Posts Tagged 'Number Theory'

The Ever-growing Goals of Googology

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 ].

OneGoogol

Continue reading ‘The Ever-growing Goals of Googology’

The Online Encyclopedia of Integer Sequences

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.

OEIS-Homepage

The Home page of OEIS: https://oeis.org

Continue reading ‘The Online Encyclopedia of Integer Sequences’

John Horton Conway: a Charismatic Genius

J-H-Conway-2009-Denise-Applewhite-Princeton-Univ

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.

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Bang! Bang! Bang! Explosively Large Numbers

Bang-1224578

Typical Comic-book `bang’ mark [Image from vectorstock ].

Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is

\displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

which is approximately {8\times 10^{53}}. The number of atoms in the universe is estimated to be about {10^{80}}. When we consider permutations of large sets, even more breadth-taking numbers emerge.

Continue reading ‘Bang! Bang! Bang! Explosively Large Numbers’

The “extraordinary talent and superior genius” of Sophie Germain

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].

Sophie-Germain-Stamp

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

SacksSpiral100k

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”.

Continue reading ‘Spiralling Primes’

Closing the Gap between Prime Numbers

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-Colour

Yitang Zhang

Continue reading ‘Closing the Gap between Prime Numbers’

Massive Collaboration in Maths: the Polymath Project

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-2012-Half

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

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].

Axel Thue (1863-1922) and Marston Morse (1892-1977)
Continue reading ‘Multiple Discoveries of the Thue-Morse Sequence’

Really, 0.999999… is equal to 1. Surreally, this is not so!

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.

999999

[Image Wikimedia Commons]

Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’

Random Numbers Plucked from the Atmosphere

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].

Lightning-01

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Listing the Rational Numbers III: The Calkin-Wilf Tree

Calkin-Wilf-TreeThe 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

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.

Mediant-red Continue reading ‘Listing the Rational Numbers II: The Stern-Brocot Tree’

Listing the Rational Numbers: I. Farey Sequences

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.

Rational-Numbers-Small

Continue reading ‘Listing the Rational Numbers: I. Farey Sequences’

Hardy’s Apology

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.

Hardy-MathApology

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Kaprekar’s Number 6174

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.

KaprekarFlowGraph495

Kaprekar process for three digit numbers converging to 495 [Wikimedia Commons].

Continue reading ‘Kaprekar’s Number 6174′

More on Moduli

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.

Z12-Addition-Table

Addition table for numbers modulo 12.

Continue reading ‘More on Moduli’

Fractions of Fractions of Fractions

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 {x} can be expanded as a continued fraction:

\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 ]

where all {a_n} are integers, all positive except perhaps {a_0}. If {a_n=1} we add it to {a_{n-1}}; then the expansion is unique.

Continue reading ‘Fractions of Fractions of Fractions’

It’s as Easy as Pi

Pi-SymbolEvery 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).

Continue reading ‘It’s as Easy as Pi’

A Remarkable Pair of Sequences

The terms of the two integer sequences below are equal for all {n} such that {1<n<777{,}451{,}915{,}729{,}368},  but equality is violated for this enormous value and, intermittently, for larger values of {n}.

TicTacToe-2-Sequences

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A Geometric Sieve for the Prime Numbers

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.

MS-Sieve-Detail

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Numerical Coincidences

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!

Continue reading ‘Numerical Coincidences’

Brun’s Constant and the Pentium Bug

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:

\displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty

Obviously, this could not happen if there were only finitely many primes.

Continue reading ‘Brun’s Constant and the Pentium Bug’

The Shaky Foundations of Mathematics

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].

platoaristotlepythagoraseuclid

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.

Continue reading ‘The Shaky Foundations of Mathematics’

A Ton of Wonders

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.

tonofwonders

Continue reading ‘A Ton of Wonders’

Negative Number Names

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.

really

Continue reading ‘Negative Number Names’

Random Harmonic Series

We consider the convergence of the random harmonic series

\displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n}

where {\sigma_n\in\{-1,+1\}} is chosen randomly with probability {1/2} of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

RandomHarmonicSeriesDistribution

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Lecture sans paroles: the factors of M67

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.

FrankNelsonColePrise

Continue reading ‘Lecture sans paroles: the factors of M67’

Prime Generating Formulae

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”.

Jones-Prime-Generator

A remarkable polynomial: Theorem 1 from Jones et al., 1976.

Continue reading ‘Prime Generating Formulae’

Andrew Wiles wins 2016 Abel Prize

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].

Abel-Prize-MedalNext 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.

Continue reading ‘Andrew Wiles wins 2016 Abel Prize’

Ramanujan’s Astonishing Knowledge of 1729

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.

K3-Surfaces-Google

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

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.

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.

Continue reading ‘Prime Number Record Smashed Again’

How many Christmas Gifts?

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.

Bauble-Tetrahedron

Here is a Christmas puzzle: what is the total number of gifts over the twelve days? [TM083, or search for “thatsmaths” at irishtimes.com]

Continue reading ‘How many Christmas Gifts?’

The Tragic Demise of a Beautiful Mind

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.

Russell Crowe as John Nash in the movie A Beautiful Mind.

Continue reading ‘The Tragic Demise of a Beautiful Mind’

Fermat’s Christmas Theorem

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)

Left: Albert Girard (1595-1632). Right: Pierre de Fermat (1601-1665)

Continue reading ‘Fermat’s Christmas Theorem’

Waring’s Problem & Lagrange’s Four-Square Theorem

\displaystyle \mathrm{num}\ = \square+\square+\square+\square

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

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]

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]

Continue reading ‘Old Octonions may rule the World’

Triangular Numbers: EYPHKA

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?

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

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

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.

Continue reading ‘The Unity of Mathematics’

The Langlands Program

An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures [image from Wikimedia Commons].

Continue reading ‘The Langlands Program’

The Prime Number Theorem

God may not play dice with the Universe, but something strange is going on with the prime numbers  [Paul Erdös, paraphrasing Albert Einstein]

The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’

Experiment and Proof

Many mathematicians spend their time proving results. The (very old) joke is that they are machines for turning coffee into theorems. A theorem is a statement that has been shown, by a sequence of irrefutable steps, to follow logically from a set of fundamental assumptions known as axioms.

These axioms themselves may be self-evident, or may simply be assumed to be true. Given this, the statement contained in a theorem is known with certainty to be true.

Continue reading ‘Experiment and Proof’

The remarkable BBP Formula

Information that is declared to be forever inaccessible is sometimes revealed within a short period. Until recently, it seemed impossible that we would ever know the value of the quintillionth decimal digit of pi. But a remarkable formula has been found that allows the computation of binary digits starting from an arbitrary  position without the need to compute earlier digits. This is known as the BBP formula.
Continue reading ‘The remarkable BBP Formula’

The Ups and Downs of Hailstone Numbers

Hailstones, in the process of formation, make repeated excursions up and down within a cumulonimbus cloud until finally they fall to the ground. We look at sequences of numbers that oscillate in a similarly erratic manner until they finally reach the value 1. They are called hailstone numbers.
Continue reading ‘The Ups and Downs of Hailstone Numbers’

Amazing Normal Numbers

For any randomly chosen decimal number, we might expect that all the digits, 0, 1 , … , 9, occur with equal frequency. Likewise, digit pairs such as 21 or 59 or 83 should all be equally likely to crop up.  Similarly for triplets of digits. Indeed, the probability of finding any finite string of digits should depend only on its length. And, sooner or later, we should find any string. That’s “normal”!
Continue reading ‘Amazing Normal Numbers’

Prime Secrets Revealed

This week, That’s Maths in the Irish Times ( TM022 ) reports on two exciting recent breakthroughs in prime number theory.

The mathematics we study at school gives the impression that all the big questions have been answered: most of what we learn has been known for centuries, and new developments are nowhere in evidence. In fact, research in maths has never been more intensive and advances are made on a regular basis.

Continue reading ‘Prime Secrets Revealed’

Dis, Dat, Dix & Douze

How many fingers has Mickey Mouse? A glance at the figure shows that he has three fingers and a thumb on each hand, so eight in all. Thus, we may expect Mickey to reckon in octal numbers, with base eight. We use decimals, with ten symbols from 0 to 9 for the smallest numbers and larger numbers denoted by several digits, whose position is significant. Thus, 47 means four tens plus seven units.
Continue reading ‘Dis, Dat, Dix & Douze’

Happy Pi Day 2013

Today, 14th March, is Pi Day. In the month/day format it is 3/14, corresponding to 3.14, the first three digits of π. So, have a Happy Pi Day. Larry Shaw of San Francisco’s Exploratorium came up with the Pi Day idea in 1988. About ten years later, the U.S. House of Representatives passed a resolution recognizing March 14 as National Pi Day.

Today is also the birthday anniversary of Albert Einstein, giving us another reason to celebrate. He was born on 14 March 1879, just 134 years ago today.
Continue reading ‘Happy Pi Day 2013’


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