## Posts Tagged 'Arithmetic'

### 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.

OSi Mapviewer. XY coordinates indicated at bottom left.

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].

When you use a map of the underground network, you are doing topology.