## Posts Tagged 'Arithmetic'

### 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 ${k}$-th entry in row ${n}$ is the binomial coefficient ${\binom{n}{k}}$ (read ${n}$-choose- ${k}$), the number of ways of selecting ${k}$ elements from a set of ${n}$ distinct elements.

### 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 for “thatsmaths” at irishtimes.com]. Pascal’s triangle as found in Zhu Shiji’s treatise The Precious Mirror of the Four Elements (1303).

### 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]. Shot from “A Suitable Boy” with Maan Kapoor (Ishaan Khatter), Mrs. Mahesh Kapoor (Geeta Agarwal) and Bhaskar (Yusuf Akhtar), covered in colours during the Holi festival [image from Instagram.  See also here].

### 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 ]. ### 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. ### 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]. Roman aqueduct at Segovia, Spain.

### 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

### 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 Ardnacrusha, on the Shannon near Limerick. Data from the Electricity Supply Board (ESB).

While the exact answers may be elusive, we can make reasonable guesstimates using basic knowledge and simple mathematical reasoning. And we will be walking in the footsteps of some of the world’s greatest thinkers.

### 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 another tree structure, 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. 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. ### 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 like any other. [see TM143, or search for “thatsmaths” at irishtimes.com]. A selection of images of zero (google images).

### 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.

### “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] Joyce in Zurich: did he meet Zermelo?

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]. Joyce’s examination marks [archives of the National University of Ireland].

### 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. Addition table for numbers modulo 12.

### 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]. We use modular arithmetic for timekeeping with a 12-hour clock [Image Wikimedia Commons]

### 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 numbers yields a perfect square:

1 + 3 + 5 + … + (2n – 1) = n 2

This is easily demonstrated in a geometric way. We start with a unit square, and repeatedly add an additional row and column on the “east” and “north” sides and a unit square at the “north-east” corner. This amounts to adding the next odd number and, at each stage, a new square is produced.

### 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.

### 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 for “thatsmaths” at irishtimes.com]. Image from here.

It is said that an inscription over the entrance to Plato’s Academy read “Let None But Geometers Enter Here”. This indicated that the Quadrivium was a prerequisite for the study of philosophy in ancient Greece.

### 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. With three steps, there are three possibilities. We can now proceed in an inductive manner. ### 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.

### Metallic Means The golden mean occurs repeatedly in the pentagram [image Wikimedia Commons]

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 ${\phi}$ and is the positive root of the quadratic equation $\displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)$

with the value ${\phi = (1+\sqrt{5})/2 \approx 1.618}$.

There is no doubt that ${\phi}$ is significant in many biological contexts and has also been an inspiration for artists. Called the Divine Proportion, it  was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

### 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 [TM107 or search for “thatsmaths” at irishtimes.com]. Artist’s impression of the Library of Babel [Image from Here].

Long before the Internet, the renowned Argentine writer, poet, translator and literary critic Jorge Luis Borges (1889 – 1986) envisaged the Universe as a vast information bank in the form of a library. The Library of Babel was imagined to contain every book that ever was or ever could be written.

### The Mathematics of Voting

Selection of leaders by voting has a history reaching back to the Athenian democracy. Elections are essentially arithmetical exercises, but they involve more than simple counting, and have some subtle mathematical aspects [TM085, or search for “thatsmaths” at irishtimes.com]. Rock-paper-scissors, a zero-sum game. There is a cyclic relationship: rock beats paper, paper beats scissors and scissors beats rock [Image: Wikimedia Commons].

### 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. ### 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 of land.

French philosopher, scientist and mathematician René Descartes demonstrated the power of coordinates and his method of algebraic geometry revolutionized mathematics. It had a profound, unifying effect on pure mathematics and greatly increased the ability of maths to model the physical world.

### You Can Do Maths

Bragging about mathematical ineptitude is not cool. There is nothing admirable about ignorance and incompetence. Moreover, everyone thinks mathematically all the time, even if they are not aware of it. Can we all do maths? Yes, we can!  [See this week’s That’s Maths column (TM064) or search for “thatsmaths” at irishtimes.com].

### 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.

### 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 achieve an arbitrarily large hangover — most students find this out for themselves!  In practice, at more than about one coin diameter it starts to become difficult to maintain balance.

### 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 [image from Wikimedia Commons].

### 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.

### 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 by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).

### 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’

### Joyce’s Number

With Bloomsday looming, it is time to re-Joyce. We reflect on some properties of a large number occurring in Ulysses.