at the University of Marburg. He spent the rest of his career there, retiring in 1930.
Hensel is best known for his introduction of the p-adic numbers. They evoked little interest at first but later became increasingly important in number theory and other fields. Today, p-adics are considered by number theorists as being “just as good as the real numbers”. Hensel’s p-adics were first described in 1897, and much more completely in his books, Theorie der algebraischen Zahlen, published in 1908 and Zahlentheorie published in 1913.
The p-Adic Numbers (Part I)
Published October 22, 2020 Occasional Leave a CommentTags: Number Theory
Image from Cover of Katok, 2007.
The motto of the Pythagoreans was “All is Number”. They saw numbers as the essence and foundation of the physical universe. For them, numbers meant the positive whole numbers, or natural numbers 
If we arrange the rational numbers on a line, there are gaps everywhere. We can fill these gaps by introducing additional numbers, which are the limits of sequences of rational numbers. This process of completion gives us the real numbers 
Think of a Number: What are the Odds that it is Even?
Published August 13, 2020 Occasional Leave a CommentTags: Number Theory, Probability
Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime?
Probability distribution 
Continue reading ‘Think of a Number: What are the Odds that it is Even?’
The Ever-growing Goals of Googology
Published July 2, 2020 Irish Times Leave a CommentTags: Arithmetic, Number Theory
In 1920, a kindergarten class was asked to describe the biggest number that they could imagine. One child proposed to “write down digits until you get tired”. A more concrete idea was to write a one followed by 100 zeros. This number, which scientists would express as ten to the power 100, was given the name “googol” by its inventor [TM190; or search for “thatsmaths” at irishtimes.com ].
The Online Encyclopedia of Integer Sequences
Published June 25, 2020 Occasional Leave a CommentTags: Analysis, Arithmetic, Number Theory
Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.
The Home page of OEIS: https://oeis.org
Continue reading ‘The Online Encyclopedia of Integer Sequences’
John Horton Conway: a Charismatic Genius
Published April 23, 2020 Occasional Leave a CommentTags: Games, Group Theory, Number Theory, Recreational Maths, Topology
John H Conway in 2009
[image Denise Applewhite, Princeton University].
John Horton Conway was a charismatic character, something of a performer, always entertaining his fellow-mathematicians with clever magic tricks, memory feats and brilliant mathematics. A Liverpudlian, interested from early childhood in mathematics, he studied at Gonville & Caius College in Cambridge, earning a BA in 1959. He obtained his PhD five years later, after which he was appointed Lecturer in Pure Mathematics.
In 1986, Conway moved to Princeton University, where he was Professor of Mathematics and John Von Neumann Professor in Applied and Computational Mathematics. He was awarded numerous honours during his career. Conway enjoyed emeritus status from 2013 until his death just two weeks ago on 11 April.
Bang! Bang! Bang! Explosively Large Numbers
Published March 26, 2020 Occasional Leave a CommentTags: Algebra, Number Theory
Typical Comic-book `bang’ mark [Image from vectorstock ].
which is approximately 
Continue reading ‘Bang! Bang! Bang! Explosively Large Numbers’
The “extraordinary talent and superior genius” of Sophie Germain
Published January 16, 2020 Irish Times Leave a CommentTags: History, Mechanics, Number Theory
When a guitar string is plucked, we don’t see waves travelling along the string. This is because the ends are fixed. Instead, we see a standing-wave pattern. Standing waves are also found on drum-heads and on the sound-boxes of violins. The shape of a violin strongly affects the quality and purity of the sound, as it determines the mixture of standing wave harmonics that it can sustain [TM179 or search for “thatsmaths” at irishtimes.com].
French postage stamp, issued in 2016, to commemorate the
250th anniversary of the birth of Sophie Germain (1776-1831).
Continue reading ‘The “extraordinary talent and superior genius” of Sophie Germain’
Spiralling Primes
Published September 12, 2019 Occasional 1 CommentTags: Number Theory, Recreational Maths
The Sacks Spiral.
The prime numbers have presented mathematicians with some of their most challenging problems. They continue to play a central role in number theory, and many key questions remain unsolved.
Order and Chaos
The primes have many intriguing properties. In his article “The first 50 million prime numbers”, Don Zagier noted two contradictory characteristics of the distribution of prime numbers. The first is the erratic and seemingly chaotic way in which the primes “grow like weeds among the natural numbers”. The second is that, when they are viewed in the large, they exhibit “stunning regularity”.
Closing the Gap between Prime Numbers
Published April 18, 2019 Irish Times Leave a CommentTags: Arithmetic, Number Theory
Occasionally, a major mathematical discovery comes from an individual working in isolation, and this gives rise to great surprise. Such an advance was announced by Yitang Zhang six years ago. [TM161 or search for “thatsmaths” at irishtimes.com].
Yitang Zhang
Massive Collaboration in Maths: the Polymath Project
Published April 11, 2019 Occasional Leave a CommentTags: Number Theory
Sometimes proofs of long-outstanding problems emerge without prior warning. In the 1990s, Andrew Wiles proved Fermat’s Last Theorem. More recently, Yitang Zhang announced a key result on bounded gaps in the prime numbers. Both Wiles and Zhang had worked for years in isolation, keeping abreast of developments but carrying out intensive research programs unaided by others. This ensured that they did not have to share the glory of discovery, but it may not be an optimal way of making progress in mathematics.
Polymath
Timothy Gowers in 2012 [image Wikimedia Commons].
Is massively collaborative mathematics possible? This was the question posed in a 2009 blog post by Timothy Gowers, a Cambridge mathematician and Fields Medal winner. Gowers suggested completely new ways in which mathematicians might work together to accelerate progress in solving some really difficult problems in maths. He envisaged a forum for the online discussion of problems. Anybody interested could contribute to the discussion. Contributions would be short, and could include false routes and failures; these are normally hidden from view so that different workers repeat the same mistakes.
Continue reading ‘Massive Collaboration in Maths: the Polymath Project’
Multiple Discoveries of the Thue-Morse Sequence
Published February 21, 2019 Irish Times Leave a CommentTags: Algorithms, Number Theory
It is common practice in science to name important advances after the first discoverer or inventor. However, this process often goes awry. A humorous principle called Stigler’s Law holds that no scientific result is named after its original discoverer. This law was formulated by Professor Stephen Stigler of the University of Chicago in his publication “Stigler’s law of eponymy”. He pointed out that his “law” had been proposed by others before him so it was, in a sense, self-verifying. [TM157 or search for “thatsmaths” at irishtimes.com].
Really, 0.999999… is equal to 1. Surreally, this is not so!
Published January 10, 2019 Occasional Leave a CommentTags: Analysis, Number Theory
The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory.
[Image Wikimedia Commons]
Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’
Random Numbers Plucked from the Atmosphere
Published December 6, 2018 Irish Times Leave a CommentTags: Number Theory
Randomness is a slippery concept, defying precise definition. A simple example of a random series is provided by repeatedly tossing a coin. Assigning “1” for heads and “0” for tails, we generate a random sequence of binary digits or bits. Ten tosses might produce a sequence such as 1001110100. Continuing thus, we can generate a sequence of any length having no discernible pattern [TM152 or search for “thatsmaths” at irishtimes.com].
Continue reading ‘Random Numbers Plucked from the Atmosphere’
Listing the Rational Numbers III: The Calkin-Wilf Tree
Published November 8, 2018 Occasional Leave a CommentTags: Arithmetic, Number Theory
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. In previous articles we showed how the rationals can be presented as a list that includes each rational precisely once. One approach leads to the Farey Sequences. A second, related, approach gives us the Stern-Brocot Tree. Here, we introduce another tree structure, The Calkin-Wilf Tree.
Continue reading ‘Listing the Rational Numbers III: The Calkin-Wilf Tree’
Listing the Rational Numbers II: The Stern-Brocot Tree
Published October 11, 2018 Occasional Leave a CommentTags: Arithmetic, Number Theory
The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. But it is not obvious how to construct a list that is sure to contain every rational number precisely once. In a previous post we described the Farey Sequences. Here we examine another, related, approach.
Continue reading ‘Listing the Rational Numbers II: The Stern-Brocot Tree’
Listing the Rational Numbers: I. Farey Sequences
Published September 27, 2018 Occasional Leave a CommentTags: Arithmetic, Number Theory
We know, thanks to Georg Cantor, that the rational numbers — ratios of integers — are countable: they can be put into one-to-one correspondence with the natural numbers.
Continue reading ‘Listing the Rational Numbers: I. Farey Sequences’
Godfrey Harold Hardy’s memoir, A Mathematician’s Apology, was published when he was 63 years old. It is a slight volume at just 90 pages, but is replete with interesting observations and not a few controversial opinions. After 78 years, it is still in print and is available in virtually every mathematics library. Though many of Hardy’s opinions are difficult to support and some of his predictions have turned out to be utterly wrong, the book is still well worth reading.
Kaprekar’s Number 6174
Published January 25, 2018 Occasional 1 CommentTags: Number Theory, Recreational Maths
The Indian mathematician D. R. Kaprekar spent many happy hours during his youth solving mathematical puzzles. He graduated from Fergusson College in Pune in 1929 and became a mathematical teacher at a school in Devlali, north-east of Mumbai.
Kaprekar process for three digit numbers converging to 495 [Wikimedia Commons].
More on Moduli
Published November 6, 2017 Occasional Leave a CommentTags: Arithmetic, Number Theory
We wrote last week on modular arithmetic, the arithmetic of remainders. Here we will examine a few other aspects of this huge subject. Modular arithmetic was advanced by Gauss in his Disquisitiones Arithmeticae. In this system, number wrap around when they reach a point known as the modulus. Numbers that differ by a multiple of the modulus are called congruent. Thus 4, 11 and 18 are all congruent modulo 7.
Addition table for numbers modulo 12.
Fractions of Fractions of Fractions
Published August 10, 2017 Occasional 1 CommentTags: Arithmetic, Number Theory, Recreational Maths
Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number 
![\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x+%3D+a_0+%2B+%5Ccfrac%7B1%7D%7B+a_1+%2B+%5Ccfrac%7B1%7D%7B+a_2+%2B+%5Ccfrac%7B1%7D%7B+a_3+%2B+%5Cdotsb+%2B+%5Ccfrac%7B1%7D%7Ba_n%7D+%7D+%7D%7D+%3D+%5B+a_0+%3B+a_1+%2C+a_2+%2C+a_3+%2C+%5Cdots+%2C+a_n+%5D+&bg=ffffff&fg=000000&s=0&c=20201002)
where all 
It’s as Easy as Pi
Published August 3, 2017 Irish Times Leave a CommentTags: Archimedes, Geometry, Number Theory, Pi
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see TM120 or search for “thatsmaths” at irishtimes.com].
The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).
A Remarkable Pair of Sequences
Published June 8, 2017 Occasional Leave a CommentTags: Number Theory
The terms of the two integer sequences below are equal for all 
A Geometric Sieve for the Prime Numbers
Published April 27, 2017 Occasional Leave a CommentTags: Number Theory, Primes
In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.
Numerical Coincidences
Published March 23, 2017 Occasional Leave a CommentTags: Number Theory, Recreational Maths
A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.
Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!
Brun’s Constant and the Pentium Bug
Published March 9, 2017 Occasional 1 CommentTags: Arithmetic, Euler, Number Theory
Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:
Obviously, this could not happen if there were only finitely many primes.
The Shaky Foundations of Mathematics
Published December 1, 2016 Irish Times Leave a CommentTags: Algorithms, Logic, Number Theory
The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a shadow of uncertainty. We can never be absolutely sure that the foundations of our subject are rock-solid [TM104 or search for “thatsmaths” at irishtimes.com].
Left: Plato and Aristotle. Centre: Pythagoras. Right: Euclid [Raphael, The School of Athens]
The ancient Greeks put geometry on a firm footing. Euclid set down a list of axioms, or basic intuitive assumptions. Upon these, the entire edifice of Euclidean geometry is constructed. This axiomatic approach has been the model for mathematics ever since.
Every number is interesting. Suppose there were uninteresting numbers. Then there would be a smallest one. But this is an interesting property, contradicting the supposition. By reductio ad absurdum, there are none!
This is the hundredth “That’s Maths” article to appear in The Irish Times [TM100, or search for “thatsmaths” at irishtimes.com]. To celebrate the event, we have composed an ode to the number 100.
The counting numbers that we learn as children are so familiar that using them is second nature. They bear the appropriate name natural numbers. From then on, names of numbers become less and less apposite.
Random Harmonic Series
Published July 28, 2016 Occasional Leave a CommentTags: Analysis, Number Theory, Probability
We consider the convergence of the random harmonic series
where 
Lecture sans paroles: the factors of M67
Published June 30, 2016 Occasional Leave a CommentTags: Number Theory
In 1903 Frank Nelson Cole delivered an extraordinary lecture to the American Mathematical Society. For almost an hour he performed a calculation on the chalkboard without uttering a single word. When he finished, the audience broke into enthusiastic applause.
The prime numbers have challenged and perplexed the greatest mathematicians for millennia. Shortly before he died, the brilliant Hungarian number theorist Paul Erdös said “it will be another million years, at least, before we understand the primes”.
A remarkable polynomial: Theorem 1 from Jones et al., 1976.
Andrew Wiles wins 2016 Abel Prize
Published May 19, 2016 Irish Times Leave a CommentTags: Algebra, Number Theory
A recent post described the Abel Prize, effectively the Nobel Prize for Mathematics, and promised a further post when the 2016 winner was announced. This is the follow-up post [also at TM091, or search for “thatsmaths” at irishtimes.com].
Next Tuesday, HRH Crown Prince Haakon will present the Abel Medal to Sir Andrew Wiles at a ceremony in Oslo. The Abel Prize, comparable to a Nobel Prize, is awarded for outstanding work in mathematics. Wiles has won the award for his “stunning proof of Fermat’s Last Theorem” with his research “opening a new era in number theory”. Wiles’ proof made international headlines in 1994 when he cracked one of the most famous and long-standing unsolved problems in mathematics.
Pierre de Fermat, a French lawyer and amateur mathematician, stated the theorem in 1637, writing in the margin of a maths book that he had “a truly marvellous proof”. But for more than 350 years no proof was found despite the efforts of many of the most brilliant mathematicians.
Ramanujan’s Astonishing Knowledge of 1729
Published May 12, 2016 Occasional 1 CommentTags: Number Theory, Ramanujan
Question: What is the connection between Ramanujan’s number 1729 and Fermat’s Last Theorem? For the answer, read on.
The story of how Srinivasa Ramanujan responded to G. H. Hardy’s comment on the number of a taxi is familiar to all mathematicians. With the recent appearance of the film The Man who Knew Infinity, this curious incident is now more widely known.
Result of a Google image search for “K3 Surface”.
Visiting Ramanujan in hospital, Hardy remarked that the number of the taxi he had taken was 1729, which he thought to be rather dull. Ramanujan replied “No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
Continue reading ‘Ramanujan’s Astonishing Knowledge of 1729′
Prime Number Record Smashed Again
Published January 28, 2016 Occasional Leave a CommentTags: Algorithms, Number Theory
Once again the record for the largest prime number has been shattered. As with all recent records, the new number is a Mersenne prime, a number of the form
Mp = 2p – 1
where p itself is a prime. Participants in a distributed computing project called GIMPS (Great Internet Mersenne Prime Search) continue without rest to search for ever-larger primes of this form.
Image from http://primes.utm.edu
Most of the recent large primes have been found in the GIMPS project (for an earlier post on GIMPS, click Mersennery Quest. The project uses a search algorithm called the Lucas-Lehmer primality test, which is particularly suitable for finding Mersenne primes. The test, which was originally devised by Edouard Lucas in the nineenth century and extended by Derek Lehmer in 1930, is very efficient on binary computers.
How many Christmas Gifts?
Published December 17, 2015 Irish Times Leave a CommentTags: Number Theory, Recreational Maths
We all know the festive carol The Twelve Days of Christmas. Each day, “my true love” receives an increasing number of gifts. On the first day there is one gift, a partridge in a pear tree. On the second, two turtle doves and another partridge, making three. There are six gifts on the third day, ten on the fourth, fifteen on the fifth, and so on.
Here is a Christmas puzzle: what is the total number of gifts over the twelve days? [TM083, or search for “thatsmaths” at irishtimes.com]
The Tragic Demise of a Beautiful Mind
Published June 4, 2015 Irish Times 1 CommentTags: Algorithms, Games, Number Theory
John Nash, who was the subject of the book and film A Beautiful Mind, won the Abel Prize recently. But his journey home from the award ceremony in Norway ended in tragedy [see this week’s That’s Maths column (TM069): search for “thatsmaths” at irishtimes.com].
Russell Crowe as John Nash in the movie A Beautiful Mind.
Fermat’s Christmas Theorem
Published December 25, 2014 Occasional Leave a CommentTags: Arithmetic, Number Theory, Pi, Primes, Spherical Trigonometry
Albert Girard (1595-1632) was a French-born mathematician who studied at the University of Leiden. He was the first to use the abbreviations ‘sin’, ‘cos’ and ‘tan’ for the trigonometric functions.
Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)
Waring’s Problem & Lagrange’s Four-Square Theorem
Published October 23, 2014 Occasional Leave a CommentTags: Gauss, Number Theory, Primes, Ramanujan
Introduction
We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.
More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.
Continue reading ‘Waring’s Problem & Lagrange’s Four-Square Theorem’
Old Octonions may rule the World
Published October 16, 2014 Irish Times Leave a CommentTags: Algebra, Hamilton, Number Theory
This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton.
![Multiplication table for octonions, of the formz=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]](https://thatsmaths.files.wordpress.com/2014/10/octonions.jpg)
Multiplication table for octonions, of the form z=a+bi+cj+dk+eE+fI+gJ+hK [Source: http://jmc2008.wurzel.org/index.php/Main/Logo]
Triangular Numbers: EYPHKA
Published October 9, 2014 Occasional Leave a CommentTags: Gauss, Number Theory, Recreational Maths
The maths teacher was at his wits’ end. To get some respite, he set the class a task:
Add up the first one hundred numbers.
“That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians. Continue reading ‘Triangular Numbers: EYPHKA’
Invention or Discovery?
Published July 24, 2014 Occasional Leave a CommentTags: Analysis, Logic, Number Theory, Social attitudes
Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:
Conjecture: Definitions are invented. Theorems are discovered.
The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’
Breaking Weather Records
Published April 24, 2014 Occasional Leave a CommentTags: Geophysics, Number Theory, Probability
In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each term is a fixed multiple of the previous one. Again, there is a formula for the sum of the first N terms of such a series. Continue reading ‘Breaking Weather Records’
The Unity of Mathematics
Published March 20, 2014 Irish Times Leave a CommentTags: Algebra, Analysis, Geometry, Number Theory
This week, That’s Maths in The Irish Times ( TM041 ) is about an ambitious program to unify mathematics.
Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have a comprehensive understanding of even a single field of mathematics: the web of knowledge grows so fast that no-one can master it all.
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