The Sieve of Eratosthenes and a Partition of the Natural Numbers

The sieve of Eratosthenes is a method for finding all the prime numbers less than some maximum value $latex {M}&fg=000000$ by repeatedly removing multiples of the smallest remaining prime until no composite numbers less than or equal to $latex {M}&fg=000000$ remain. The sieve provides a means of partitioning the natural numbers. We examine this partition … Continue reading The Sieve of Eratosthenes and a Partition of the Natural Numbers →

Maths in the Time of the Pharaohs

Why would the Ancient Egyptians have any interest in or need for mathematics? There are many reasons. They had a well-organised and developed civilisation extending over millennia. Science and maths must have played important or even essential roles in this culture. They needed measurement for land surveying and for designing irrigation canals, arithmetic for accounting … Continue reading Maths in the Time of the Pharaohs →

Numbers Without Ones: Chorisenic Sets

There is no end to the variety of sets of natural numbers. Sets having all sorts of properties have been studied and many more remain to be discovered. In this note we study the set of natural numbers for which the decimal digit 1 does not occur. Google Translate on my mobile phone gives the … Continue reading Numbers Without Ones: Chorisenic Sets →

Amusical Permutations and Unsettleable Problems

In a memorial tribute in the Notices of the American Mathematical Society (Ryba, et al, 2022), Dierk Schleicher wrote of how he convinced John Conway to publish a paper, ``On unsettleable arithmetical problems'', which included a discussion of the Amusical Permutations. This paper, which discusses arithmetical statements that are almost certainly true but likely unprovable, … Continue reading Amusical Permutations and Unsettleable Problems →

Summing the Fibonacci Sequence

The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation, $latex \displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)&fg=000000$ and two initial values, $latex {F_0 = 0}&fg=000000$ and $latex {F_1 = 1}&fg=000000$. This immediately yields the well-known sequence $latex \displaystyle … Continue reading Summing the Fibonacci Sequence →

What’s the Next Number?

We are all familiar with simple mathematical puzzles that give a short sequence and ask ``What is the next number in the sequence''. Simple examples would be $latex \displaystyle \begin{array}{rcl} && 1, 3, 5, 7, 9, 11, \dots \\ && 1, 4, 9, 16, 25, \dots \\ && 1, 1, 2, 3, 5, 8, \dots … Continue reading What’s the Next Number? →

The Arithmetic Triangle is Analytical too

Pascal's triangle is one of the most famous of all mathematical diagrams. It is simple to construct and rich in mathematical patterns. There is always a chance of finding something never seen before, and the discovery of new patterns is very satisfying. Not too long ago, Harlan Brothers found Euler's number $latex {e}&fg=000000$ in the … Continue reading The Arithmetic Triangle is Analytical too →

The Spine of Pascal’s Triangle

We are all familiar with Pascal's Triangle, also known as the Arithmetic Triangle (AT). Each entry in the AT is the sum of the two closest entries in the row above it. The $latex {k}&fg=000000$-th entry in row $latex {n}&fg=000000$ is the binomial coefficient $latex {\binom{n}{k}}&fg=000000$ (read $latex {n}&fg=000000$-choose-$latex {k}&fg=000000$), the number of ways of … Continue reading The Spine of Pascal’s Triangle →

The Simple Arithmetic Triangle is full of Surprises

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 … Continue reading The Simple Arithmetic Triangle is full of Surprises →

Suitable Names for Large Numbers

One year ago, there were just two centibillionaires, Jeff Bezos and Bill Gates. Recently, Facebook's Mark Zuckerberg has joined the Amazon and Microsoft founders. Elon Musk, CEO of Tesla and SpaceX, is tipped to be next to join this exclusive club [TM194 or search for “thatsmaths” at irishtimes.com]. The word centibillionaire has slithered into common usage for … Continue reading Suitable Names for Large Numbers →

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 … 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 … Continue reading The Online Encyclopedia of Integer Sequences →

What did the Romans ever do for Maths?

The ancient Romans developed many new techniques for engineering and architecture. The citizens of Rome enjoyed fountains, public baths, central heating, underground sewage systems and public toilets. All right, but apart from sanitation, medicine, education, irrigation, roads and aqueducts, what did the Romans ever do for maths? [TM166 or search for “thatsmaths” at irishtimes.com]. It might … Continue reading What did the Romans ever do for Maths? →

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]. After completing his doctorate at Purdue in 1991, Zhang had great difficulty finding an academic position and worked at various … Continue reading Closing the Gap between Prime Numbers →

Rambling and Reckoning

A walk on the beach, in the hills or along a river bank provides great opportunities for mathematical reflection. How high is the mountain? How many grains of sand are on the beach? How much water is flowing in the river? [TM156 or search for “thatsmaths” at irishtimes.com]. Daily average flow (cubic metres per second) at … Continue reading Rambling and Reckoning →

Listing the Rational Numbers III: The Calkin-Wilf Tree

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 … 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. The Stern-Brocot Tree We … 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. How can we make a list that includes all rationals? For the present, let us consider rationals in the interval $latex {[0,1]}&fg=000000$. It would be nice if … Continue reading Listing the Rational Numbers: I. Farey Sequences →

The Empty Set is Nothing to Worry About

Today's article is about nothing: nothing at all, as encapsulated in the number zero and the empty set. It took humanity millennia to move beyond the counting numbers. Zero emerged in several civilizations, first as a place-holder to denote a space or gap between digits, and later as a true number, which could be manipulated … Continue reading The Empty Set is Nothing to Worry About →

Numbers with Nines

What proportion of all numbers less than a given size N have a 9 in their decimal expansion? A naive argument would be that, since 9 is one of ten distinct digits, the answer must be about 10%. But this is not "remotely close" to the true answer. Counting the Nines It is a simple … Continue reading Numbers with Nines →

“Dividends and Divisors Ever Diminishing”

Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week's ThatsMaths post] In Ithaca, the narrator takes every … Continue reading “Dividends and Divisors Ever Diminishing” →

Leopold Bloom’s Arithmetical Adventures

As Bloomsday approaches, we reflect on James Joyce and mathematics. Joyce entered UCD in September 1898. His examination marks are recorded in the archives of the National University of Ireland, and summarized in a table in Richard Ellmann's biography of Joyce (reproduced below) [TM140 or search for “thatsmaths” at irishtimes.com]. The marks fluctuate widely, suggesting some lack of … Continue reading Leopold Bloom’s Arithmetical Adventures →

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 … Continue reading More on Moduli →

Modular Arithmetic: from Clock Time to High Tech

You may never have heard of modular arithmetic, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders [TM126 or search for “thatsmaths” at irishtimes.com]. When reckoning hours, we count up to twelve and start again … Continue reading Modular Arithmetic: from Clock Time to High Tech →

Moessner’s Magical Method

Amongst the many joys of mathematics are the beautiful patterns that appear unexpectedly from time to time. In 1951, Alfred Moessner discovered a delightful and ingenious way of generating sequences of powers of natural numbers. It is surprisingly simple and offers great opportunities for development and generalization. It is well-known that the sum of odd … Continue reading Moessner’s Magical Method →

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 $latex {x}&fg=000000$ can be expanded as a continued fraction: $latex \displaystyle x = a_0 + \cfrac{1}{ a_1 … Continue reading Fractions of Fractions of Fractions →

Quadrivium: The Noble Fourfold Way

According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato's Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD [see TM119 or search … Continue reading Quadrivium: The Noble Fourfold Way →

Patterns in Poetry, Music and Morse Code

Suppose we have to ascent a flight of stairs and can take only one or two steps at a time. How many different patterns of ascent are there? We start with the simplest cases. With one step there is only one way; with two, there are two: take two single steps or one double step. … Continue reading Patterns in Poetry, Music and Morse Code →

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: $latex \displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty &fg=000000$ Obviously, this could not happen if there were … Continue reading Brun’s Constant and the Pentium Bug →

Metallic Means

Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by $latex {\phi}&fg=000000$ and is the positive root of the quadratic equation $latex \displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)&fg=000000$ with the value $latex {\phi … Continue reading Metallic Means →

The Library of Babel and the Information Explosion

The world has been transformed by the Internet. Google, founded just 20 years ago, is a major force in online information. The company name is a misspelt version of "googol", the number one followed by one hundred zeros. This name echoes the vast quantities of information available through the search engines of the company … Continue reading The Library of Babel and the Information Explosion →

Factorial 52: A Stirling Problem

How many ways can a deck of cards be arranged? It is very easy to calculate the answer, but very difficult to grasp its significance. There are 52 cards. Thus, the first one may be chosen in 52 ways. The next one can be any of the remaining 51 cards. For the third, there are … Continue reading Factorial 52: A Stirling Problem →

Who Needs EirCode?

The idea of using two numbers to identify a position on the Earth's surface is very old. The Greek astronomer Hipparchus (190–120 BC) was the first to specify location using latitude and longitude. However, while latitude could be measured relatively easily, the accurate determination of longitude was more difficult, especially for sailors out of site … Continue reading Who Needs EirCode? →

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. Girard also showed how the area of a spherical triangle depends on its interior angles. If the angles of a triangle on the unit sphere … Continue reading Fermat’s Christmas Theorem →

Biscuits, Books, Coins and Cards: Massive Hangovers

Have you ever tried to build a high stack of coins? In theory it's fine: as long as the centre of mass of the coins above each level remains over the next coin, the stack should stand. But as the height grows, it becomes increasingly trickier to avoid collapse. In theory it is possible to … Continue reading Biscuits, Books, Coins and Cards: Massive Hangovers →

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 … 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 … Continue reading Experiment and Proof →

Ternary Variations

Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor's Ternary Set. In fact, the ternary set had been studied some ten years earlier … Continue reading Ternary Variations →

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. The Collatz Conjecture There are many simply-stated … Continue reading The Ups and Downs of Hailstone Numbers →

Joyce’s Number

With Bloomsday looming, it is time to re-Joyce. We reflect on some properties of a large number occurring in Ulysses. The Largest Three-digit Number What is the largest number that can be written using only three decimal digits? An initial guess might be 999. But soon we realize that factorials permit much greater numbers, and … Continue reading Joyce’s Number →

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 … Continue reading Dis, Dat, Dix & Douze →