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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Computers Speaking in Irish

Most of us use computer terminals, tablets and smart phones, absorbing information quickly and easily. How do the many thousands of Irish people who are blind or visually impaired manage to interact with computers? For them, entering data by keyboard or voice is easy, but special software is needed to convert the text on screen into a form for output to a loudspeaker or headphones, or to drive a refreshable Braille display [TM095, or search for “thatsmaths” at irishtimes.com].

Braille-Display

Braille display (www.humanware.com)

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Squircles

You can put a square peg in a round hole.

Shapes between circles and squares have proved invaluable to engineers and have also found their way onto our dinner tables. A plate in the shape of a `squircle’ is shown in this figure .

Squircle-Plate.jpg

Squircular plate: holds more food and is easier to store.

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Lateral Thinking in Mathematics

Many problems in mathematics that appear difficult to solve turn out to be remarkably simple when looked at from a new perspective. George Pólya, a Hungarian-born mathematician, wrote a popular book, How to Solve It, in which he discussed the benefits of attacking problems from a variety of angles [see TM094, or search for “thatsmaths” at irishtimes.com].

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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

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Bending the Rules to Square the Circle

Squaring the circle was one of the famous Ancient Greek mathematical problems. Although studied intensively for millennia by many brilliant scholars, no solution was ever found. The problem requires the construction of a square having area equal to that of a given circle. This must be done in a finite number of steps, using only ruler and compass.

SquareCircle-00

Taking unit radius for the circle, the area is π, so the square must have a side length of √π. If we could construct a line segment of length π, we could also draw one of length √π. However, the only constructable numbers are those arising from a unit length by addition, subtraction, multiplication and division, together with the extraction of square roots.

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Bloom’s attempt to Square the Circle

The quadrature of the circle is one of the great problems posed by the ancient Greeks. This “squaring of the circle” was also an issue of particular interest to Leopold Bloom, the central character in James Joyce’s novel Ulysses, whom we celebrate today, Bloomsday, 16 June 2016 [see TM093, or search for “thatsmaths” at irishtimes.com].

Joyces-Tower

Joyce’s Tower, Sandycove, Co Dublin.

The challenge is to construct a square with area equal to that of a given circle using only the methods of classical geometry. Thus, only a ruler and compass may be used in the construction and the process must terminate in a finite number of steps.

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