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Slingshot Orbit to Asteroid Bennu

The Voyager 1 and Voyager 2 spacecraft have now left the solar system and will continue into deep space. How did we manage to send them so far? The Voyager spacecraft used gravity assists to visit Jupiter, Saturn, Uranus and Neptune in the late 1970s and 1980s. Gravity assist manoeuvres, known as slingshots, are essential for interplanetary missions. They were first used in the Soviet Luna-3 mission in 1959, when images of the far side of the Moon were obtained. Space mission planners use them because they require no fuel and the gain in speed dramatically shortens the time of missions to the outer planets.

OSIRIS-REx

Artist’s impression of OSIRIS-REx orbiting Bennu [Photo Credit: NASA]

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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].

Clock-Arithmetic

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

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Learning Maths has never been Easier

Maths is hard: many people find it inscrutable and have negative attitudes towards maths. They may have bad memories of school maths or have been told they lack mathematical talents. This is unfortunate: we all have the capacity to apply reasoning and logic and we can all do maths. Given the vital role mathematics plays in modern society, there is an urgent need to help young people to become more numerate and comfortable with mathematics. With a wealth of online resources, learning maths has never been easier. [TM125 or search for “thatsmaths” at irishtimes.com].

MathsWeek2017-ShaneONeill

Eoin Gill and Sheila Donegan with Jadine Rock of Rutland National School, Dublin , at the launch of Maths Week Ireland. Image: Shane O’Neill, SON Photographic.

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From Sailing on a Rhumb to Flying on a Geodesic

If you fly 14,500 km due westward from New York you will come to Beijing. The two cities are on the fortieth parallel of latitude. However, by flying a great circle route over the Arctic, you can reach Beijing in 11,000 km, saving 3,500 km and much time and aviation fuel.  [TM124 or search for “thatsmaths” at irishtimes.com].

JFK-PEK-Gnomonic

Great circle route from New York to Beijing (gnomonic projection).

On a gnomonic projection (as above) each point on the Earth’s surface is projected from the centre of the Earth onto a plane tangent to the globe. On this map, great circles appear as straight lines.

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Andrey Markov’s Brilliant Ideas are still a Driving Force

AA-Markov-SMALL

A A Markov (1856-1922)

Imagine examining the first 20,000 letters of a book, counting frequencies and studying patterns. This is precisely what Andrey Markov did when he analyzed the text of Alexander Pushkin’s verse novel Eugene Onegin. This work comprises almost 400 stanzas of iambic tetrameter and is a classic of Russian literature. Markov studied the way vowels and consonants alternate and deduced the probabilities of a vowel being followed by a another vowel, by a consonant, and so on. He was applying a statistical model that he had developed in 1906 and that we now call a Markov Process or Markov chain. [TM123 or search for “thatsmaths” at irishtimes.com].

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Euler and the Fountains of Sanssouci

When Frederick the Great was crowned King of Prussia in 1740 he immediately revived the Berlin Academy of Sciences and invited scholars from throughout Europe to Berlin. The most luminous of these was Leonhard Euler, who arrived at the academy in 1741. Euler was an outstanding genius, brilliant in both mathematics and physics. Yet, a myth persists that he failed spectacularly to solve a problem posed by Frederick. Euler is reputed to have bungled his mathematical analysis. In truth, there was much bungling, but the responsibility lay elsewhere. [TM122 or search for “thatsmaths” at irishtimes.com].

Sanssouci-Fountain

Sanssouci Palace, the summer home of Frederick the Great in Potsdam.
PHOTO: https://www.flickr.com/photos/b_hurwitz/4064337397/

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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).

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